--- /srv/rebuilderd/tmp/rebuilderdHjiBzg/inputs/macaulay2-common_1.25.11+ds-3_all.deb +++ /srv/rebuilderd/tmp/rebuilderdHjiBzg/out/macaulay2-common_1.25.11+ds-3_all.deb ├── file list │ @@ -1,3 +1,3 @@ │ -rw-r--r-- 0 0 0 4 2026-01-19 15:01:18.000000 debian-binary │ --rw-r--r-- 0 0 0 540560 2026-01-19 15:01:18.000000 control.tar.xz │ --rw-r--r-- 0 0 0 31296244 2026-01-19 15:01:18.000000 data.tar.xz │ +-rw-r--r-- 0 0 0 540440 2026-01-19 15:01:18.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 31297700 2026-01-19 15:01:18.000000 data.tar.xz ├── control.tar.xz │ ├── control.tar │ │ ├── ./control │ │ │ @@ -1,13 +1,13 @@ │ │ │ Package: macaulay2-common │ │ │ Source: macaulay2 │ │ │ Version: 1.25.11+ds-3 │ │ │ Architecture: all │ │ │ Maintainer: Debian Math Team │ │ │ -Installed-Size: 305286 │ │ │ +Installed-Size: 305298 │ │ │ Depends: fonts-katex (>= 0.16.10+~cs6.1.0), libjs-bootsidemenu (>= 2.2.1), libjs-bootstrap5 (>= 5.3.8+dfsg), libjs-d3 (>= 3.5.17), libjs-jquery (>= 3.7.1+dfsg+~3.5.33), libjs-katex (>= 0.16.10+~cs6.1.0), libjs-nouislider (>= 15.8.1+ds), libjs-three (>= 111+dfsg1), node-clipboard (>= 2.0.11+ds+~cs9.6.11), node-fortawesome-fontawesome-free (>= 6.7.2+ds1) │ │ │ Section: math │ │ │ Priority: optional │ │ │ Multi-Arch: foreign │ │ │ Homepage: http://macaulay2.com │ │ │ Description: Software system for algebraic geometry research (common files) │ │ │ Macaulay 2 is a software system for algebraic geometry research, written by │ │ ├── ./md5sums │ │ │ ├── ./md5sums │ │ │ │┄ Files differ ├── data.tar.xz │ ├── data.tar │ │ ├── file list │ │ │ @@ -3353,25 +3353,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 47016 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/A1BrouwerDegrees/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15458 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/A1BrouwerDegrees/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 41444 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1026 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___A__Infinity.out │ │ │ --rw-r--r-- 0 root (0) root (0) 918 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 917 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4374 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_a__Infinity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 56403 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_burke__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3427 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_display__Blocks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3016 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_extract__Blocks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1740 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_golod__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 832 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_is__Golod__A__Inf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2183 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_picture.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 7249 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7248 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14672 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_a__Infinity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 67508 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_burke__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9657 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_display__Blocks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10392 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_extract__Blocks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9306 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_golod__Betti.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5933 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_has__Minimal__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6465 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_is__Golod__A__Inf.html │ │ │ @@ -3465,26 +3465,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11993 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7615 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4178 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37097 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1946 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1748 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1945 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1747 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_expected__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_linear__System__On__Rational__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1889 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1272 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_rational__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1596 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_slow__Adjunction__Calculation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2944 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_special__Families__Of__Sommese__Vande__Ven.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 9249 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11329 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9248 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11328 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6620 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_expected__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7258 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_linear__System__On__Rational__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9580 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9741 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_rational__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9319 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_slow__Adjunction__Calculation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12483 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_special__Families__Of__Sommese__Vande__Ven.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13802 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/index.html │ │ │ @@ -3723,18 +3723,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 76682 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4226 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2909 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2927 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 422 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 433 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 29 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5573 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5584 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5233 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4242 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 2912 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Benchmark/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BernsteinSato/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 289851 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BernsteinSato/example-output/ │ │ │ @@ -3998,15 +3998,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10474 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Bertini/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 141706 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 6933 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8038 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6929 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6928 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2267 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp4.out │ │ │ -rw-r--r-- 0 root (0) root (0) 758 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Character_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1576 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Equality_spchecks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1927 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Labels.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2080 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Sub.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3164 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/_action_lp__Complex_cm__List_cm__List_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2099 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/_action_lp__Module_cm__List_cm__List_rp.out │ │ │ @@ -4028,15 +4028,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 595 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 62 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7189 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6096 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action__On__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8643 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action__On__Graded__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15258 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14939 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18504 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 18503 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8439 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp4.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9931 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6339 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6812 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4366 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character_spoperations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6855 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Equality_spchecks.html │ │ │ @@ -4365,15 +4365,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 422 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Simplex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Well__Defined_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 595 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Well__Defined_lp__Cell_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_max__Cells.out │ │ │ -rw-r--r-- 0 root (0) root (0) 275 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_new__Cell.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_new__Simplex__Cell.out │ │ │ --rw-r--r-- 0 root (0) root (0) 733 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 734 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_ring_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 520 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2473 2026-01-19 15:01:18.000000 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2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Well__Defined_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8974 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Well__Defined_lp__Cell_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6426 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_max__Cells.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8432 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_new__Cell.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7358 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_new__Simplex__Cell.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8605 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8606 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5874 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_ring_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6841 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_scarf__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8430 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_skeleton_lp__Z__Z_cm__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9468 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_subcomplex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6493 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_taylor__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23005 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19645 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/master.html │ │ │ @@ -4433,15 +4433,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1448 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism.out │ │ │ -rw-r--r-- 0 root (0) root (0) 771 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Quasi__Isomorphism_lp..._cm__Length__Limit_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 278 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_koszul__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_nonzero__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_prepend__Zero__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 899 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_remove__Zero__Trailing__Terms.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3452 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3451 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2570 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_substitute_lp__Chain__Complex_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1333 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1351 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_trivial__Homological__Truncation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/ │ │ │ @@ -4464,15 +4464,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5473 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7220 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10096 2026-01-19 15:01:18.000000 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root (0) root (0) 7373 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9135 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10433 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_scarf__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5755 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_substitute_lp__Chain__Complex_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6750 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7255 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ @@ -4501,49 +4501,49 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12258 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8418 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4775 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 123136 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4377 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4375 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1503 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3419 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Toric__Variety__Valid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3442 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chern.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2404 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chow__Ring.out │ │ │ -rw-r--r-- 0 root 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./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Complete__Intersection__Resolutions.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4600 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4594 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4116 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash__Total.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2743 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1023 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module__Data.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2102 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./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_splittings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5224 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_stable__Hom.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6048 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5573 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_tensor__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4869 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_to__Array.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6365 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6364 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 56031 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35373 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15036 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Complexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Complexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 706402 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Complexes/example-output/ │ │ │ @@ -5340,29 +5340,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 21320 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16146 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9518 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50933 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 36619 2026-01-19 15:01:18.000000 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./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5611 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_base__Fraction__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6834 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6189 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7537 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7266 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Transform.html │ │ │ @@ -5586,134 +5586,134 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 16572 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 239171 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2309 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Codim__Bs__Inv.out │ │ │ --rw-r--r-- 0 root (0) root (0) 19789 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 19783 2026-01-19 15:01:18.000000 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./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 42549 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1530 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_coefficients_lp__Rational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 33237 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 33238 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_describe_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 515 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_exceptional__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1047 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15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_to__Map.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 607 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4634 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Blow__Up__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6106 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Certify.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11860 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5447 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Codim__Bs__Inv.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4085 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./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/_quot__Scheme.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 10955 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10957 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6153 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_affine__Part.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7448 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_co1__Fitting.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7377 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_gauss__Col.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8186 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_gotzmann__Test.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7246 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_next__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5849 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/_quot__Scheme.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9538 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/index.html │ │ │ @@ -6867,15 +6867,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 124 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_foreign__Symbol.out │ │ │ -rw-r--r-- 0 root (0) root (0) 688 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_foreign__Union__Type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 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./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pullback_lp__Equivariant__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 615 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pushforward.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_set__Indicator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 437 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_trivial__K__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 401 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_underlying__Graph_lp__Moment__Graph_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/ │ │ │ @@ -7168,15 +7168,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7893 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.html 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2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_underlying__Graph_lp__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26820 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/index.html │ │ │ @@ -8244,32 +8244,32 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33164 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1505 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Constraints.out │ │ │ -rw-r--r-- 0 root (0) root (0) 598 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Vector.out │ │ │ -rw-r--r-- 0 root (0) root (0) 30368 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2709 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Stratum.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_linear__Part.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13267 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ --rw-r--r-- 0 root (0) root (0) 14736 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6625 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13044 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 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15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_is__Torus__Fixed.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4657 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_put__Weyl__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2252 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out │ │ │ @@ -8435,15 +8435,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6663 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Appell__F1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14273 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Canonical_sp__Series_sp__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3427 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Theta__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3323 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Wto__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3758 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_create__Theta__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6985 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Expts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7875 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Expts__Mult.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11149 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11150 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12251 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_diff__Ops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8520 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_distraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7705 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_euler__Operators.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8983 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_gkz.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7671 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_indicial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6418 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_is__Torus__Fixed.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3906 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_nilsson__Support.html │ │ │ @@ -8457,21 +8457,21 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 21857 2026-01-19 15:01:18.000000 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./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ring__From__Fractions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2043 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_test__Huneke__Question.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/ │ │ │ @@ -8679,18 +8679,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6424 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_ic__P__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8990 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_idealizer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5007 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_idealizer_lp..._cm__Index_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6571 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_idealizer_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) 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./usr/share/doc/Macaulay2/InvariantRing/example-output/_secondary__Invariants_lp..._cm__Print__Degree__Polynomial_eq_gt..._rp.out │ │ │ @@ -8771,38 +8771,38 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6553 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_generators_lp__Finite__Group__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6507 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_generators_lp__Ring__Of__Invariants_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6798 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_group.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6565 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_group__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10288 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hilbert__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6898 2026-01-19 15:01:18.000000 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-8983,62 +8983,62 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9828 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Jets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7299 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Jets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101880 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1949 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_all__Gradings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2785 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2786 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5835 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_canonical__Homotopies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1036 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2262 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3428 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1007 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ --rw-r--r-- 0 root (0) root (0) 269 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ 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./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1638 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution__Twists.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2221 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2218 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ -rw-r--r-- 0 root (0) root (0) 129 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 758 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_scheme__In__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3037 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_schreyer__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2026-01-19 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15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Rev__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Assign__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 371 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Release__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6869 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Grassmannian.out │ │ │ @@ -9820,15 +9820,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 238 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Family_sp_us_st.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1401 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Map.out │ │ │ 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5127 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Schreyer_sporders.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1024 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Schubert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Self__Initializing__Type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 950 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 205 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp-_sp__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp_sh_qu_sp__Thing.out │ │ │ @@ -9965,15 +9965,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apply_lp__Z__Z_cm__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1347 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apropos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ascii.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asinh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 436 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assigning_spvalues.out │ │ │ --rw-r--r-- 0 root (0) root (0) 235 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 234 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 154 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan.out │ │ │ -rw-r--r-- 0 root (0) root (0) 245 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 207 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atanh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_augmented_spassignment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1258 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_autoload.out │ │ │ -rw-r--r-- 0 root (0) root (0) 137 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Name.out │ │ │ @@ -9988,23 +9988,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_between.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_binomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_block__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_borel_lp__Matrix_rp.out │ │ │ 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ceiling_lp__Number_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_center__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Base.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 196 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_characters.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1523 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1522 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ -rw-r--r-- 0 root (0) root (0) 303 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1307 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_clean.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2015 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Monomial__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Quotient__Ring_rp.out │ │ │ @@ -10020,22 +10020,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_columnate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_combine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 198 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_command__Interpreter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 149 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comments.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_common__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_commonest.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1535 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1547 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compact__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compare__Exchange.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compose.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2477 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compositions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compress.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4286 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4287 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ -rw-r--r-- 0 root (0) root (0) 640 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_spsyzygies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1767 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenating_spmatrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 392 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conditional_spexecution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 148 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate_lp__Partition_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_constructing_spmaps_spbetween_spmodules.out │ │ │ @@ -10115,15 +10115,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1720 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eagon__Northcott.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvalues.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvectors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 366 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elements.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21208 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21210 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 782 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3683 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_equality_spand_spcontainment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erfc.out │ │ │ @@ -10264,15 +10264,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1260 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inheritance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1187 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inputting_spa_spmatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 592 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_insert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 686 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spassignment_spmethods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 936 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spaugmented_spassignment_spmethods.out │ │ │ --rw-r--r-- 0 root (0) root (0) 932 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 933 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integers_spmodulo_spa_spprime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integrate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1162 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 189 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Set_cm__Set_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 190 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_spof_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inverse__Erf.out │ │ │ @@ -10404,18 +10404,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 769 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 125 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max.out │ │ │ --rw-r--r-- 0 root (0) root (0) 82 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 83 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ -rw-r--r-- 0 root (0) root (0) 83 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 311 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Position.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1639 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1638 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1730 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_merge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2673 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 970 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method__Options_lp__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6748 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2721 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods_spfor_spnormal_spforms_spand_spremainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Exponent.out │ │ │ @@ -10501,16 +10501,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_override.out │ │ │ -rw-r--r-- 0 root (0) root (0) 681 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pack.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_package.out │ │ │ -rw-r--r-- 0 root 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8673 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parsing_spprecedence_cm_spin_spdetail.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3030 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1297 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1097 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partitions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 652 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parts.out │ │ │ 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2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 133490 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Monodromy__Solver.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Monodromy__Solver__Options.out │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_complete__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 243 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_complete__Graph__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 418 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_compute__Mixed__Volume.out │ │ │ -rw-r--r-- 0 root (0) root (0) 407 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_create__Seed__Pair.out │ │ │ --rw-r--r-- 0 root (0) root (0) 944 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 943 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9570 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out │ │ │ -rw-r--r-- 0 root (0) root (0) 427 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_cm__Abstract__Point_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1395 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 956 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__E.out │ │ │ -rw-r--r-- 0 root (0) root (0) 442 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__Lower__Bound.out │ │ │ @@ -13609,15 +13609,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5869 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/___Point__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4500 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_append__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4552 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_append__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4875 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4697 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5190 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_compute__Mixed__Volume.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7814 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_create__Seed__Pair.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7626 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7625 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5009 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4670 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4889 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6199 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_homotopy__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4606 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_indices_lp__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4847 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_is__Member_lp__Abstract__Point_cm__Point__Array_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4736 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_length_lp__Point__Array_rp.html │ │ │ @@ -13855,28 +13855,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 19109 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12849 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7429 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 41382 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2242 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2241 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 719 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_compute__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 723 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_interpolate__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1000 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_max__Grading.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_trim__Basis__In__Degree.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5318 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Coefficient__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4235 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4491 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Previous__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4750 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Return__Target__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5012 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Interpolation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4647 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Matroid.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11856 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11855 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6602 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9224 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_compute__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10098 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_interpolate__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8008 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_max__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9470 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_trim__Basis__In__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14051 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13133 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/master.html │ │ │ @@ -13884,30 +13884,30 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 35822 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 369 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/___N__P.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_gr__Gr.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1249 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_hilbert__Sequence.out │ │ │ --rw-r--r-- 0 root (0) root (0) 412 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ --rw-r--r-- 0 root (0) root (0) 261 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 414 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 262 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ -rw-r--r-- 0 root (0) root (0) 573 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Reduction.out │ │ │ --rw-r--r-- 0 root (0) root (0) 596 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 595 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_multiplicity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1239 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_print__Hilbert__Sequence.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 47 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6190 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/___N__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6706 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_gr__Gr.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8885 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_hilbert__Sequence.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6039 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5705 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6041 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5706 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6831 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Reduction.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6478 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6477 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9585 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_multiplicity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6921 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_print__Hilbert__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15265 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9576 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4371 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplierIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiplierIdeals/dump/ │ │ │ @@ -13973,16 +13973,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 728 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_bs_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 507 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pc_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pl_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 905 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1340 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1277 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1341 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1275 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1200 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_lt_lt_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 693 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1848 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 772 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 803 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 763 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 744 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multirational__Map.out │ │ │ @@ -14001,42 +14001,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_coefficient__Ring_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3115 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cone__Of__Lines.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_conormal__Variety_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cycle__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 671 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_decompose_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 158 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1007 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out │ │ │ +-rw-r--r-- 0 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│ │ -rw-r--r-- 0 root (0) root (0) 7139 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Nauty/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6373 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Nauty/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7363 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Nauty/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8940 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Nauty/html/_relabel__Graph.html │ │ │ @@ -14622,48 +14622,48 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8104 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Nauty/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 131317 2026-01-19 15:01:18.000000 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./usr/share/doc/Macaulay2/NautyGraphs/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1498 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_build__Graph__Filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 124 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 498 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 497 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ -rw-r--r-- 0 root (0) root (0) 323 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__To__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_is__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 849 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_neighborhood__Complements.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_new__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 247 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_only__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 322 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_relabel__Bipartite.out │ │ │ -rw-r--r-- 0 root (0) root (0) 381 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_relabel__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Isomorphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 203 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6195 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5706 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8548 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8545 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9550 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7633 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13153 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7800 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8080 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9132 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9062 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8413 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6700 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6470 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph6__To__Sparse6.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7732 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7731 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8914 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__To__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5929 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_is__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7363 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_neighborhood__Complements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6390 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6301 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6955 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9029 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Graph.html │ │ │ @@ -14691,15 +14691,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179468 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 431 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Dependent__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 791 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op.out │ │ │ -rw-r--r-- 0 root (0) root (0) 553 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op_sp__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_amult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 931 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_coordinate__Change__Ops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 354 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 403 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_differential__Primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 349 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_eliminating__Dual.out │ │ │ -rw-r--r-- 0 root (0) root (0) 270 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_evaluate_lp__Diff__Op_cm__Abstract__Point_rp.out │ │ │ @@ -14732,15 +14732,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 75 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8524 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Dependent__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9032 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7316 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op_sp__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5026 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Sampler.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7406 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7805 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7814 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7815 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4027 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Tolerance_sp_lp__Noetherian__Operators_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6738 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_amult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5314 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_colon.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8444 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_coordinate__Change__Ops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4120 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7027 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6518 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op_lp__Matrix_rp.html │ │ │ @@ -14829,15 +14829,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8147 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3915 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NonminimalComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 608995 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 3510 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3513 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3270 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 908 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___O__O_sp__Toric__Divisor.out │ │ │ @@ -14901,15 +14901,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1271 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_lattice__Points_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Simplicial_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1568 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Smooth_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 601 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 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root (0) 13553 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9792 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12084 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ @@ -15340,34 +15340,34 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 146200 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 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15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_extracting_spinformation_spfrom_spchain_spcomplexes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_free_spresolutions_spof_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_graded__Module__Map.out │ │ │ @@ -15790,15 +15790,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4997 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6585 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5865 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6228 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5591 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_complete_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4290 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_complete_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4214 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_components_lp__Chain__Complex_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7475 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7473 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7151 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_cone_lp__Chain__Complex__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dd.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5785 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dual_lp__Chain__Complex__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3841 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dual_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8237 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_extend_lp__Chain__Complex_cm__Chain__Complex_cm__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7530 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_extracting_spinformation_spfrom_spchain_spcomplexes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5900 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_free_spresolutions_spof_spmodules.html │ │ │ @@ -16254,56 +16254,56 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6360 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OpenMath/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4999 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OpenMath/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3143 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/OpenMath/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84707 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 17050 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 17048 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1080 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2543 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2081 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5401 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 16187 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Oscillators.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3812 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3811 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1682 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_all__Unique__Principal__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10260 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_find__Real__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Angles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 429 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2330 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_identify__Stability.out │ │ │ -rw-r--r-- 0 root (0) root (0) 459 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_is__Stable__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10664 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Jacobian.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1211 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Quadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 625 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 23259 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__System.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2051 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2050 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_standard__Sols.out │ │ │ -rw-r--r-- 0 root (0) root (0) 236 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_vertex__Spanning__Polynomial.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 25409 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25407 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6234 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7736 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7644 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10645 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6691 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3563 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Harrington-__Schenck-__Stillman.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8191 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8190 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8807 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_all__Unique__Principal__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19710 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_find__Real__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6864 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Angles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6738 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10573 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_identify__Stability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7493 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_is__Stable__Solution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20003 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Jacobian.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7847 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Quadrics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8735 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 32787 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__System.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9935 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9934 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7891 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_standard__Sols.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6388 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_vertex__Spanning__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 40232 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18097 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8240 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Oscillators/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PHCpack/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PHCpack/dump/ │ │ │ @@ -16524,15 +16524,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12104 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Parsing/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9154 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Parsing/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7244 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Parsing/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 136324 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2842 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2841 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ -rw-r--r-- 0 root (0) root (0) 456 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___C__Axis__Tensor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1364 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___C__Mon__Tensor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2634 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Path.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1530 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Path__Signatures.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1392 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Polynomial_sppaths_spof_spdegree_spm.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11000 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___The_spuniversal_spvariety_spand_sptoric_spcoordinates.out │ │ │ @@ -16558,15 +16558,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 204 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_tensor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 886 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_tensor__Parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2023 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 387 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__Format.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__String.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 38 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 11402 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11401 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7073 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Array_sp_us_sp__N__C__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7959 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___C__Axis__Tensor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8285 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___C__Mon__Tensor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3941 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Computing_sp__Path_sp__Varieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7451 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___N__C__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21351 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___N__C__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15444 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Path.html │ │ │ @@ -16962,15 +16962,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 14765 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12202 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4887 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 47371 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1730 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1728 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 619 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Make__Ring__Maps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1456 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1010 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 463 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points__Mat.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_expected__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 234 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_min__Max__Resolution.out │ │ │ @@ -16982,15 +16982,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_projective__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_random__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 651 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/example-output/_random__Points__Mat.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 14 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4517 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/___All__Random.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4654 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/___Verify__Points.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10167 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10165 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7047 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points__By__Intersection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6451 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Make__Ring__Maps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8176 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7009 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points__By__Intersection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6970 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points__Mat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6070 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/_expected__Betti.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5252 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Points/html/_min__Max__Resolution.html │ │ │ @@ -17404,15 +17404,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2469 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__Hibi_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__Intersection_splattices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__L__C__M-lattices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 123 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_st.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2468 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2465 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_adjoin__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 306 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_adjoin__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 668 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_all__Relations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_antichains.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_atoms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 253 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_boolean__Lattice.out │ │ │ @@ -17443,15 +17443,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_filtration.out │ │ │ -rw-r--r-- 0 root (0) root (0) 525 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_flag__Chains.out │ │ │ -rw-r--r-- 0 root (0) root (0) 842 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_flag__Poset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 210 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_flagf__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 244 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_flagh__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_gap__Convert__Poset.out │ │ │ --rw-r--r-- 0 root (0) root (0) 589 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_greene__Kleitman__Partition.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 591 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_greene__Kleitman__Partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_h__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 290 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_hasse__Diagram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 96 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_height_lp__Poset_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 258 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_hibi__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 907 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_hibi__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_incomparability__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 269 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_index__Labeling.out │ │ │ @@ -17531,15 +17531,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9486 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__Hibi_spideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7037 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__Intersection_splattices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6041 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__L__C__M-lattices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 38795 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5707 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_sp__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5830 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5493 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_st.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8668 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8665 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6042 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_adjoin__Max.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6073 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_adjoin__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6593 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_all__Relations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6851 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_antichains.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7024 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5632 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_atoms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5773 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_augment__Poset.html │ │ │ @@ -17572,15 +17572,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6119 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7393 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_filtration.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6942 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_flag__Chains.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7182 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_flag__Poset.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6827 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_flagf__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6830 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_flagh__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7695 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_gap__Convert__Poset.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8202 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8204 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6368 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_h__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6162 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_hasse__Diagram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5339 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_height_lp__Poset_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6756 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_hibi__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9660 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_hibi__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6039 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_incomparability__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6934 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Posets/html/_index__Labeling.html │ │ │ @@ -17694,30 +17694,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 81139 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2696 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/___Primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3736 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated__Primes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated_spprimes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 411 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_irreducible__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_is__Primary.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2222 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2223 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 627 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_localize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1734 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1390 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 423 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_remove__Lowest__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2249 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_strategies_spfor_spcomputing_spprimary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_top__Components.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 17690 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4670 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated_spprimes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5762 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_irreducible__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8232 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_is__Primary.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8732 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8733 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10172 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_localize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8209 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11813 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21071 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5735 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary_spdecomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9886 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7321 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_remove__Lowest__Dimension.html │ │ │ @@ -18142,15 +18142,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2633 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300c].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2039 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Half_spcanonical_spdegree_sp20.out │ │ │ --rw-r--r-- 0 root (0) root (0) 22026 2026-01-19 15:01:18.000000 ./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) 22546 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Noether-__Lefschetz_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7051 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Pfaffians_spon_spquadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Singularities_spof_splifting_spof_sptype_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 4091 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[331]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ @@ -18184,15 +18184,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9061 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 9848 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300c].html │ │ │ -rw-r--r-- 0 root (0) root (0) 6828 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16172 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19856 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9268 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8693 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Half_spcanonical_spdegree_sp20.html │ │ │ --rw-r--r-- 0 root (0) root (0) 47674 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 48194 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10984 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Noether-__Lefschetz_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4123 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12440 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Pfaffians_spon_spquadrics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8586 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Singularities_spof_splifting_spof_sptype_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 14692 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12230 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9284 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.html │ │ │ @@ -18328,27 +18328,27 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2678 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_disturb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1358 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_histogram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1315 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_maximal__Entry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1522 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_normalize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5055 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 510 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Simplicial__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1161 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1157 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3937 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Discrete.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4295 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___With__L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4151 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Zero__Mean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9992 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_disturb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7241 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_histogram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7257 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_maximal__Entry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6973 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14308 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Simplicial__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8072 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8068 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11390 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9528 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4800 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1330 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/ │ │ │ @@ -18395,25 +18395,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6319 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4166 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 85866 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Finding_sp__Extreme_sp__Examples.out │ │ │ --rw-r--r-- 0 root (0) root (0) 486 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 448 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 481 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__Chain__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_is__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Addition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 894 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) 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./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11976 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13012 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15236 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_are__Pseudo__Inverses.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3541 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_check__S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10080 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14702 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14335 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_laplacians.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3882 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_new__Chain__Complex__Map.html │ │ │ @@ -19539,24 +19539,24 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52965 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_annihilator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_ideal_spquotients_spand_spsaturation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_is__Supported__In__Zero__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 807 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1836 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1835 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2582 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 837 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_saturate.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 69 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8747 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/_annihilator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6560 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/_ideal_spquotients_spand_spsaturation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6876 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/_is__Supported__In__Zero__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7263 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17356 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17355 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15598 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp__Module_cm__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11944 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12655 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13853 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11866 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4273 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Saturation/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/ │ │ │ @@ -19590,15 +19590,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1688 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Hom_lp__Abstract__Sheaf_cm__Abstract__Sheaf_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1585 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1584 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Abstract__Variety.out │ │ │ 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(0) root (0) 5605 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Examples_spfrom_sp__Schubert_cm_sptranslated.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8983 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Flag__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6370 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Hom_lp__Abstract__Sheaf_cm__Abstract__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5551 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Ideas_spfor_spfuture_spdevelopment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6351 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Incidence__Correspondence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4163 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Intersection__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4202 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Isotropic.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8062 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8061 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4976 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Abstract__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6353 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4277 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Pull__Back.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6529 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Quotient__Bundles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5154 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spcurve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7875 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spsurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11386 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spwithout_spdenominators.html │ │ │ @@ -19997,29 +19997,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Multi__Hom.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Product__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 453 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2026-01-19 15:01:18.000000 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0 root (0) root (0) 6810 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44566 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/example-output/ │ │ │ @@ -20541,18 +20541,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1122 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Multidimensional__Matrix_sp_st_sp__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 646 2026-01-19 15:01:18.000000 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./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1362 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 973 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dim_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 494 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_entries_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 394 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents_lp__Sparse__Discriminant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 471 2026-01-19 15:01:18.000000 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./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6902 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8190 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8264 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8705 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6122 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dim_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5711 2026-01-19 15:01:18.000000 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15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 65533 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 20152 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 65532 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7079 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sylvester__Matrix_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21933 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20968 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11034 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root (0) 439 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_first__Row__Descent_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_garnir__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 557 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_generalized__Vandermonde__Matrix_lp__List_cm__List_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 787 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_generate__Permutation__Group_lp__List_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3114 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3116 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11027 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 325 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_hook__Length__Formula_lp__Partition_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 538 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_index__Monomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_index__Tableau_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2026-01-19 15:01:18.000000 ./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 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_list__To__Tableau_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 993 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_matrix__Representation.out │ │ │ @@ -20708,15 +20708,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5388 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_cycle__Decomposition_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5828 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_elementary__Symmetric__Polynomials_lp__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5058 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_entries_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6351 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_first__Row__Descent_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8083 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_garnir__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6758 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_generalized__Vandermonde__Matrix_lp__List_cm__List_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6340 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_generate__Permutation__Group_lp__List_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13548 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13550 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20867 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6155 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_hook__Length__Formula_lp__Partition_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7235 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_index__Monomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6024 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_index__Tableau_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6746 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_inner__Product_lp__Z__Z_cm__Mutable__Matrix_cm__Mutable__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5269 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_list__To__Tableau_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8110 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_matrix__Representation.html │ │ │ @@ -20755,48 +20755,48 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 203984 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 395 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/___Congruence__Of__Curves_sp__Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 914 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/___G__Mtables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1325 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_ambient__Fivefold.out │ │ │ --rw-r--r-- 0 root (0) root (0) 954 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 953 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1005 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1191 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_beauville__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 351 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_clean_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1173 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1172 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1555 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Cubic__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 287 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 286 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 599 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_from__Ordinary__To__Gushel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible__G__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 716 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_map_lp__Congruence__Of__Curves_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1356 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_mirror__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_normal__Sheaf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 814 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 786 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 973 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 653 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1972 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6386 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6383 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 393 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold_lp__String_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1499 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Fourfold.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1519 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1520 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 762 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__Array_cm__Array_cm__String_cm__Thing_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 469 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 530 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold_lp__String_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 413 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2427 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 770 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__External__String_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5324 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5325 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4324 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 531 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6372 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Congruence__Of__Curves.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6988 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Congruence__Of__Curves_sp__Embedded__Projective__Variety.html │ │ │ @@ -20804,26 +20804,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9888 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Hodge__Special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6265 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Intersection__Of__Three__Quadrics__In__P7.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4602 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Singular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9519 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Special__Cubic__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10070 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Special__Gushel__Mukai__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8773 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Verbose.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7793 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_ambient__Fivefold.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9596 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9595 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4945 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10359 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10574 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6841 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_beauville__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5897 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_check_lp__Z__Z_cm__Congruence__Of__Curves_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5533 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_clean_lp__Hodge__Special__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6192 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8687 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8686 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9574 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6654 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Cubic__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6746 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6745 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6642 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5909 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_from__Ordinary__To__Gushel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5585 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2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_normal__Sheaf.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8483 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7955 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8182 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8186 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8747 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize_lp__Hodge__Special__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15460 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15457 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6948 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7039 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold_lp__String_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10491 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6591 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold_lp__String_cm__Z__Z_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12073 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12074 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11528 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__Array_cm__Array_cm__String_cm__Thing_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7954 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7745 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold_lp__String_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7348 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7730 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9302 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6144 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__External__String_lp__Hodge__Special__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11932 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11933 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10746 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5400 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_trisecant__Flop.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7020 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37413 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 33496 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14735 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/SpectralSequences/ │ │ │ @@ -21787,25 +21787,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 806 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_compatible__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 378 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_decompose__Fraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_descend__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_floor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Preimage.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1329 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1328 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1117 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Trace__On__Canonical__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 488 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1822 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1820 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 637 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Pure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Rational.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1492 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1491 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_multiplicative__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_parameter__Test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1514 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1512 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2756 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Module.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4326 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Ascent__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4385 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__C__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5130 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__Domain.html │ │ │ @@ -21838,25 +21838,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9151 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_compatible__Ideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7168 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_decompose__Fraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9334 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_descend__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5225 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_floor__Log.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10611 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12408 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5237 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Preimage.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16746 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16745 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8554 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Trace__On__Canonical__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8216 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15809 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15807 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10346 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Pure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9661 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Rational.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16392 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16391 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5824 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_multiplicative__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7890 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_parameter__Test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6302 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Element.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14994 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14992 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17708 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37577 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34814 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12484 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Text/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Text/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 136721 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump │ │ │ @@ -22056,30 +22056,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 37607 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27702 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11513 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24388 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1510 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1429 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6776 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 924 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1155 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1989 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1034 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1143 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3848 2026-01-19 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root (0) root (0) 6665 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6866 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7158 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12692 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6976 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7146 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14551 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7430 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21626 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6806 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4627 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Topcom/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Topcom/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 66598 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump │ │ │ @@ -22186,25 +22186,25 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 20320 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 944 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Volumes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_dual__Deg__Codim.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1537 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1536 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1032 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_polar__Degrees.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 86 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4388 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Force__Amat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4214 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4206 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Text__Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8847 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7510 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Volumes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7471 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_dual__Deg__Codim.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8818 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8817 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8849 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_polar__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9974 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7189 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4667 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricTopology/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 38537 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/rawdocumentation.dump │ │ │ @@ -22373,17 +22373,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 17380 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11888 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7217 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 31886 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2273 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2275 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 18048 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_all__Triangulations_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 75224 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 75242 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 714 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_is__Regular__Triangulation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1654 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_triangulation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4460 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3951 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Cone__Index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7308 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/___Triangulation.html │ │ │ @@ -22391,29 +22391,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 30386 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_all__Triangulations_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3970 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_bistellar__Flip.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3884 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3827 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_delaunay__Subdivision.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3823 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_delaunay__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3867 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_fine__Star__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3781 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_flips.html │ │ │ --rw-r--r-- 0 root (0) root (0) 86004 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 86022 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3911 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_gkz__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3873 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Fine.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9223 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Regular__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3883 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_is__Star.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3814 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_naive__Chirotope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4176 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_naive__Is__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3807 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_neighbors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3950 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Fine__Star__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3970 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Fine__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4144 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Triangulation__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9719 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3727 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_vectors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3837 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_volume__Vector.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24697 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24699 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10341 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6323 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triangulations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triplets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triplets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49146 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/___Betti1_lp__Triplet_rp.out │ │ │ @@ -23204,15 +23204,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11114 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9177 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 226382 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Def__Param.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1215 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1212 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_check__Comparison__Theorem.out │ │ │ -rw-r--r-- 0 root (0) root (0) 406 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_check__Tangent__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 862 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_correct__Deformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_cotangent__Cohomology1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 774 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_cotangent__Cohomology2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 562 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_ext__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3316 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_families.out │ │ │ @@ -23241,15 +23241,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5713 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Correction__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7708 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Def__Param.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5603 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Degree__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4933 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Highest__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9463 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Nested__Deformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5058 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Polynomial__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6067 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Sanity__Check.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8083 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8080 2026-01-19 15:01:18.000000 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│ │ @@ -23444,15 +23444,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6439 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_ceiling_lp__R__Weil__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5314 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_clean__Support.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6048 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_clear__Cache.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6403 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_coefficient_lp__Basic__List_cm__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6550 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_coefficient_lp__Ideal_cm__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8179 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_coefficients_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19737 2026-01-19 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./usr/share/doc/Macaulay2/WeilDivisors/html/_to__R__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7207 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6826 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_torsion__Submodule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6176 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_trim_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4924 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_zero__Divisor.html │ │ │ @@ -23502,15 +23502,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 365 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/___Dsingular__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 380 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeylAlgebras/example-output/___Dtransposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2187 2026-01-19 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./usr/share/doc/Macaulay2/WeylAlgebras/html/_make__Cyclic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7465 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeylAlgebras/html/_make__Weyl__Algebra.html │ │ │ @@ -23815,15 +23815,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 36900 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WeylGroups/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 62174 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 658 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_conormal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1236 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_euler__Obs__Matrix.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4499 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4498 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1164 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify__Pol.out │ │ │ -rw-r--r-- 0 root (0) root (0) 785 2026-01-19 15:01:18.000000 ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_min__Coarsen__W__S.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1601 2026-01-19 15:01:18.000000 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15:01:18.000000 ./usr/share/info/SumsOfSquares.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5366 2026-01-19 15:01:18.000000 ./usr/share/info/SuperLinearAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2753 2026-01-19 15:01:18.000000 ./usr/share/info/SwitchingFields.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14147 2026-01-19 15:01:18.000000 ./usr/share/info/SymbolicPowers.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14145 2026-01-19 15:01:18.000000 ./usr/share/info/SymbolicPowers.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2157 2026-01-19 15:01:18.000000 ./usr/share/info/SymmetricPolynomials.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 14151 2026-01-19 15:01:18.000000 ./usr/share/info/TSpreadIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 17554 2026-01-19 15:01:18.000000 ./usr/share/info/Tableaux.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1559 2026-01-19 15:01:18.000000 ./usr/share/info/TangentCone.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 47114 2026-01-19 15:01:18.000000 ./usr/share/info/TateOnProducts.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 47117 2026-01-19 15:01:18.000000 ./usr/share/info/TateOnProducts.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 22604 2026-01-19 15:01:18.000000 ./usr/share/info/TensorComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2389 2026-01-19 15:01:18.000000 ./usr/share/info/TerraciniLoci.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 31169 2026-01-19 15:01:18.000000 ./usr/share/info/TestIdeals.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 31184 2026-01-19 15:01:18.000000 ./usr/share/info/TestIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 15054 2026-01-19 15:01:18.000000 ./usr/share/info/Text.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 22883 2026-01-19 15:01:18.000000 ./usr/share/info/ThinSincereQuivers.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8072 2026-01-19 15:01:18.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8441 2026-01-19 15:01:18.000000 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+-rw-r--r-- 0 root (0) root (0) 10702 2026-01-19 15:01:18.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 33226 2026-01-19 15:01:18.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14775 2026-01-19 15:01:18.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14774 2026-01-19 15:01:18.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8865 2026-01-19 15:01:18.000000 ./usr/share/info/XML.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 49483 2026-01-19 15:01:18.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 49481 2026-01-19 15:01:18.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/lintian/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/lintian/overrides/ │ │ │ -rw-r--r-- 0 root (0) root (0) 11489 2026-01-19 03:44:31.000000 ./usr/share/lintian/overrides/macaulay2-common │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/Macaulay2/Style/katex/contrib/auto-render.min.js -> ../../../../javascript/katex/contrib/auto-render.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/Macaulay2/Style/katex/contrib/copy-tex.min.js -> ../../../../javascript/katex/contrib/copy-tex.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/Macaulay2/Style/katex/contrib/render-a11y-string.min.js -> ../../../../javascript/katex/contrib/render-a11y-string.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-01-19 15:01:18.000000 ./usr/share/Macaulay2/Style/katex/fonts/KaTeX_AMS-Regular.ttf -> ../../../../fonts/truetype/katex/KaTeX_AMS-Regular.ttf │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ @@ -10,25 +10,25 @@ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ - -- 1.87056s elapsed │ │ │ + -- 1.4482s 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) │ │ │ - -- 1.98785s elapsed │ │ │ + -- 1.84748s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ @@ -90,28 +90,28 @@ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i3 : elapsedTime burkeResolution(M, 7, Check => false)
│ │ │ - -- 1.87056s elapsed
│ │ │ + -- 1.4482s 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)
│ │ │ - -- 1.98785s elapsed
│ │ │ + -- 1.84748s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o4 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o4 : Complex
│ │ │ ├── html2text {} │ │ │ │ @@ -23,24 +23,24 @@ │ │ │ │ i2 : M = coker vars R │ │ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ │ │ 1 │ │ │ │ o2 : R-module, quotient of R │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ │ - -- 1.87056s elapsed │ │ │ │ + -- 1.4482s 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) │ │ │ │ - -- 1.98785s elapsed │ │ │ │ + -- 1.84748s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ @@ -49,15 +49,15 @@ │ │ │ o8 : BettiTally │ │ │ │ │ │ i9 : c=codim I │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ i10 : elapsedTime fI=res I │ │ │ - -- .0787826s elapsed │ │ │ + -- .030287s elapsed │ │ │ │ │ │ 1 14 33 28 8 │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn <-- 0 │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ o10 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ @@ -87,30 +87,30 @@ │ │ │ o13 : BettiTally │ │ │ │ │ │ i14 : phi=map(P2,Pn,H); │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ i15 : elapsedTime betti(I'=trim ker phi) │ │ │ - -- .641723s elapsed │ │ │ + -- .60542s elapsed │ │ │ │ │ │ 0 1 │ │ │ o15 = total: 1 11 │ │ │ 0: 1 . │ │ │ 1: . 3 │ │ │ 2: . 8 │ │ │ │ │ │ o15 : BettiTally │ │ │ │ │ │ i16 : I'== I │ │ │ │ │ │ o16 = true │ │ │ │ │ │ i17 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 5.96256s elapsed │ │ │ + -- 5.38966s 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) │ │ │ - -- .0128168s elapsed │ │ │ + -- .0153536s 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) │ │ │ - -- .0546974s elapsed │ │ │ + -- .0663602s elapsed │ │ │ │ │ │ 0 1 │ │ │ o16 = total: 1 12 │ │ │ 0: 1 . │ │ │ 1: . 12 │ │ │ │ │ │ o16 : BettiTally │ │ │ │ │ │ i17 : I'== I │ │ │ │ │ │ o17 = true │ │ │ │ │ │ i18 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 1.68941s elapsed │ │ │ + -- 1.61418s elapsed │ │ │ │ │ │ i19 : tally apply(basePts,c->(dim c, degree c, betti c)) │ │ │ │ │ │ 0 1 │ │ │ o19 = Tally{(0, 34, total: 1 15) => 1} │ │ │ 0: 1 . │ │ │ 1: . . │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ @@ -149,15 +149,15 @@ │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i10 : elapsedTime fI=res I
│ │ │ - -- .0787826s elapsed
│ │ │ + -- .030287s elapsed
│ │ │  
│ │ │          1       14       33       28       8
│ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn  <-- 0
│ │ │                                                    
│ │ │        0       1        2        3        4       5
│ │ │  
│ │ │  o10 : ChainComplex
│ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ 2: . 12 │ │ │ │ │ │ │ │ o8 : BettiTally │ │ │ │ i9 : c=codim I │ │ │ │ │ │ │ │ o9 = 4 │ │ │ │ i10 : elapsedTime fI=res I │ │ │ │ - -- .0787826s elapsed │ │ │ │ + -- .030287s elapsed │ │ │ │ │ │ │ │ 1 14 33 28 8 │ │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn <-- 0 │ │ │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ │ │ o10 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ @@ -217,15 +217,15 @@ │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .641723s elapsed
│ │ │ + -- .60542s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o15 = total: 1 11
│ │ │            0: 1  .
│ │ │            1: .  3
│ │ │            2: .  8
│ │ │  
│ │ │ @@ -238,15 +238,15 @@
│ │ │  
│ │ │  o16 = true
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 5.96256s elapsed
│ │ │ + -- 5.38966s 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)
│ │ │ │ - -- .641723s elapsed
│ │ │ │ + -- .60542s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o15 = total: 1 11
│ │ │ │            0: 1  .
│ │ │ │            1: .  3
│ │ │ │            2: .  8
│ │ │ │  
│ │ │ │  o15 : BettiTally
│ │ │ │  i16 : I'== I
│ │ │ │  
│ │ │ │  o16 = true
│ │ │ │  i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 5.96256s elapsed
│ │ │ │ + -- 5.38966s elapsed
│ │ │ │  i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                            0 1
│ │ │ │  o18 = Tally{(1, 1, total: 1 2) => 5}
│ │ │ │                         0: 1 2
│ │ │ │                            0 1
│ │ │ │              (1, 3, total: 1 3) => 8
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html
│ │ │ @@ -193,15 +193,15 @@
│ │ │  
│ │ │  o13 : BettiTally
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : elapsedTime sub(I,H)
│ │ │ - -- .0128168s elapsed
│ │ │ + -- .0153536s elapsed
│ │ │  
│ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o14 : Ideal of P2
│ │ │ │ │ │ │ │ │ │ │ │ @@ -210,15 +210,15 @@ │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .0546974s elapsed
│ │ │ + -- .0663602s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o16 = total: 1 12
│ │ │            0: 1  .
│ │ │            1: . 12
│ │ │  
│ │ │  o16 : BettiTally
│ │ │ @@ -230,15 +230,15 @@ │ │ │ │ │ │ o17 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 1.68941s elapsed
│ │ │ + -- 1.61418s 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)
│ │ │ │ - -- .0128168s elapsed
│ │ │ │ + -- .0153536s 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)
│ │ │ │ - -- .0546974s elapsed
│ │ │ │ + -- .0663602s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o16 = total: 1 12
│ │ │ │            0: 1  .
│ │ │ │            1: . 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ │ │  i17 : I'== I
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 1.68941s elapsed
│ │ │ │ + -- 1.61418s elapsed
│ │ │ │  i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                             0  1
│ │ │ │  o19 = Tally{(0, 34, total: 1 15) => 1}
│ │ │ │                          0: 1  .
│ │ │ │                          1: .  .
│ │ │ │                          2: .  .
│ │ ├── ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out
│ │ │ @@ -114,26 +114,26 @@
│ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │  
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │  
│ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.451781s (cpu); 0.373051s (thread); 0s (gc)
│ │ │ + -- used 0.526854s (cpu); 0.442563s (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 0.495643s (cpu); 0.418777s (thread); 0s (gc)
│ │ │ + -- used 0.621266s (cpu); 0.531415s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html
│ │ │ @@ -253,15 +253,15 @@
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.451781s (cpu); 0.373051s (thread); 0s (gc)
│ │ │ + -- used 0.526854s (cpu); 0.442563s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │ @@ -272,15 +272,15 @@
│ │ │          

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

│ │ │          

│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,26 +192,26 @@
│ │ │ │  o18 : ActionOnComplex
│ │ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │ │  
│ │ │ │  o19 = Complex with 6 actors
│ │ │ │  
│ │ │ │  o19 : ActionOnComplex
│ │ │ │  i20 : elapsedTime a1 = character A1
│ │ │ │ - -- .966261s elapsed
│ │ │ │ + -- .732098s elapsed
│ │ │ │  
│ │ │ │  o20 = Character over R
│ │ │ │  
│ │ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │        (2, {11}) => | 1 1 1 1 1 1 |
│ │ │ │        (2, {13}) => | 1 1 1 1 1 1 |
│ │ │ │  
│ │ │ │  o20 : Character
│ │ │ │  i21 : elapsedTime a2 = character A2
│ │ │ │ - -- 33.2037s elapsed
│ │ │ │ + -- 26.984s elapsed
│ │ │ │  
│ │ │ │  o21 = Character over R
│ │ │ │  
│ │ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ │ @@ -308,15 +308,15 @@
│ │ │ │  i30 : M = Is2 / I2;
│ │ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │ │  
│ │ │ │  o31 = Module with 6 actors
│ │ │ │  
│ │ │ │  o31 : ActionOnGradedModule
│ │ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ │ - -- 15.7832s elapsed
│ │ │ │ + -- 12.2335s elapsed
│ │ │ │  
│ │ │ │  o32 = Character over R
│ │ │ │  
│ │ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │ │  
│ │ │ │  o32 : Character
│ │ │ │  i33 : b/T
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out
│ │ │ @@ -230,15 +230,15 @@
│ │ │            0: 1 . . . .
│ │ │            1: . 4 2 . .
│ │ │            2: . 1 6 5 1
│ │ │  
│ │ │  o22 : BettiTally
│ │ │  
│ │ │  i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.225031s (cpu); 0.172702s (thread); 0s (gc)
│ │ │ + -- used 0.312088s (cpu); 0.224665s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
│ │ │  
│ │ │  i24 : betti res j
│ │ │  
│ │ │               0 1 2 3 4
│ │ │  o24 = total: 1 3 6 5 1
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html
│ │ │ @@ -380,15 +380,15 @@
│ │ │  
│ │ │  o22 : BettiTally
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │          

│ │ │                
i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.495643s (cpu); 0.418777s (thread); 0s (gc)
│ │ │ + -- used 0.621266s (cpu); 0.531415s (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 0.451781s (cpu); 0.373051s (thread); 0s (gc)
│ │ │ │ + -- used 0.526854s (cpu); 0.442563s (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 0.495643s (cpu); 0.418777s (thread); 0s (gc)
│ │ │ │ + -- used 0.621266s (cpu); 0.531415s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o15 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out
│ │ │ @@ -1,10 +1,10 @@
│ │ │  -- -*- M2-comint -*- hash: 1330545576567
│ │ │  
│ │ │  i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Mon Jan 19 17:35:01 UTC 2026
│ │ │ --- Linux sbuild 6.12.63+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1 (2025-12-30) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249  
│ │ │ +-- beginning computation Sun Jan 25 00:48:10 UTC 2026
│ │ │ +-- Linux sbuild 6.12.63+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1 (2025-12-30) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .12166 seconds
│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .165499 seconds
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html
│ │ │ @@ -75,19 +75,19 @@
│ │ │          

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

│ │ │          

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

│ │ │                
i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Mon Jan 19 17:35:01 UTC 2026
│ │ │ --- Linux sbuild 6.12.63+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1 (2025-12-30) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249  
│ │ │ +-- beginning computation Sun Jan 25 00:48:10 UTC 2026
│ │ │ +-- Linux sbuild 6.12.63+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1 (2025-12-30) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .12166 seconds

│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .165499 seconds
│ │ │              

│ │ │        
│ │ │        

│ │ │          

│ │ │            
For the programmer

│ │ │ ├── html2text {}
│ │ │ │ @@ -23,18 +23,18 @@
│ │ │ │  "gb3445" -- gb of an ideal with elements of degree 3,4,4,5 in 8 variables
│ │ │ │  "gbB148" -- gb of Bayesian graph ideal #148
│ │ │ │  "res39" -- res of a generic 3 by 9 matrix over ZZ/101
│ │ │ │  "resG25" -- res of the coordinate ring of Grassmannian(2,5)
│ │ │ │  "yang-gb1" -- an example of Yang-Hui He arising in string theory
│ │ │ │  "yang-subring" -- an example of Yang-Hui He
│ │ │ │  i1 : runBenchmarks "res39"
│ │ │ │ --- beginning computation Mon Jan 19 17:35:01 UTC 2026
│ │ │ │ --- Linux sbuild 6.12.63+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1
│ │ │ │ -(2025-12-30) x86_64 GNU/Linux
│ │ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.249
│ │ │ │ +-- beginning computation Sun Jan 25 00:48:10 UTC 2026
│ │ │ │ +-- Linux sbuild 6.12.63+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.63-1 (2025-12-30) x86_64 GNU/Linux
│ │ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998
│ │ │ │  -- Macaulay2 1.25.11, compiled with gcc 15.2.0
│ │ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .12166 seconds
│ │ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .165499 seconds
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_u_n_B_e_n_c_h_m_a_r_k_s is a _c_o_m_m_a_n_d.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Benchmark.m2:297:0.
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -515,15 +515,15 @@
│ │ │  Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbUNvbXBsZXgifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57fX0sU1BBTntUTzJ7bmV3
│ │ │  IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pVXNlckhvbW90b3B5LFJhbmRvbVJlYWxdLCJiZXJ0
│ │ │  aW5pVXNlckhvbW90b3B5KC4uLixSYW5kb21SZWFsPT4uLi4pIiwiQmVydGluaSJ9LCJSYW5kb21S
│ │ │  ZWFsIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZhbHVlICIsInt9
│ │ │  In0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHsiVG9wRGlyZWN0
│ │ │  b3J5IiwiVG9wRGlyZWN0b3J5IiwiQmVydGluaSJ9LCJUb3BEaXJlY3RvcnkifSxUVHsiID0+ICJ9
│ │ │ -LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTI4NjgwLTAv
│ │ │ +LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTQwOTEwLTAv
│ │ │  MFwiIn0sIiwgIixTUEFOeyJPcHRpb24gdG8gY2hhbmdlIGRpcmVjdG9yeSBmb3IgZmlsZSBzdG9y
│ │ │  YWdlLiJ9fSxTUEFOe1RPMntuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2JlcnRpbmlVc2VySG9tb3Rv
│ │ │  cHksVmVyYm9zZV0sImJlcnRpbmlVc2VySG9tb3RvcHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0
│ │ │  aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQg
│ │ │  dmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwg
│ │ │  b3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsi
│ │ │  YmVydGluaVVzZXJIb21vdG9weSIsImJlcnRpbmlVc2VySG9tb3RvcHkiLCJCZXJ0aW5pIn0sIEtl
│ │ │ @@ -1100,15 +1100,15 @@
│ │ │  ZXJ0aW5pUGFyYW1ldGVySG9tb3RvcHkoLi4uLFJhbmRvbVJlYWw9Pi4uLikiLCJCZXJ0aW5pIn0s
│ │ │  IlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFs
│ │ │  dWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGljaCBkZXNpZ25hdGVzIHN5bWJvbHMv
│ │ │  c3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0byBiZSBhIHJhbmRvbSByZWFsIG51
│ │ │  bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BBTntUTzJ7bmV3IERvY3VtZW50VGFn
│ │ │  IGZyb20geyJUb3BEaXJlY3RvcnkiLCJUb3BEaXJlY3RvcnkiLCJCZXJ0aW5pIn0sIlRvcERpcmVj
│ │ │  dG9yeSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIsICIsU1BBTnsiZGVmYXVsdCB2YWx1ZSAiLCJc
│ │ │ -Ii90bXAvTTItMjg2ODAtMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │ +Ii90bXAvTTItNDA5MTAtMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │  b3J5IGZvciBmaWxlIHN0b3JhZ2UuIn19LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVBhcmFtZXRlckhvbW90b3B5LFZlcmJvc2VdLCJiZXJ0aW5pUGFyYW1ldGVySG9tb3Rv
│ │ │  cHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRU
│ │ │  eyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9w
│ │ │  dGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwgb3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiYmVydGluaVBhcmFtZXRlckhvbW90b3B5IiwiYmVy
│ │ │  dGluaVBhcmFtZXRlckhvbW90b3B5IiwiQmVydGluaSJ9LCBLZXkgPT4gYmVydGluaVBhcmFtZXRl
│ │ │ @@ -2449,15 +2449,15 @@
│ │ │  YWw9Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGlj
│ │ │  aCBkZXNpZ25hdGVzIHN5bWJvbHMvc3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0
│ │ │  byBiZSBhIHJhbmRvbSByZWFsIG51bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BB
│ │ │  TntUTzJ7bmV3IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFRvcERpcmVj
│ │ │  dG9yeV0sImJlcnRpbmlaZXJvRGltU29sdmUoLi4uLFRvcERpcmVjdG9yeT0+Li4uKSIsIkJlcnRp
│ │ │  bmkifSwiVG9wRGlyZWN0b3J5In0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZh
│ │ │ -dWx0IHZhbHVlICIsIlwiL3RtcC9NMi0yODY4MC0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │ +dWx0IHZhbHVlICIsIlwiL3RtcC9NMi00MDkxMC0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │  IGNoYW5nZSBkaXJlY3RvcnkgZm9yIGZpbGUgc3RvcmFnZS4ifX0sU1BBTntUTzJ7bmV3IERvY3Vt
│ │ │  ZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFVzZVJlZ2VuZXJhdGlvbl0sImJlcnRp
│ │ │  bmlaZXJvRGltU29sdmUoLi4uLFVzZVJlZ2VuZXJhdGlvbj0+Li4uKSIsIkJlcnRpbmkifSwiVXNl
│ │ │  UmVnZW5lcmF0aW9uIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZh
│ │ │  bHVlICIsIi0xIn0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVplcm9EaW1Tb2x2ZSxWZXJib3NlXSwiYmVydGluaVplcm9EaW1Tb2x2ZSguLi4sVmVy
│ │ │  Ym9zZT0+Li4uKSIsIkJlcnRpbmkifSwiVmVyYm9zZSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │              
  • HomVariableGroup => ..., default value {}, an option to group variables and use multihomogeneous homotopies
    • 
│ │ │              
  • M2Precision (missing documentation)
│ │ │   => ..., default value 53, 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-28680-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-40910-0/0", 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/M2-28680-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-40910-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list whose entries are lists of solutions for each
│ │ │ │              target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__User__Homotopy.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │   => ..., default value 53, 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │              
  • RandomReal (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-28680-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-40910-0/0", 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/M2-28680-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-40910-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S0, a _l_i_s_t, a list of solutions to the target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This method calls Bertini to track a user-defined homotopy. The user needs to
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │   => ..., default value "main_data", 
│ │ │              
  • NameSolutionsFile (missing documentation)
│ │ │   => ..., default value "raw_solutions", 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-28680-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-40910-0/0", 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/M2-28680-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-40910-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o UseRegeneration (missing documentation) => ..., default value -1,
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list of points that are contained in the variety of F
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out
│ │ │ @@ -76,15 +76,15 @@
│ │ │  i8 : A = action(RI,S7)
│ │ │  
│ │ │  o8 = Complex with 15 actors
│ │ │  
│ │ │  o8 : ActionOnComplex
│ │ │  
│ │ │  i9 : elapsedTime c = character A
│ │ │ - -- .437329s elapsed
│ │ │ + -- .375566s elapsed
│ │ │  
│ │ │  o9 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {2}) => | 0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14 |
│ │ │       (2, {3}) => | 0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35 |
│ │ │       (3, {4}) => | 0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out
│ │ │ @@ -100,15 +100,15 @@
│ │ │  i6 : A=action(RI,S6)
│ │ │  
│ │ │  o6 = Complex with 11 actors
│ │ │  
│ │ │  o6 : ActionOnComplex
│ │ │  
│ │ │  i7 : elapsedTime c=character A
│ │ │ - -- .461365s elapsed
│ │ │ + -- .443902s elapsed
│ │ │  
│ │ │  o7 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {5}) => | 0 1 0 2 0 1 3 0 2 4 6 |
│ │ │       (1, {7}) => | 0 0 0 0 0 1 3 0 4 16 60 |
│ │ │       (1, {9}) => | 0 0 0 0 2 2 2 0 4 8 20 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out
│ │ │ @@ -187,27 +187,27 @@
│ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │  
│ │ │  o19 = Complex with 6 actors
│ │ │  
│ │ │  o19 : ActionOnComplex
│ │ │  
│ │ │  i20 : elapsedTime a1 = character A1
│ │ │ - -- .966261s elapsed
│ │ │ + -- .732098s elapsed
│ │ │  
│ │ │  o20 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {11}) => | 1 1 1 1 1 1 |
│ │ │        (2, {13}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o20 : Character
│ │ │  
│ │ │  i21 : elapsedTime a2 = character A2
│ │ │ - -- 33.2037s elapsed
│ │ │ + -- 26.984s elapsed
│ │ │  
│ │ │  o21 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ @@ -297,15 +297,15 @@
│ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │  
│ │ │  o31 = Module with 6 actors
│ │ │  
│ │ │  o31 : ActionOnGradedModule
│ │ │  
│ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ - -- 15.7832s elapsed
│ │ │ + -- 12.2335s elapsed
│ │ │  
│ │ │  o32 = Character over R
│ │ │         
│ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o32 : Character
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o8 : ActionOnComplex
    • 
│ │ │              
    
│ │ │                
    i9 : elapsedTime c = character A
│ │ │ - -- .437329s elapsed
│ │ │ + -- .375566s elapsed
│ │ │  
│ │ │  o9 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {2}) => | 0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14 |
│ │ │       (2, {3}) => | 0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35 |
│ │ │       (3, {4}) => | 0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │  o7 : List
│ │ │ │  i8 : A = action(RI,S7)
│ │ │ │  
│ │ │ │  o8 = Complex with 15 actors
│ │ │ │  
│ │ │ │  o8 : ActionOnComplex
│ │ │ │  i9 : elapsedTime c = character A
│ │ │ │ - -- .437329s elapsed
│ │ │ │ + -- .375566s elapsed
│ │ │ │  
│ │ │ │  o9 = Character over R
│ │ │ │  
│ │ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │ │       (1, {2}) => | 0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14 |
│ │ │ │       (2, {3}) => | 0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35 |
│ │ │ │       (3, {4}) => | 0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html
│ │ │ @@ -180,15 +180,15 @@
│ │ │  
│ │ │  o6 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime c=character A
│ │ │ - -- .461365s elapsed
│ │ │ + -- .443902s elapsed
│ │ │  
│ │ │  o7 = Character over R
│ │ │        
│ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │       (1, {5}) => | 0 1 0 2 0 1 3 0 2 4 6 |
│ │ │       (1, {7}) => | 0 0 0 0 0 1 3 0 4 16 60 |
│ │ │       (1, {9}) => | 0 0 0 0 2 2 2 0 4 8 20 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -113,15 +113,15 @@
│ │ │ │  o5 : List
│ │ │ │  i6 : A=action(RI,S6)
│ │ │ │  
│ │ │ │  o6 = Complex with 11 actors
│ │ │ │  
│ │ │ │  o6 : ActionOnComplex
│ │ │ │  i7 : elapsedTime c=character A
│ │ │ │ - -- .461365s elapsed
│ │ │ │ + -- .443902s elapsed
│ │ │ │  
│ │ │ │  o7 = Character over R
│ │ │ │  
│ │ │ │       (0, {0}) => | 1 1 1 1 1 1 1 1 1 1 1 |
│ │ │ │       (1, {5}) => | 0 1 0 2 0 1 3 0 2 4 6 |
│ │ │ │       (1, {7}) => | 0 0 0 0 0 1 3 0 4 16 60 |
│ │ │ │       (1, {9}) => | 0 0 0 0 2 2 2 0 4 8 20 |
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html
│ │ │ @@ -310,30 +310,30 @@
│ │ │  
│ │ │  o19 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime a1 = character A1
│ │ │ - -- .966261s elapsed
│ │ │ + -- .732098s elapsed
│ │ │  
│ │ │  o20 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {8}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {11}) => | 1 1 1 1 1 1 |
│ │ │        (2, {13}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o20 : Character
    
│ │ │              
    
│ │ │                
    i21 : elapsedTime a2 = character A2
│ │ │ - -- 33.2037s elapsed
│ │ │ + -- 26.984s elapsed
│ │ │  
│ │ │  o21 = Character over R
│ │ │         
│ │ │        (0, {0}) => | 1 1 1 1 1 1 |
│ │ │        (1, {16}) => | 6 2 0 0 -1 -1 |
│ │ │        (2, {19}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │        (2, {21}) => | 3 -1 0 1 a4+a2+a -a4-a2-a-1 |
│ │ │ @@ -467,15 +467,15 @@
│ │ │  
│ │ │  o31 : ActionOnGradedModule
    
│ │ │              
    
│ │ │                
    i32 : elapsedTime b = character(B,21)
│ │ │ - -- 15.7832s elapsed
│ │ │ + -- 12.2335s elapsed
│ │ │  
│ │ │  o32 = Character over R
│ │ │         
│ │ │        (0, {21}) => | 1 1 1 1 1 1 |
│ │ │  
│ │ │  o32 : Character
    
│ │ │              
    
│ │ │                
    i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.225031s (cpu); 0.172702s (thread); 0s (gc)
│ │ │ + -- used 0.312088s (cpu); 0.224665s (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 0.225031s (cpu); 0.172702s (thread); 0s (gc)
│ │ │ │ + -- used 0.312088s (cpu); 0.224665s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of S
│ │ │ │  i24 : betti res j
│ │ │ │  
│ │ │ │               0 1 2 3 4
│ │ │ │  o24 = total: 1 3 6 5 1
│ │ │ │            0: 1 . . . .
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_boundary.out
│ │ │ @@ -34,14 +34,14 @@
│ │ │  
│ │ │  i12 : f = (cells(2,C))#0;
│ │ │  
│ │ │  i13 : boundary(f)
│ │ │  
│ │ │  o13 = {(Cell of dimension 1 with label 1, 1), (Cell of dimension 1 with label
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │ +      1, -1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │        -----------------------------------------------------------------------
│ │ │ -      with label 1, -1)}
│ │ │ +      with label 1, 1)}
│ │ │  
│ │ │  o13 : List
│ │ │  
│ │ │  i14 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Polyhedral__Complex_rp.out
│ │ │ @@ -12,27 +12,27 @@
│ │ │  
│ │ │  i6 : F = polyhedralComplex {P1,P2,P3,P4};
│ │ │  
│ │ │  i7 : C = cellComplex(R,F);
│ │ │  
│ │ │  i8 : facePoset C
│ │ │  
│ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ -                      | 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ -                      | 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 |
│ │ │ +                      | 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 |
│ │ │ +                      | 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 |
│ │ │ +                      | 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o8 : Poset
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.out
│ │ │ @@ -24,15 +24,15 @@
│ │ │  
│ │ │  o7 : CellComplex
│ │ │  
│ │ │  i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │  
│ │ │                        5   4    3 2   2 3     4   5
│ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
│ │ │ -                      5    3 3   5 2   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4
│ │ │ -               1 => {x y, x y , x y , x y , x y , x y , x y, x y , x y , x y , x y , x y }
│ │ │ +                      2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4   5    3 3   5 2
│ │ │ +               1 => {x y , x y , x y , x y, x y , x y , x y , x y , x y , x y, x y , x y }
│ │ │                        5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │  
│ │ │  o8 : HashTable
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cells.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │  
│ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │  
│ │ │  i7 : cells(C)
│ │ │  
│ │ │ -o7 = HashTable{0 => {Cell of dimension 0 with label y, Cell of dimension 0 with label z, Cell of dimension 0 with label x}}
│ │ │ +o7 = HashTable{0 => {Cell of dimension 0 with label x, Cell of dimension 0 with label z, Cell of dimension 0 with label y}}
│ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │  
│ │ │  o7 : HashTable
│ │ │  
│ │ │  i8 : R = QQ;
│ │ │  
│ │ │  i9 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}};
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cells_lp__Z__Z_cm__Cell__Complex_rp.out
│ │ │ @@ -10,17 +10,17 @@
│ │ │  
│ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │  
│ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │  
│ │ │  i7 : cells(0,C)
│ │ │  
│ │ │ -o7 = {Cell of dimension 0 with label y, Cell of dimension 0 with label x,
│ │ │ +o7 = {Cell of dimension 0 with label z, Cell of dimension 0 with label x,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Cell of dimension 0 with label z}
│ │ │ +     Cell of dimension 0 with label y}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : cells(1,C)
│ │ │  
│ │ │  o8 = {Cell of dimension 1 with label x*y}
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out
│ │ │ @@ -29,13 +29,13 @@
│ │ │          2      2   2
│ │ │  o13 = {a b, b*c , b , a*c}
│ │ │  
│ │ │  o13 : List
│ │ │  
│ │ │  i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -        2   2   2 2   2 2       2     2
│ │ │ -o14 = {a b*c , a b , b c , a*b*c , a*b c}
│ │ │ +        2 2   2   2     2    2 2       2
│ │ │ +o14 = {b c , a b*c , a*b c, a b , a*b*c }
│ │ │  
│ │ │  o14 : List
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_boundary.html
│ │ │ @@ -145,17 +145,17 @@
│ │ │            
    
│ │ │                
    i13 : boundary(f)
│ │ │  
│ │ │  o13 = {(Cell of dimension 1 with label 1, 1), (Cell of dimension 1 with label
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │ +      1, -1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │        -----------------------------------------------------------------------
│ │ │ -      with label 1, -1)}
│ │ │ +      with label 1, 1)}
│ │ │  
│ │ │  o13 : List
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,17 +42,17 @@
│ │ │ │  i10 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}};
│ │ │ │  i11 : C = cellComplex(R,P);
│ │ │ │  i12 : f = (cells(2,C))#0;
│ │ │ │  i13 : boundary(f)
│ │ │ │  
│ │ │ │  o13 = {(Cell of dimension 1 with label 1, 1), (Cell of dimension 1 with label
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      1, 1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │ │ +      1, -1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      with label 1, -1)}
│ │ │ │ +      with label 1, 1)}
│ │ │ │  
│ │ │ │  o13 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _b_o_u_n_d_a_r_y_C_e_l_l_s_(_C_e_l_l_) -- returns the boundary cells of the given cell
│ │ │ │  ********** WWaayyss ttoo uussee bboouunnddaarryy:: **********
│ │ │ │      * boundary(Cell)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Polyhedral__Complex_rp.html
│ │ │ @@ -112,27 +112,27 @@
│ │ │                
    i7 : C = cellComplex(R,F);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : facePoset C
│ │ │  
│ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ -                      | 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ -                      | 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 |
│ │ │ +                      | 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 |
│ │ │ +                      | 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 |
│ │ │ +                      | 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o8 : Poset
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,27 +26,27 @@
│ │ │ │  i3 : P2 = convexHull matrix {{2,-2,0},{1,1,0}};
│ │ │ │  i4 : P3 = convexHull matrix {{-2,-2,0},{1,-1,0}};
│ │ │ │  i5 : P4 = convexHull matrix {{-2,2,0},{-1,-1,0}};
│ │ │ │  i6 : F = polyhedralComplex {P1,P2,P3,P4};
│ │ │ │  i7 : C = cellComplex(R,F);
│ │ │ │  i8 : facePoset C
│ │ │ │  
│ │ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ │ -                      | 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ │ -                      | 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ │ -                      | 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ │ -                      | 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 |
│ │ │ │ +                      | 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 |
│ │ │ │ +                      | 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 |
│ │ │ │ +                      | 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 |
│ │ │ │ +                      | 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1 |
│ │ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │ │  
│ │ │ │  o8 : Poset
│ │ │ │  The labels on the vertices can be controlled via the optional parameter Labels
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.html
│ │ │ @@ -129,16 +129,16 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │  
│ │ │                        5   4    3 2   2 3     4   5
│ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
│ │ │ -                      5    3 3   5 2   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4
│ │ │ -               1 => {x y, x y , x y , x y , x y , x y , x y, x y , x y , x y , x y , x y }
│ │ │ +                      2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4   5    3 3   5 2
│ │ │ +               1 => {x y , x y , x y , x y, x y , x y , x y , x y , x y , x y, x y , x y }
│ │ │                        5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │  
│ │ │  o8 : HashTable
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,17 +41,17 @@
│ │ │ │  
│ │ │ │  o7 : CellComplex
│ │ │ │  i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │ │  
│ │ │ │                        5   4    3 2   2 3     4   5
│ │ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }
│ │ │ │  }
│ │ │ │ -                      5    3 3   5 2   2 4   5 3   5 4   5    5 2   5 3   5 4
│ │ │ │ -4 2   4 4
│ │ │ │ -               1 => {x y, x y , x y , x y , x y , x y , x y, x y , x y , x y ,
│ │ │ │ +                      2 4   5 3   5 4   5    5 2   5 3   5 4   4 2   4 4   5
│ │ │ │ +3 3   5 2
│ │ │ │ +               1 => {x y , x y , x y , x y, x y , x y , x y , x y , x y , x y,
│ │ │ │  x y , x y }
│ │ │ │                        5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │ │  
│ │ │ │  o8 : HashTable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_e_l_l_C_o_m_p_l_e_x -- create a cell complex
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cells.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │                
    i6 : C = cellComplex(R,{exy,vz});
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : cells(C)
│ │ │  
│ │ │ -o7 = HashTable{0 => {Cell of dimension 0 with label y, Cell of dimension 0 with label z, Cell of dimension 0 with label x}}
│ │ │ +o7 = HashTable{0 => {Cell of dimension 0 with label x, Cell of dimension 0 with label z, Cell of dimension 0 with label y}}
│ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │  
│ │ │  o7 : HashTable
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,16 +19,16 @@
│ │ │ │  i2 : vx = newSimplexCell({},x);
│ │ │ │  i3 : vy = newSimplexCell({},y);
│ │ │ │  i4 : vz = newSimplexCell({},z);
│ │ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │ │  i7 : cells(C)
│ │ │ │  
│ │ │ │ -o7 = HashTable{0 => {Cell of dimension 0 with label y, Cell of dimension 0 with
│ │ │ │ -label z, Cell of dimension 0 with label x}}
│ │ │ │ +o7 = HashTable{0 => {Cell of dimension 0 with label x, Cell of dimension 0 with
│ │ │ │ +label z, Cell of dimension 0 with label y}}
│ │ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │ │  
│ │ │ │  o7 : HashTable
│ │ │ │  i8 : R = QQ;
│ │ │ │  i9 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}};
│ │ │ │  i10 : C = cellComplex(R,P);
│ │ │ │  i11 : cells C
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cells_lp__Z__Z_cm__Cell__Complex_rp.html
│ │ │ @@ -103,17 +103,17 @@
│ │ │                
    i6 : C = cellComplex(R,{exy,vz});
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : cells(0,C)
│ │ │  
│ │ │ -o7 = {Cell of dimension 0 with label y, Cell of dimension 0 with label x,
│ │ │ +o7 = {Cell of dimension 0 with label z, Cell of dimension 0 with label x,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Cell of dimension 0 with label z}
│ │ │ +     Cell of dimension 0 with label y}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : cells(1,C)
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,17 +19,17 @@
│ │ │ │  i2 : vx = newSimplexCell({},x);
│ │ │ │  i3 : vy = newSimplexCell({},y);
│ │ │ │  i4 : vz = newSimplexCell({},z);
│ │ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │ │  i7 : cells(0,C)
│ │ │ │  
│ │ │ │ -o7 = {Cell of dimension 0 with label y, Cell of dimension 0 with label x,
│ │ │ │ +o7 = {Cell of dimension 0 with label z, Cell of dimension 0 with label x,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Cell of dimension 0 with label z}
│ │ │ │ +     Cell of dimension 0 with label y}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : cells(1,C)
│ │ │ │  
│ │ │ │  o8 = {Cell of dimension 1 with label x*y}
│ │ │ │  
│ │ │ │  o8 : List
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html
│ │ │ @@ -146,16 +146,16 @@
│ │ │  o13 : List
    
│ │ │              
    
│ │ │                
    i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -        2   2   2 2   2 2       2     2
│ │ │ -o14 = {a b*c , a b , b c , a*b*c , a*b c}
│ │ │ +        2 2   2   2     2    2 2       2
│ │ │ +o14 = {b c , a b*c , a*b c, a b , a*b*c }
│ │ │  
│ │ │  o14 : List
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,16 +37,16 @@
│ │ │ │  
│ │ │ │          2      2   2
│ │ │ │  o13 = {a b, b*c , b , a*c}
│ │ │ │  
│ │ │ │  o13 : List
│ │ │ │  i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │ │  
│ │ │ │ -        2   2   2 2   2 2       2     2
│ │ │ │ -o14 = {a b*c , a b , b c , a*b*c , a*b c}
│ │ │ │ +        2 2   2   2     2    2 2       2
│ │ │ │ +o14 = {b c , a b*c , a*b c, a b , a*b*c }
│ │ │ │  
│ │ │ │  o14 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_e_l_l_L_a_b_e_l -- return the label of a cell
│ │ │ │      * _R_i_n_g_M_a_p_ _*_*_ _C_e_l_l_C_o_m_p_l_e_x -- tensors labels via a ring map
│ │ │ │  ********** WWaayyss ttoo uussee rreellaabbeellCCeellllCCoommpplleexx:: **********
│ │ │ │      * relabelCellComplex(CellComplex,HashTable)
│ │ ├── ./usr/share/doc/Macaulay2/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 0.243853s (cpu); 0.199615s (thread); 0s (gc)
│ │ │ + -- used 0.485969s (cpu); 0.378305s (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 0.0970818s (cpu); 0.0970811s (thread); 0s (gc)
│ │ │ + -- used 0.157359s (cpu); 0.157356s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.211561s (cpu); 0.14771s (thread); 0s (gc)
│ │ │ + -- used 0.286229s (cpu); 0.190562s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : betti source m
│ │ │  
│ │ │               0  1  2   3   4   5   6   7
│ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │            0: 1  3  3   1   .   .   .   .
│ │ │            1: .  .  1   3   3   .   .   .
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │          
    
│ │ │            
    Now we minimize the result. The free summand we added to the end maps to zero, and thus is part of the minimization.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -301,15 +301,15 @@
│ │ │  
│ │ │  o21 : A
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -160,15 +160,15 @@
│ │ │ │  
│ │ │ │                2              2
│ │ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │ │                0 3    1 2 4   2 5
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : time csmK=CSM(A,K)
│ │ │ │ - -- used 0.501371s (cpu); 0.332722s (thread); 0s (gc)
│ │ │ │ + -- used 1.3148s (cpu); 0.423824s (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 0.0577868s (cpu); 0.056827s (thread); 0s (gc)
│ │ │ │ + -- used 0.110792s (cpu); 0.0787295s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2     2         2    2            2
│ │ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │ │  
│ │ │ │  o22 : A
│ │ │ │  In the case where the ambient space is a toric variety which is not a product
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.243853s (cpu); 0.199615s (thread); 0s (gc)
    
│ │ │ + -- used 0.485969s (cpu); 0.378305s (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 0.243853s (cpu); 0.199615s (thread); 0s (gc)
│ │ │ │ + -- used 0.485969s (cpu); 0.378305s (thread); 0s (gc)
│ │ │ │  i14 : isQuasiIsomorphism m
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : E[1] == source m
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  i16 : E' = target m
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html
│ │ │ @@ -129,21 +129,21 @@
│ │ │                  		
│ │ │                
    i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-1),mods_i,substitute(matrix C.dd_i,S));
    
│ │ │              
    
│ │ │                
    i8 : time m = resolutionOfChainComplex C;
│ │ │ - -- used 0.0970818s (cpu); 0.0970811s (thread); 0s (gc)
    
│ │ │ + -- used 0.157359s (cpu); 0.157356s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.211561s (cpu); 0.14771s (thread); 0s (gc)
    
│ │ │ + -- used 0.286229s (cpu); 0.190562s (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 0.0970818s (cpu); 0.0970811s (thread); 0s (gc)
│ │ │ │ + -- used 0.157359s (cpu); 0.157356s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.211561s (cpu); 0.14771s (thread); 0s (gc)
│ │ │ │ + -- used 0.286229s (cpu); 0.190562s (thread); 0s (gc)
│ │ │ │  i10 : betti source m
│ │ │ │  
│ │ │ │               0  1  2   3   4   5   6   7
│ │ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │ │            0: 1  3  3   1   .   .   .   .
│ │ │ │            1: .  .  1   3   3   .   .   .
│ │ │ │            2: .  1  3   3   2   .   .   .
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out
│ │ │ @@ -83,15 +83,15 @@
│ │ │                2              2
│ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │                0 3    1 2 4   2 5
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.501371s (cpu); 0.332722s (thread); 0s (gc)
│ │ │ + -- used 1.3148s (cpu); 0.423824s (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 0.0577868s (cpu); 0.056827s (thread); 0s (gc)
│ │ │ + -- used 0.110792s (cpu); 0.0787295s (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 0.302401s (cpu); 0.222388s (thread); 0s (gc)
│ │ │ + -- used 0.323885s (cpu); 0.207539s (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 0.359803s (cpu); 0.286202s (thread); 0s (gc)
│ │ │ + -- used 0.537862s (cpu); 0.415074s (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 0.367045s (cpu); 0.279357s (thread); 0s (gc)
│ │ │ + -- used 0.852629s (cpu); 0.429037s (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 2.34563s (cpu); 1.98719s (thread); 0s (gc)
│ │ │ + -- used 2.55234s (cpu); 2.18873s (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 0.278694s (cpu); 0.220482s (thread); 0s (gc)
│ │ │ + -- used 0.362214s (cpu); 0.242344s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
│ │ │  
│ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0857724s (cpu); 0.0857799s (thread); 0s (gc)
│ │ │ + -- used 0.116896s (cpu); 0.116907s (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 0.0395215s (cpu); 0.0374236s (thread); 0s (gc)
│ │ │ + -- used 0.0908457s (cpu); 0.0525768s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.206093s (cpu); 0.142389s (thread); 0s (gc)
│ │ │ + -- used 0.362502s (cpu); 0.198863s (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 0.079323s (cpu); 0.0753477s (thread); 0s (gc)
│ │ │ + -- used 0.20781s (cpu); 0.112184s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.167518s (cpu); 0.0900969s (thread); 0s (gc)
│ │ │ + -- used 0.273319s (cpu); 0.112632s (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 0.0522926s (cpu); 0.0517551s (thread); 0s (gc)
│ │ │ + -- used 0.0775611s (cpu); 0.0664675s (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 1.56431s (cpu); 1.21073s (thread); 0s (gc)
│ │ │ + -- used 5.74094s (cpu); 1.51588s (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 1.75651s (cpu); 1.34853s (thread); 0s (gc)
│ │ │ + -- used 6.08413s (cpu); 1.51657s (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 0.522973s (cpu); 0.393972s (thread); 0s (gc)
│ │ │ + -- used 1.20106s (cpu); 0.563452s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0326577s (cpu); 0.032314s (thread); 0s (gc)
│ │ │ + -- used 0.110343s (cpu); 0.0507232s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o4 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i5 : time Chern I
│ │ │ - -- used 0.0323977s (cpu); 0.0319015s (thread); 0s (gc)
│ │ │ + -- used 0.0950639s (cpu); 0.0438652s (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 1.14505s (cpu); 0.889383s (thread); 0s (gc)
│ │ │ + -- used 3.25235s (cpu); 1.27754s (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 0.288446s (cpu); 0.224228s (thread); 0s (gc)
│ │ │ + -- used 0.693426s (cpu); 0.291132s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html
│ │ │ @@ -234,15 +234,15 @@
│ │ │  
│ │ │  o14 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.501371s (cpu); 0.332722s (thread); 0s (gc)
│ │ │ + -- used 1.3148s (cpu); 0.423824s (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 0.0577868s (cpu); 0.056827s (thread); 0s (gc)
│ │ │ + -- used 0.110792s (cpu); 0.0787295s (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 0.302401s (cpu); 0.222388s (thread); 0s (gc)
│ │ │ + -- used 0.323885s (cpu); 0.207539s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ @@ -88,15 +88,15 @@
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
    
│ │ │              
    
│ │ │                
    i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.359803s (cpu); 0.286202s (thread); 0s (gc)
│ │ │ + -- used 0.537862s (cpu); 0.415074s (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 0.302401s (cpu); 0.222388s (thread); 0s (gc)
│ │ │ │ + -- used 0.323885s (cpu); 0.207539s (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 0.359803s (cpu); 0.286202s (thread); 0s (gc)
│ │ │ │ + -- used 0.537862s (cpu); 0.415074s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.367045s (cpu); 0.279357s (thread); 0s (gc)
│ │ │ + -- used 0.852629s (cpu); 0.429037s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -109,15 +109,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.34563s (cpu); 1.98719s (thread); 0s (gc)
│ │ │ + -- used 2.55234s (cpu); 2.18873s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o9 : Ideal of S
    
│ │ │              
    
│ │ │                
    i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.278694s (cpu); 0.220482s (thread); 0s (gc)
│ │ │ + -- used 0.362214s (cpu); 0.242344s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │ @@ -159,15 +159,15 @@
│ │ │          h
│ │ │           1
    
│ │ │              
    
│ │ │                
    i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0857724s (cpu); 0.0857799s (thread); 0s (gc)
│ │ │ + -- used 0.116896s (cpu); 0.116907s (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 0.367045s (cpu); 0.279357s (thread); 0s (gc)
│ │ │ │ + -- used 0.852629s (cpu); 0.429037s (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 2.34563s (cpu); 1.98719s (thread); 0s (gc)
│ │ │ │ + -- used 2.55234s (cpu); 2.18873s (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 0.278694s (cpu); 0.220482s (thread); 0s (gc)
│ │ │ │ + -- used 0.362214s (cpu); 0.242344s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o10 = 3h  + 5h
│ │ │ │          1     1
│ │ │ │  
│ │ │ │        ZZ[h ]
│ │ │ │            1
│ │ │ │  o10 : ------
│ │ │ │           4
│ │ │ │          h
│ │ │ │           1
│ │ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ │ - -- used 0.0857724s (cpu); 0.0857799s (thread); 0s (gc)
│ │ │ │ + -- used 0.116896s (cpu); 0.116907s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o11 = 3H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html
│ │ │ @@ -125,23 +125,23 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0395215s (cpu); 0.0374236s (thread); 0s (gc)
│ │ │ + -- used 0.0908457s (cpu); 0.0525768s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : time Euler I
│ │ │ - -- used 0.206093s (cpu); 0.142389s (thread); 0s (gc)
│ │ │ + -- used 0.362502s (cpu); 0.198863s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
    
│ │ │              
    
│ │ │                
    i6 : EulerIHash=Euler(I,Output=>HashForm);
    
│ │ │ @@ -189,23 +189,23 @@
│ │ │          
    
│ │ │            
    Note that the ideal J above is a complete intersection, thus we may change the method option which may speed computation in some cases. We may also note that the ideal generated by the first 2 generators of I defines a smooth scheme and input this information into the method. This may also improve computation speed.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.079323s (cpu); 0.0753477s (thread); 0s (gc)
│ │ │ + -- used 0.20781s (cpu); 0.112184s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    
│ │ │              
    
│ │ │                
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.167518s (cpu); 0.0900969s (thread); 0s (gc)
│ │ │ + -- used 0.273319s (cpu); 0.112632s (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 0.0395215s (cpu); 0.0374236s (thread); 0s (gc)
│ │ │ │ + -- used 0.0908457s (cpu); 0.0525768s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : time Euler I
│ │ │ │ - -- used 0.206093s (cpu); 0.142389s (thread); 0s (gc)
│ │ │ │ + -- used 0.362502s (cpu); 0.198863s (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 0.079323s (cpu); 0.0753477s (thread); 0s (gc)
│ │ │ │ + -- used 0.20781s (cpu); 0.112184s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 2
│ │ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ │ - -- used 0.167518s (cpu); 0.0900969s (thread); 0s (gc)
│ │ │ │ + -- used 0.273319s (cpu); 0.112632s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = 2
│ │ │ │  Now consider an example in \PP^2 \times \PP^2.
│ │ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │ │  
│ │ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │              
    
│ │ │                
    i4 : time EulerAffine I
│ │ │ - -- used 0.0522926s (cpu); 0.0517551s (thread); 0s (gc)
│ │ │ + -- used 0.0775611s (cpu); 0.0664675s (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 0.0522926s (cpu); 0.0517551s (thread); 0s (gc)
│ │ │ │ + -- used 0.0775611s (cpu); 0.0664675s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  Observe that the algorithm is a probabilistic algorithm and may give a wrong
│ │ │ │  answer with a small but nonzero probability. Read more under _p_r_o_b_a_b_i_l_i_s_t_i_c
│ │ │ │  _a_l_g_o_r_i_t_h_m.
│ │ │ │  ********** WWaayyss ttoo uussee EEuulleerrAAffffiinnee:: **********
│ │ │ │      * EulerAffine(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 1.56431s (cpu); 1.21073s (thread); 0s (gc)
│ │ │ + -- used 5.74094s (cpu); 1.51588s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ @@ -87,15 +87,15 @@
│ │ │        (h , h )
│ │ │          1   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 1.75651s (cpu); 1.34853s (thread); 0s (gc)
│ │ │ + -- used 6.08413s (cpu); 1.51657s (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 1.56431s (cpu); 1.21073s (thread); 0s (gc)
│ │ │ │ + -- used 5.74094s (cpu); 1.51588s (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 1.75651s (cpu); 1.34853s (thread); 0s (gc)
│ │ │ │ + -- used 6.08413s (cpu); 1.51657s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html
│ │ │ @@ -66,15 +66,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM I
│ │ │ - -- used 0.522973s (cpu); 0.393972s (thread); 0s (gc)
│ │ │ + -- used 1.20106s (cpu); 0.563452s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -83,15 +83,15 @@
│ │ │         h
│ │ │          1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0326577s (cpu); 0.032314s (thread); 0s (gc)
│ │ │ + -- used 0.110343s (cpu); 0.0507232s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -105,15 +105,15 @@
│ │ │          
    
│ │ │            
    Note that one could, equivalently, use the command Chern instead in this case.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -433,26 +433,26 @@
│ │ │  
│ │ │  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)
│ │ │ │ - -- 3.43714s elapsed
│ │ │ │ + -- 3.27568s 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)
│ │ │ │ - -- .333914s elapsed
│ │ │ │ + -- .537631s 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))
│ │ │ │ - -- 11.9253s elapsed
│ │ │ │ + -- 10.4332s 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)
│ │ │ │ - -- .373602s elapsed
│ │ │ │ + -- .521561s 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))
│ │ │ │ - -- .592041s elapsed
│ │ │ │ - -- .592074s elapsed
│ │ │ │ + -- .507702s elapsed
│ │ │ │ + -- .507734s elapsed
│ │ │ │  
│ │ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o28 : List
│ │ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ │ │  _c_o_h_o_m_C_a_l_g computes cohomology vectors by calling CohomCalg. It also stashes
│ │ │ │  it's results in the toric variety's cache table, so computations need not be
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : len = 10
│ │ │  
│ │ │  o6 = 10
│ │ │  
│ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.79661s (cpu); 5.23664s (thread); 0s (gc)
│ │ │ + -- used 9.49261s (cpu); 6.56571s (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 0.0719777s (cpu); 0.0709228s (thread); 0s (gc)
│ │ │ + -- used 0.266277s (cpu); 0.154776s (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 0.968442s (cpu); 0.740402s (thread); 0s (gc)
│ │ │ + -- used 1.40843s (cpu); 1.0806s (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 0.920761s (cpu); 0.729766s (thread); 0s (gc)
│ │ │ + -- used 1.30916s (cpu); 0.999135s (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 0.371964s (cpu); 0.318637s (thread); 0s (gc)
│ │ │ + -- used 0.603772s (cpu); 0.427245s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.158094s (cpu); 0.11446s (thread); 0s (gc)
│ │ │ + -- used 0.334544s (cpu); 0.169248s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 3.576e-06s (cpu); 1.5809e-05s (thread); 0s (gc)
│ │ │ + -- used 6.376e-06s (cpu); 3.941e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out
│ │ │ @@ -2,23 +2,23 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : twoMonomials(2,3)
│ │ │ - -- used 0.791017s (cpu); 0.599646s (thread); 0s (gc)
│ │ │ + -- used 1.3338s (cpu); 0.768908s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.413458s (cpu); 0.366394s (thread); 0s (gc)
│ │ │ + -- used 0.687455s (cpu); 0.446493s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.19005s (cpu); 0.139974s (thread); 0s (gc)
│ │ │ + -- used 0.330523s (cpu); 0.172093s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o6 = 10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time Chern I
│ │ │ - -- used 0.0323977s (cpu); 0.0319015s (thread); 0s (gc)
│ │ │ + -- used 0.0950639s (cpu); 0.0438652s (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 0.522973s (cpu); 0.393972s (thread); 0s (gc)
│ │ │ │ + -- used 1.20106s (cpu); 0.563452s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o3 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.0326577s (cpu); 0.032314s (thread); 0s (gc)
│ │ │ │ + -- used 0.110343s (cpu); 0.0507232s (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 0.0323977s (cpu); 0.0319015s (thread); 0s (gc)
│ │ │ │ + -- used 0.0950639s (cpu); 0.0438652s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o5 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 1.14505s (cpu); 0.889383s (thread); 0s (gc)
│ │ │ + -- used 3.25235s (cpu); 1.27754s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -87,15 +87,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.288446s (cpu); 0.224228s (thread); 0s (gc)
│ │ │ + -- used 0.693426s (cpu); 0.291132s (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 1.14505s (cpu); 0.889383s (thread); 0s (gc)
│ │ │ │ + -- used 3.25235s (cpu); 1.27754s (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 0.288446s (cpu); 0.224228s (thread); 0s (gc)
│ │ │ │ + -- used 0.693426s (cpu); 0.291132s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o4 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out
│ │ │ @@ -184,15 +184,15 @@
│ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, -1, -1}}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.43714s elapsed
│ │ │ + -- 3.27568s 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)
│ │ │ - -- .333914s elapsed
│ │ │ + -- .537631s 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))
│ │ │ - -- 11.9253s elapsed
│ │ │ + -- 10.4332s 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)
│ │ │ - -- .373602s elapsed
│ │ │ + -- .521561s 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))
│ │ │ - -- .592041s elapsed
│ │ │ - -- .592074s elapsed
│ │ │ + -- .507702s elapsed
│ │ │ + -- .507734s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
│ │ │  
│ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/html/index.html
│ │ │ @@ -309,15 +309,15 @@
│ │ │  
│ │ │  o19 : List
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.43714s elapsed
│ │ │ + -- 3.27568s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -399,25 +399,25 @@
│ │ │  
│ │ │  o22 : List
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .333914s elapsed
│ │ │ + -- .537631s 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))
│ │ │ - -- 11.9253s elapsed
│ │ │ + -- 10.4332s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .373602s elapsed
│ │ │ + -- .521561s 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))
│ │ │ - -- .592041s elapsed
│ │ │ - -- .592074s elapsed
│ │ │ + -- .507702s elapsed
│ │ │ + -- .507734s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
    
│ │ │              
    
│ │ │                
    i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.79661s (cpu); 5.23664s (thread); 0s (gc)
│ │ │ + -- used 9.49261s (cpu); 6.56571s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -295,28 +295,28 @@
│ │ │  
│ │ │  o19 : QuotientRing
    
│ │ │              
    
│ │ │                
    i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.0719777s (cpu); 0.0709228s (thread); 0s (gc)
│ │ │ + -- used 0.266277s (cpu); 0.154776s (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 0.968442s (cpu); 0.740402s (thread); 0s (gc)
│ │ │ + -- used 1.40843s (cpu); 1.0806s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │ @@ -328,15 +328,15 @@
│ │ │          
    
│ │ │            
    The function also deals correctly with complexes F where min F is not 0:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.920761s (cpu); 0.729766s (thread); 0s (gc)
│ │ │ + -- used 1.30916s (cpu); 0.999135s (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 6.79661s (cpu); 5.23664s (thread); 0s (gc)
│ │ │ │ + -- used 9.49261s (cpu); 6.56571s (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 0.0719777s (cpu); 0.0709228s (thread); 0s (gc)
│ │ │ │ + -- used 0.266277s (cpu); 0.154776s (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 0.968442s (cpu); 0.740402s (thread); 0s (gc)
│ │ │ │ + -- used 1.40843s (cpu); 1.0806s (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 0.920761s (cpu); 0.729766s (thread); 0s (gc)
│ │ │ │ + -- used 1.30916s (cpu); 0.999135s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        -2      -1      0        1        2
│ │ │ │  
│ │ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html
│ │ │ @@ -79,23 +79,23 @@
│ │ │  
│ │ │  o1 = 0
│ │ │              
    
│ │ │                
    i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.371964s (cpu); 0.318637s (thread); 0s (gc)
│ │ │ + -- used 0.603772s (cpu); 0.427245s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.158094s (cpu); 0.11446s (thread); 0s (gc)
│ │ │ + -- used 0.334544s (cpu); 0.169248s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 3.576e-06s (cpu); 1.5809e-05s (thread); 0s (gc)
│ │ │ + -- used 6.376e-06s (cpu); 3.941e-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 0.371964s (cpu); 0.318637s (thread); 0s (gc)
│ │ │ │ + -- used 0.603772s (cpu); 0.427245s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │ │  
│ │ │ │ - -- used 0.158094s (cpu); 0.11446s (thread); 0s (gc)
│ │ │ │ + -- used 0.334544s (cpu); 0.169248s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  
│ │ │ │ - -- used 3.576e-06s (cpu); 1.5809e-05s (thread); 0s (gc)
│ │ │ │ + -- used 6.376e-06s (cpu); 3.941e-06s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ssuummTTwwooMMoonnoommiiaallss:: **********
│ │ │ │      * sumTwoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html
│ │ │ @@ -83,25 +83,25 @@
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │                
    i2 : twoMonomials(2,3)
│ │ │ - -- used 0.791017s (cpu); 0.599646s (thread); 0s (gc)
│ │ │ + -- used 1.3338s (cpu); 0.768908s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.413458s (cpu); 0.366394s (thread); 0s (gc)
│ │ │ + -- used 0.687455s (cpu); 0.446493s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.19005s (cpu); 0.139974s (thread); 0s (gc)
│ │ │ + -- used 0.330523s (cpu); 0.172093s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,25 +20,25 @@
│ │ │ │  that is, for an appropriate syzygy M of M0 = R/(m1, m2) where m1 and m2 are
│ │ │ │  monomials of the same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │   -- setting random seed to 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : twoMonomials(2,3)
│ │ │ │ - -- used 0.791017s (cpu); 0.599646s (thread); 0s (gc)
│ │ │ │ + -- used 1.3338s (cpu); 0.768908s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{1, 1}} => 2        }
│ │ │ │        {{2, 2}, {1, 2}} => 4
│ │ │ │  
│ │ │ │ - -- used 0.413458s (cpu); 0.366394s (thread); 0s (gc)
│ │ │ │ + -- used 0.687455s (cpu); 0.446493s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │ │        {{3, 3}, {2, 3}} => 1
│ │ │ │  
│ │ │ │ - -- used 0.19005s (cpu); 0.139974s (thread); 0s (gc)
│ │ │ │ + -- used 0.330523s (cpu); 0.172093s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ttwwooMMoonnoommiiaallss:: **********
│ │ │ │      * twoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out
│ │ │ @@ -27,18 +27,18 @@
│ │ │       - ϵ*z*dy + 2ϵ  - ϵ, x*dx + y*dy + z*dz - 2ϵ)
│ │ │  
│ │ │  o7 : Ideal of D
│ │ │  
│ │ │  i8 : assert(holonomicRank I == 4)
│ │ │  
│ │ │  i9 : elapsedTime A = connectionMatrices I;
│ │ │ - -- 2.8198s elapsed
│ │ │ + -- 2.48499s elapsed
│ │ │  
│ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.998s elapsed
│ │ │ + -- 4.33407s 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);
│ │ │ - -- .69513s elapsed
│ │ │ + -- .51843s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
│ │ │  
│ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.47197s elapsed
│ │ │ + -- 1.17717s elapsed
│ │ │  
│ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .974659s elapsed
│ │ │ + -- .968332s 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);
│ │ │ - -- .460552s elapsed
│ │ │ + -- .348487s elapsed
│ │ │  
│ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- .79831s elapsed
│ │ │ + -- .740042s elapsed
│ │ │  
│ │ │  i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |                                     |
│ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |                                     |
│ │ │        || 0       0           ϵ/(x+z) (-yϵ+zϵ)/(x2+xy+xz+yz) |                                     |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out
│ │ │ @@ -16,18 +16,18 @@
│ │ │  
│ │ │                     2
│ │ │  o6 = {1, dx, dy, dy }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime A = connectionMatrices I;
│ │ │ - -- .189937s elapsed
│ │ │ + -- .207219s elapsed
│ │ │  
│ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .152326s elapsed
│ │ │ + -- .182903s elapsed
│ │ │  
│ │ │  i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o9 = || 0                                                       1                                               0                                                                     0                                                                      ||
│ │ │       || 0                                                       -1/x                                            1/x                                                                   y/x                                                                    ||
│ │ │       || -1/2xy                                                  -1/y                                            (-x-3y+1)/2xy                                                         (-x-y+1)/2x                                                            ||
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html
│ │ │ @@ -118,21 +118,21 @@
│ │ │          
    
│ │ │            
    Then, we compute the system in connection form and verify that it meets the integrability conditions.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,15 +82,15 @@
│ │ │ │                2                 2
│ │ │ │       - a*c + e         - b*c + f
│ │ │ │       ----------*v, x + ----------*v)
│ │ │ │        d*e - a*f         d*e - a*f
│ │ │ │  
│ │ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │ │  i6 : time phi^** q
│ │ │ │ - -- used 0.158186s (cpu); 0.158148s (thread); 0s (gc)
│ │ │ │ + -- used 0.182363s (cpu); 0.182363s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                  e        d        c        b        a
│ │ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │ │                  f        f        f        f        f
│ │ │ │  
│ │ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │ │  i7 : oo == p
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html
│ │ │ @@ -134,59 +134,59 @@
│ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1 6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
    
│ │ │                
    i9 : elapsedTime A = connectionMatrices I;
│ │ │ - -- 2.8198s elapsed
    
│ │ │ + -- 2.48499s elapsed
│ │ │              
    
│ │ │                
    i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.998s elapsed
    
│ │ │ + -- 4.33407s elapsed
│ │ │              
    
│ │ │                
    i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -167,30 +167,30 @@
│ │ │              
    		
│ │ │                
    i13 : B = {1_D,dx,dy,dx*dy};
    
│ │ │              
    
│ │ │                
    i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .69513s elapsed
│ │ │ + -- .51843s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.47197s elapsed
    
│ │ │ + -- 1.17717s elapsed
│ │ │              
    
│ │ │                
    i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .974659s elapsed
    
│ │ │ + -- .968332s elapsed
│ │ │              
    
│ │ │                
    i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -227,21 +227,21 @@
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .460552s elapsed
    
│ │ │ + -- .348487s elapsed
│ │ │              
    
│ │ │                
    i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- .79831s elapsed
    
│ │ │ + -- .740042s elapsed
│ │ │              
    
│ │ │                
    i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,17 +41,17 @@
│ │ │ │  
│ │ │ │  o7 : Ideal of D
│ │ │ │  First, we check that the system has finite holonomic rank using _h_o_l_o_n_o_m_i_c_R_a_n_k.
│ │ │ │  i8 : assert(holonomicRank I == 4)
│ │ │ │  Then, we compute the system in connection form and verify that it meets the
│ │ │ │  integrability conditions.
│ │ │ │  i9 : elapsedTime A = connectionMatrices I;
│ │ │ │ - -- 2.8198s elapsed
│ │ │ │ + -- 2.48499s elapsed
│ │ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ │ - -- 5.998s elapsed
│ │ │ │ + -- 4.33407s 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);
│ │ │ │ - -- .69513s elapsed
│ │ │ │ + -- .51843s elapsed
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o14 : Matrix F  <-- F
│ │ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ │ - -- 1.47197s elapsed
│ │ │ │ + -- 1.17717s elapsed
│ │ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ │ - -- .974659s elapsed
│ │ │ │ + -- .968332s 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);
│ │ │ │ - -- .460552s elapsed
│ │ │ │ + -- .348487s elapsed
│ │ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ │ - -- .79831s elapsed
│ │ │ │ + -- .740042s elapsed
│ │ │ │  i21 : netList A2
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------+
│ │ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |
│ │ │ │  |
│ │ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html
│ │ │ @@ -100,21 +100,21 @@
│ │ │          
    
│ │ │            
    Finally, we can compute the connection matrices.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : elapsedTime A = connectionMatrices I;
│ │ │ - -- .189937s elapsed
    
│ │ │ + -- .207219s elapsed
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .152326s elapsed
    
│ │ │ + -- .182903s elapsed
│ │ │              
    
│ │ │                
    i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,17 +20,17 @@
│ │ │ │  
│ │ │ │                     2
│ │ │ │  o6 = {1, dx, dy, dy }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  Finally, we can compute the connection matrices.
│ │ │ │  i7 : elapsedTime A = connectionMatrices I;
│ │ │ │ - -- .189937s elapsed
│ │ │ │ + -- .207219s elapsed
│ │ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ │ - -- .152326s elapsed
│ │ │ │ + -- .182903s elapsed
│ │ │ │  i9 : netList A
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------+
│ │ │ │  o9 = || 0                                                       1
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out
│ │ │ @@ -13,27 +13,27 @@
│ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
│ │ │  
│ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 2.13317s (cpu); 1.1571s (thread); 0s (gc)
│ │ │ + -- used 2.95535s (cpu); 1.42315s (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 1.28884s (cpu); 0.908882s (thread); 0s (gc)
│ │ │ + -- used 1.78324s (cpu); 1.14735s (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 0.338074s (cpu); 0.192758s (thread); 0s (gc)
│ │ │ + -- used 0.442399s (cpu); 0.215451s (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 0.116827s (cpu); 0.0440068s (thread); 0s (gc)
│ │ │ + -- used 0.231026s (cpu); 0.0588934s (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 0.00420313s (cpu); 0.00419957s (thread); 0s (gc)
│ │ │ + -- used 0.00579333s (cpu); 0.00579207s (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 0.122781s (cpu); 0.065672s (thread); 0s (gc)
│ │ │ + -- used 0.183994s (cpu); 0.0865513s (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 0.0292919s (cpu); 0.029296s (thread); 0s (gc)
│ │ │ + -- used 0.0361924s (cpu); 0.0362006s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.617907s (cpu); 0.436744s (thread); 0s (gc)
│ │ │ + -- used 0.742939s (cpu); 0.526145s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.061967s (cpu); 0.0619758s (thread); 0s (gc)
│ │ │ + -- used 0.0730622s (cpu); 0.0730746s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00229866s (cpu); 0.00229945s (thread); 0s (gc)
│ │ │ + -- used 0.00297404s (cpu); 0.00298069s (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 0.461077s (cpu); 0.392442s (thread); 0s (gc)
│ │ │ + -- used 0.4282s (cpu); 0.428211s (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 0.00996664s (cpu); 0.00997121s (thread); 0s (gc)
│ │ │ + -- used 0.012497s (cpu); 0.0125013s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.327999s (cpu); 0.203484s (thread); 0s (gc)
│ │ │ + -- used 0.618492s (cpu); 0.307599s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 5.13464s (cpu); 4.44306s (thread); 0s (gc)
│ │ │ + -- used 6.6227s (cpu); 6.09155s (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 0.0022587s (cpu); 0.0022598s (thread); 0s (gc)
│ │ │ + -- used 0.00271443s (cpu); 0.00271919s (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 0.153174s (cpu); 0.0865693s (thread); 0s (gc)
│ │ │ + -- used 0.212159s (cpu); 0.100868s (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 0.503127s (cpu); 0.422881s (thread); 0s (gc)
│ │ │ + -- used 0.490845s (cpu); 0.490851s (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 0.313198s (cpu); 0.259718s (thread); 0s (gc)
│ │ │ + -- used 0.480484s (cpu); 0.36706s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : time degrees phi
│ │ │ - -- used 0.0182114s (cpu); 0.0177219s (thread); 0s (gc)
│ │ │ + -- used 0.0323021s (cpu); 0.0202118s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : time describe phi
│ │ │ - -- used 0.00315504s (cpu); 0.00315575s (thread); 0s (gc)
│ │ │ + -- used 0.00399366s (cpu); 0.00396264s (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 0.010093s (cpu); 0.0100937s (thread); 0s (gc)
│ │ │ + -- used 0.0121709s (cpu); 0.0121307s (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 0.00979107s (cpu); 0.00979179s (thread); 0s (gc)
│ │ │ + -- used 0.0111708s (cpu); 0.0111818s (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 1.28655s (cpu); 0.923224s (thread); 0s (gc)
│ │ │ + -- used 1.28386s (cpu); 1.04453s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
│ │ │  
│ │ │  i22 : time degree f
│ │ │ - -- used 1.6732e-05s (cpu); 1.6331e-05s (thread); 0s (gc)
│ │ │ + -- used 2.8715e-05s (cpu); 2.459e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 0.00171978s (cpu); 0.00173089s (thread); 0s (gc)
│ │ │ + -- used 0.00173732s (cpu); 0.00174327s (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 0.29196s (cpu); 0.163368s (thread); 0s (gc)
│ │ │ + -- used 0.406503s (cpu); 0.21753s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0120227s (cpu); 0.0115317s (thread); 0s (gc)
│ │ │ + -- used 0.0364921s (cpu); 0.0162617s (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 0.052799s (cpu); 0.0523954s (thread); 0s (gc)
│ │ │ + -- used 0.10504s (cpu); 0.0694934s (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 0.0337406s (cpu); 0.0333982s (thread); 0s (gc)
│ │ │ + -- used 0.0553047s (cpu); 0.0433303s (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 0.158186s (cpu); 0.158148s (thread); 0s (gc)
│ │ │ + -- used 0.182363s (cpu); 0.182363s (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 0.822525s (cpu); 0.527571s (thread); 0s (gc)
│ │ │ + -- used 0.946186s (cpu); 0.608314s (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 0.476782s (cpu); 0.305554s (thread); 0s (gc)
│ │ │ + -- used 0.754484s (cpu); 0.423151s (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 0.0214445s (cpu); 0.0208821s (thread); 0s (gc)
│ │ │ + -- used 0.0418629s (cpu); 0.0270347s (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 0.0973794s (cpu); 0.0969241s (thread); 0s (gc)
│ │ │ + -- used 0.136s (cpu); 0.12328s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -98,22 +98,22 @@
│ │ │  
│ │ │  i8 : o4 == o6 and o5 == o7
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ - = ------[x ..x ]
│ │ │ -   100003  0   6
│ │ │ +       ZZ
│ │ │ +o9 = ------[x ..x ]
│ │ │ +     100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
│ │ │  
│ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.20101s (cpu); 0.102205s (thread); 0s (gc)
│ │ │ + -- used 0.0704228s (cpu); 0.0704275s (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 0.164822s (cpu); 0.164827s (thread); 0s (gc)
│ │ │ + -- used 0.432353s (cpu); 0.282718s (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 0.521163s (cpu); 0.330863s (thread); 0s (gc)
│ │ │ + -- used 0.518404s (cpu); 0.356451s (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 1.64393s (cpu); 1.09158s (thread); 0s (gc)
│ │ │ + -- used 1.88055s (cpu); 1.0608s (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 0.000436017s (cpu); 0.000429806s (thread); 0s (gc)
│ │ │ + -- used 0.0004648s (cpu); 0.000459107s (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 0.282829s (cpu); 0.171024s (thread); 0s (gc)
│ │ │ + -- used 0.420696s (cpu); 0.221181s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.58234s (cpu); 0.452614s (thread); 0s (gc)
│ │ │ + -- used 0.502749s (cpu); 0.399617s (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 0.153309s (cpu); 0.0746711s (thread); 0s (gc)
│ │ │ + -- used 0.209072s (cpu); 0.111369s (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 3.64881s (cpu); 1.96273s (thread); 0s (gc)
│ │ │ + -- used 5.26661s (cpu); 2.46606s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 4.2339e-05s (cpu); 4.2189e-05s (thread); 0s (gc)
│ │ │ + -- used 3.8158e-05s (cpu); 3.6694e-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 5.81059s (cpu); 3.09414s (thread); 0s (gc)
│ │ │ + -- used 8.34301s (cpu); 3.9703s (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 4.56986s (cpu); 2.57125s (thread); 0s (gc)
│ │ │ + -- used 6.96262s (cpu); 3.43641s (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 0.238683s (cpu); 0.19446s (thread); 0s (gc)
│ │ │ + -- used 0.350617s (cpu); 0.253758s (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 0.185817s (cpu); 0.150042s (thread); 0s (gc)
│ │ │ + -- used 0.286801s (cpu); 0.184497s (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 2.23199s (cpu); 1.72322s (thread); 0s (gc)
│ │ │ + -- used 2.21243s (cpu); 1.82591s (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 3.12853s (cpu); 2.52738s (thread); 0s (gc)
│ │ │ + -- used 3.27801s (cpu); 2.80235s (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 0.0449938s (cpu); 0.0449687s (thread); 0s (gc)
│ │ │ + -- used 0.0559744s (cpu); 0.0559739s (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 1.1282s (cpu); 0.686245s (thread); 0s (gc)
│ │ │ + -- used 1.55683s (cpu); 0.882117s (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 0.00068776s (cpu); 0.000679514s (thread); 0s (gc)
│ │ │ + -- used 0.000884035s (cpu); 0.000876136s (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 0.0162802s (cpu); 0.0159994s (thread); 0s (gc)
│ │ │ + -- used 0.0401547s (cpu); 0.0206611s (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 0.0323697s (cpu); 0.0319186s (thread); 0s (gc)
│ │ │ + -- used 0.0659179s (cpu); 0.0437032s (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 0.00381518s (cpu); 0.00381419s (thread); 0s (gc)
│ │ │ + -- used 0.00429425s (cpu); 0.00429116s (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 0.0930291s (cpu); 0.0930347s (thread); 0s (gc)
│ │ │ + -- used 0.111615s (cpu); 0.111614s (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 0.205898s (cpu); 0.128046s (thread); 0s (gc)
│ │ │ + -- used 0.240758s (cpu); 0.134812s (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 0.471188s (cpu); 0.313205s (thread); 0s (gc)
│ │ │ + -- used 0.462064s (cpu); 0.246758s (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 0.0580093s (cpu); 0.0580102s (thread); 0s (gc)
│ │ │ + -- used 0.0659368s (cpu); 0.0658969s (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 0.017676s (cpu); 0.0176757s (thread); 0s (gc)
│ │ │ + -- used 0.022758s (cpu); 0.022757s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0144793s (cpu); 0.014089s (thread); 0s (gc)
│ │ │ + -- used 0.0295073s (cpu); 0.0170099s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}};
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │  
│ │ │  i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 2.42783s (cpu); 1.9977s (thread); 0s (gc)
│ │ │ + -- used 2.62625s (cpu); 2.23964s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : P7 = ZZ/101[x_0..x_7];
│ │ │  
│ │ │  i5 : -- hyperelliptic curve of genus 3
│ │ │       C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6-26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5-24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7-23*x_1*x_7+8*x_2*x_7-22*x_3*x_7+20*x_4*x_7-15*x_5*x_7,x_2*x_5+47*x_5^2-40*x_0*x_6+37*x_1*x_6-25*x_2*x_6-22*x_3*x_6-8*x_4*x_6+27*x_5*x_6+15*x_6^2-23*x_0*x_7-42*x_1*x_7+27*x_2*x_7+35*x_3*x_7+39*x_4*x_7+24*x_5*x_7,x_1*x_5+15*x_5^2+49*x_0*x_6+8*x_1*x_6-31*x_2*x_6+9*x_3*x_6+38*x_4*x_6-36*x_5*x_6-30*x_6^2-33*x_0*x_7+26*x_1*x_7+32*x_2*x_7+27*x_3*x_7+6*x_4*x_7+36*x_5*x_7,x_0*x_5+30*x_5^2-11*x_0*x_6-38*x_1*x_6+13*x_2*x_6-32*x_3*x_6-30*x_4*x_6+4*x_5*x_6-28*x_6^2-30*x_0*x_7-6*x_1*x_7-45*x_2*x_7+34*x_3*x_7+20*x_4*x_7+48*x_5*x_7,x_3*x_4+46*x_5^2-37*x_0*x_6+27*x_1*x_6+33*x_2*x_6+8*x_3*x_6-32*x_4*x_6+42*x_5*x_6-34*x_6^2-37*x_0*x_7-28*x_1*x_7+10*x_2*x_7-27*x_3*x_7-42*x_4*x_7-8*x_5*x_7,x_2*x_4-25*x_5^2-4*x_0*x_6+2*x_1*x_6-31*x_2*x_6-5*x_3*x_6+16*x_4*x_6-24*x_5*x_6+31*x_6^2-30*x_0*x_7+32*x_1*x_7+12*x_2*x_7-40*x_3*x_7+3*x_4*x_7-28*x_5*x_7,x_0*x_4+15*x_5^2+48*x_0*x_6-50*x_1*x_6+46*x_2*x_6-48*x_3*x_6-23*x_4*x_6-28*x_5*x_6+39*x_6^2+38*x_1*x_7-5*x_3*x_7+5*x_4*x_7-34*x_5*x_7,x_3^2-31*x_5^2+41*x_0*x_6-30*x_1*x_6-4*x_2*x_6+43*x_3*x_6+23*x_4*x_6+7*x_5*x_6+31*x_6^2-19*x_0*x_7+25*x_1*x_7-49*x_2*x_7-16*x_3*x_7-45*x_4*x_7+25*x_5*x_7,x_2*x_3+13*x_5^2-45*x_0*x_6-22*x_1*x_6+33*x_2*x_6-26*x_3*x_6-21*x_4*x_6+34*x_5*x_6-21*x_6^2-47*x_0*x_7-10*x_1*x_7+29*x_2*x_7-46*x_3*x_7-x_4*x_7+20*x_5*x_7,x_1*x_3+22*x_5^2+4*x_0*x_6+3*x_1*x_6+45*x_2*x_6+37*x_3*x_6+17*x_4*x_6+36*x_5*x_6-2*x_6^2-31*x_0*x_7+3*x_1*x_7-12*x_2*x_7+19*x_3*x_7+28*x_4*x_7+30*x_5*x_7,x_0*x_3-47*x_5^2-43*x_0*x_6+6*x_1*x_6-40*x_2*x_6+21*x_3*x_6+26*x_4*x_6-5*x_5*x_6-5*x_6^2+4*x_0*x_7-15*x_1*x_7+18*x_2*x_7-31*x_3*x_7+50*x_4*x_7-46*x_5*x_7,x_2^2+4*x_5^2+31*x_0*x_6+41*x_1*x_6+31*x_2*x_6+28*x_3*x_6+42*x_4*x_6-28*x_5*x_6-4*x_6^2-7*x_0*x_7+15*x_1*x_7-9*x_2*x_7+31*x_3*x_7+3*x_4*x_7+7*x_5*x_7,x_1*x_2-46*x_5^2-6*x_0*x_6-50*x_1*x_6+32*x_2*x_6-10*x_3*x_6+42*x_4*x_6+33*x_5*x_6+18*x_6^2-9*x_0*x_7-20*x_1*x_7+45*x_2*x_7-9*x_3*x_7+10*x_4*x_7-8*x_5*x_7,x_0*x_2-9*x_5^2+34*x_0*x_6-45*x_1*x_6+19*x_2*x_6+24*x_3*x_6+23*x_4*x_6-37*x_5*x_6-44*x_6^2+24*x_0*x_7-33*x_2*x_7+41*x_3*x_7-40*x_4*x_7+4*x_5*x_7,x_1^2+x_1*x_4+x_4^2-28*x_5^2-33*x_0*x_6-17*x_1*x_6+11*x_3*x_6+20*x_4*x_6+25*x_5*x_6-21*x_6^2-22*x_0*x_7+24*x_1*x_7-14*x_2*x_7+5*x_3*x_7-39*x_4*x_7-18*x_5*x_7,x_0*x_1-47*x_5^2-5*x_0*x_6-9*x_1*x_6-45*x_2*x_6+48*x_3*x_6+45*x_4*x_6-29*x_5*x_6+3*x_6^2+29*x_0*x_7+40*x_1*x_7+46*x_2*x_7+27*x_3*x_7-36*x_4*x_7-39*x_5*x_7,x_0^2-31*x_5^2+36*x_0*x_6-30*x_1*x_6-10*x_2*x_6+42*x_3*x_6+9*x_4*x_6+34*x_5*x_6-6*x_6^2+48*x_0*x_7-47*x_1*x_7-19*x_2*x_7+25*x_3*x_7+28*x_4*x_7+34*x_5*x_7);
│ │ │ @@ -21,12 +21,12 @@
│ │ │  
│ │ │  i6 : phi = rationalMap(C,3,2);
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │  
│ │ │  i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.61056s (cpu); 2.40993s (thread); 0s (gc)
│ │ │ + -- used 4.2377s (cpu); 2.9171s (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 0.0179336s (cpu); 0.0179291s (thread); 0s (gc)
│ │ │ + -- used 0.0207501s (cpu); 0.0207511s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
│ │ │  
│ │ │  i3 : time kernel(phi,2)
│ │ │ - -- used 0.841697s (cpu); 0.443904s (thread); 0s (gc)
│ │ │ + -- used 1.25114s (cpu); 0.564235s (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 0.00529365s (cpu); 0.00528742s (thread); 0s (gc)
│ │ │ + -- used 0.00671581s (cpu); 0.00671052s (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 0.581724s (cpu); 0.39996s (thread); 0s (gc)
│ │ │ + -- used 0.608827s (cpu); 0.454077s (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 0.401446s (cpu); 0.196261s (thread); 0s (gc)
│ │ │ + -- used 0.542618s (cpu); 0.24148s (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 0.206861s (cpu); 0.131113s (thread); 0s (gc)
│ │ │ + -- used 0.253346s (cpu); 0.145148s (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 0.0160374s (cpu); 0.0155816s (thread); 0s (gc)
│ │ │ + -- used 0.033675s (cpu); 0.0182597s (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 0.0118936s (cpu); 0.0115154s (thread); 0s (gc)
│ │ │ + -- used 0.0268169s (cpu); 0.0148575s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a rational octic surface
│ │ │       phi = map specialCremonaTransformation(7,ZZ/300007)
│ │ │ @@ -48,21 +48,21 @@
│ │ │            300007  0   6   300007  0   6     2 4    1 5          0 4          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6   2 3    0 5          1 3          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6        0 3         1 4         3 4         4          0 5         1 5         2 5          3 5          4 5         5         3 6          4 6         5 6          0 1          1         0 2          1 2         2          1 4          1 5         2 5          0 6         1 6         2 6         0          1         0 2         1 2         2         1 4          4         0 5         1 5          2 5          4 5         5         0 6         1 6          2 6          3 6         4 6         5 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │  
│ │ │  i7 : time projectiveDegrees phi
│ │ │ - -- used 4.799e-05s (cpu); 4.1858e-05s (thread); 0s (gc)
│ │ │ + -- used 7.56e-05s (cpu); 6.3659e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.7161e-05s (cpu); 2.693e-05s (thread); 0s (gc)
│ │ │ + -- used 2.1584e-05s (cpu); 2.1821e-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 0.102641s (cpu); 0.10261s (thread); 0s (gc)
│ │ │ + -- used 0.118146s (cpu); 0.118148s (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 0.0313385s (cpu); 0.0313402s (thread); 0s (gc)
│ │ │ + -- used 0.0367759s (cpu); 0.0366658s (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 1.09129s (cpu); 0.717208s (thread); 0s (gc)
│ │ │ + -- used 1.74204s (cpu); 0.718667s (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 1.46815s (cpu); 1.12516s (thread); 0s (gc)
│ │ │ + -- used 1.73689s (cpu); 1.27091s (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 0.0991676s (cpu); 0.0991684s (thread); 0s (gc)
│ │ │ + -- used 0.0959956s (cpu); 0.0959964s (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 0.0190914s (cpu); 0.019092s (thread); 0s (gc)
│ │ │ + -- used 0.0195957s (cpu); 0.019597s (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 0.0754958s (cpu); 0.0754978s (thread); 0s (gc)
│ │ │ + -- used 0.0852889s (cpu); 0.0852893s (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 0.106283s (cpu); 0.0351122s (thread); 0s (gc)
│ │ │ + -- used 0.151762s (cpu); 0.033368s (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 0.024537s (cpu); 0.0245365s (thread); 0s (gc)
│ │ │ + -- used 0.0256943s (cpu); 0.0256942s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │  
│ │ │  i5 : time describe phi'
│ │ │ - -- used 0.00548258s (cpu); 0.00548337s (thread); 0s (gc)
│ │ │ + -- used 0.00669238s (cpu); 0.00670181s (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 0.00454217s (cpu); 0.00454769s (thread); 0s (gc)
│ │ │ + -- used 0.00560702s (cpu); 0.00561764s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html
│ │ │ @@ -97,30 +97,30 @@
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
    
│ │ │              
    
│ │ │                
    i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 2.13317s (cpu); 1.1571s (thread); 0s (gc)
│ │ │ + -- used 2.95535s (cpu); 1.42315s (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 1.28884s (cpu); 0.908882s (thread); 0s (gc)
│ │ │ + -- used 1.78324s (cpu); 1.14735s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -167,30 +167,30 @@
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i9 : time ChernClass G
│ │ │ - -- used 0.338074s (cpu); 0.192758s (thread); 0s (gc)
│ │ │ + -- used 0.442399s (cpu); 0.215451s (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 0.116827s (cpu); 0.0440068s (thread); 0s (gc)
│ │ │ + -- used 0.231026s (cpu); 0.0588934s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,26 +39,26 @@
│ │ │ │                 2                           2
│ │ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │ │                          0   4
│ │ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ │ - -- used 2.13317s (cpu); 1.1571s (thread); 0s (gc)
│ │ │ │ + -- used 2.95535s (cpu); 1.42315s (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 1.28884s (cpu); 0.908882s (thread); 0s (gc)
│ │ │ │ + -- used 1.78324s (cpu); 1.14735s (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 0.338074s (cpu); 0.192758s (thread); 0s (gc)
│ │ │ │ + -- used 0.442399s (cpu); 0.215451s (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 0.116827s (cpu); 0.0440068s (thread); 0s (gc)
│ │ │ │ + -- used 0.231026s (cpu); 0.0588934s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6      5      4     3
│ │ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o10 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html
│ │ │ @@ -85,24 +85,24 @@
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i2 : time EulerCharacteristic I
│ │ │ - -- used 0.29196s (cpu); 0.163368s (thread); 0s (gc)
│ │ │ + -- used 0.406503s (cpu); 0.21753s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
    
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0120227s (cpu); 0.0115317s (thread); 0s (gc)
│ │ │ + -- used 0.0364921s (cpu); 0.0162617s (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 0.29196s (cpu); 0.163368s (thread); 0s (gc)
│ │ │ │ + -- used 0.406503s (cpu); 0.21753s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0120227s (cpu); 0.0115317s (thread); 0s (gc)
│ │ │ │ + -- used 0.0364921s (cpu); 0.0162617s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = 10
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  No test is made to see if the projective variety is smooth.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_u_l_e_r_(_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- topological Euler characteristic of a
│ │ │ │        (smooth) projective variety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i4 : time phi! ;
│ │ │ - -- used 0.052799s (cpu); 0.0523954s (thread); 0s (gc)
│ │ │ + -- used 0.10504s (cpu); 0.0694934s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
    
│ │ │              
    
│ │ │                
    i5 : describe phi
│ │ │ @@ -127,15 +127,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i9 : time phi! ;
│ │ │ - -- used 0.0337406s (cpu); 0.0333982s (thread); 0s (gc)
│ │ │ + -- used 0.0553047s (cpu); 0.0433303s (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 0.052799s (cpu); 0.0523954s (thread); 0s (gc)
│ │ │ │ + -- used 0.10504s (cpu); 0.0694934s (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 0.0337406s (cpu); 0.0333982s (thread); 0s (gc)
│ │ │ │ + -- used 0.0553047s (cpu); 0.0433303s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │ │  i10 : describe phi
│ │ │ │  
│ │ │ │  o10 = rational map defined by forms of degree 2
│ │ │ │        source variety: PP^4
│ │ │ │        target variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp^_st_st_sp__Ideal.html
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    
│ │ │              
    
│ │ │                
    i6 : time phi^** q
│ │ │ - -- used 0.158186s (cpu); 0.158148s (thread); 0s (gc)
│ │ │ + -- used 0.182363s (cpu); 0.182363s (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 0.822525s (cpu); 0.527571s (thread); 0s (gc)
│ │ │ + -- used 0.946186s (cpu); 0.608314s (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 0.476782s (cpu); 0.305554s (thread); 0s (gc)
│ │ │ + -- used 0.754484s (cpu); 0.423151s (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 0.0214445s (cpu); 0.0208821s (thread); 0s (gc)
│ │ │ + -- used 0.0418629s (cpu); 0.0270347s (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 0.0973794s (cpu); 0.0969241s (thread); 0s (gc)
│ │ │ + -- used 0.136s (cpu); 0.12328s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -203,25 +203,25 @@
│ │ │          
    
│ │ │          
    The method also accepts as input a ring map phi representing a rational map $\Phi:X\dashrightarrow Y$ between projective varieties. In this case, the method returns the push-forward to the Chow ring of the ambient projective space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it basically computes SegreClass ideal matrix phi. In the next example, we compute the Segre class of the base locus of a birational map $\mathbb{G}(1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ - = ------[x ..x ]
│ │ │ -   100003  0   6
│ │ │ +       ZZ
│ │ │ +o9 = ------[x ..x ]
│ │ │ +     100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.20101s (cpu); 0.102205s (thread); 0s (gc)
│ │ │ + -- used 0.0704228s (cpu); 0.0704275s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y , y  - y y  - y y , y y  + y y  - y y , y y  - y y , y y  - y y  - y y , y y  - y y  - y y })
│ │ │             (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )  100003  0   6     3    0 5    1 6   3 4    1 7   4    2 7    0 9   2 5    3 5    1 8   4 5    1 9   4 8    2 9    3 9   7 8    4 9    6 9
│ │ │               5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │ @@ -233,15 +233,15 @@
│ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    
│ │ │              
    
│ │ │                
    i11 : time SegreClass phi
│ │ │ - -- used 0.164822s (cpu); 0.164827s (thread); 0s (gc)
│ │ │ + -- used 0.432353s (cpu); 0.282718s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -267,30 +267,30 @@
│ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    
│ │ │              
    
│ │ │                
    i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.521163s (cpu); 0.330863s (thread); 0s (gc)
│ │ │ + -- used 0.518404s (cpu); 0.356451s (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 1.64393s (cpu); 1.09158s (thread); 0s (gc)
│ │ │ + -- used 1.88055s (cpu); 1.0608s (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 0.822525s (cpu); 0.527571s (thread); 0s (gc)
│ │ │ │ + -- used 0.946186s (cpu); 0.608314s (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 0.476782s (cpu); 0.305554s (thread); 0s (gc)
│ │ │ │ + -- used 0.754484s (cpu); 0.423151s (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 0.0214445s (cpu); 0.0208821s (thread); 0s (gc)
│ │ │ │ + -- used 0.0418629s (cpu); 0.0270347s (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 0.0973794s (cpu); 0.0969241s (thread); 0s (gc)
│ │ │ │ + -- used 0.136s (cpu); 0.12328s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o7 : -----
│ │ │ │          8
│ │ │ │ @@ -134,22 +134,22 @@
│ │ │ │  method returns the push-forward to the Chow ring of the ambient projective
│ │ │ │  space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it
│ │ │ │  basically computes SegreClass ideal matrix phi. In the next example, we compute
│ │ │ │  the Segre class of the base locus of a birational map $\mathbb{G}
│ │ │ │  (1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
│ │ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │ │  
│ │ │ │ -o9     ZZ
│ │ │ │ - = ------[x ..x ]
│ │ │ │ -   100003  0   6
│ │ │ │ +       ZZ
│ │ │ │ +o9 = ------[x ..x ]
│ │ │ │ +     100003  0   6
│ │ │ │  
│ │ │ │  o9 : PolynomialRing
│ │ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},
│ │ │ │  {x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ │ - -- used 0.20101s (cpu); 0.102205s (thread); 0s (gc)
│ │ │ │ + -- used 0.0704228s (cpu); 0.0704275s (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 0.164822s (cpu); 0.164827s (thread); 0s (gc)
│ │ │ │ + -- used 0.432353s (cpu); 0.282718s (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 0.521163s (cpu); 0.330863s (thread); 0s (gc)
│ │ │ │ + -- used 0.518404s (cpu); 0.356451s (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 1.64393s (cpu); 1.09158s (thread); 0s (gc)
│ │ │ │ + -- used 1.88055s (cpu); 1.0608s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             9       8       7      6     5
│ │ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o14 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000436017s (cpu); 0.000429806s (thread); 0s (gc)
│ │ │ + -- used 0.0004648s (cpu); 0.000459107s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │ @@ -119,23 +119,23 @@
│ │ │            
    
│ │ │          
    Now we compute first the degree of the forms defining the abstract map psi and then the corresponding concrete rational map.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.282829s (cpu); 0.171024s (thread); 0s (gc)
│ │ │ + -- used 0.420696s (cpu); 0.221181s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
    
│ │ │                
    i6 : time rationalMap psi
│ │ │ - -- used 0.58234s (cpu); 0.452614s (thread); 0s (gc)
│ │ │ + -- used 0.502749s (cpu); 0.399617s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -233,15 +233,15 @@
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
    
│ │ │              
    
│ │ │                
    i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.153309s (cpu); 0.0746711s (thread); 0s (gc)
│ │ │ + -- used 0.209072s (cpu); 0.111369s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -253,26 +253,26 @@
│ │ │            
    
│ │ │          
    The degree of the forms defining the abstract map T can be obtained by the following command:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 3.64881s (cpu); 1.96273s (thread); 0s (gc)
│ │ │ + -- used 5.26661s (cpu); 2.46606s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself is defined by linear forms:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i16 : time T2 = T * T
│ │ │ - -- used 4.2339e-05s (cpu); 4.2189e-05s (thread); 0s (gc)
│ │ │ + -- used 3.8158e-05s (cpu); 3.6694e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -281,15 +281,15 @@
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
    
│ │ │              
    
│ │ │                
    i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 5.81059s (cpu); 3.09414s (thread); 0s (gc)
│ │ │ + -- used 8.34301s (cpu); 3.9703s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself leaves a random point fixed:
    
│ │ │          
│ │ │ @@ -322,15 +322,15 @@
│ │ │            
│ │ │          
    
│ │ │          
    We now compute the concrete rational map corresponding to T:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i21 : time f = rationalMap T
│ │ │ - -- used 4.56986s (cpu); 2.57125s (thread); 0s (gc)
│ │ │ + -- used 6.96262s (cpu); 3.43641s (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 0.000436017s (cpu); 0.000429806s (thread); 0s (gc)
│ │ │ │ + -- used 0.0004648s (cpu); 0.000459107s (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 0.282829s (cpu); 0.171024s (thread); 0s (gc)
│ │ │ │ + -- used 0.420696s (cpu); 0.221181s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.58234s (cpu); 0.452614s (thread); 0s (gc)
│ │ │ │ + -- used 0.502749s (cpu); 0.399617s (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 0.153309s (cpu); 0.0746711s (thread); 0s (gc)
│ │ │ │ + -- used 0.209072s (cpu); 0.111369s (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 3.64881s (cpu); 1.96273s (thread); 0s (gc)
│ │ │ │ + -- used 5.26661s (cpu); 2.46606s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3
│ │ │ │  We verify that the composition of T with itself is defined by linear forms:
│ │ │ │  i16 : time T2 = T * T
│ │ │ │ - -- used 4.2339e-05s (cpu); 4.2189e-05s (thread); 0s (gc)
│ │ │ │ + -- used 3.8158e-05s (cpu); 3.6694e-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 5.81059s (cpu); 3.09414s (thread); 0s (gc)
│ │ │ │ + -- used 8.34301s (cpu); 3.9703s (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 4.56986s (cpu); 2.57125s (thread); 0s (gc)
│ │ │ │ + -- used 6.96262s (cpu); 3.43641s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html
│ │ │ @@ -139,15 +139,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.238683s (cpu); 0.19446s (thread); 0s (gc)
│ │ │ + -- used 0.350617s (cpu); 0.253758s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -200,15 +200,15 @@
│ │ │            
    
│ │ │                
    i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.185817s (cpu); 0.150042s (thread); 0s (gc)
│ │ │ + -- used 0.286801s (cpu); 0.184497s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i6 : assert(psi == psi')
    
│ │ │ @@ -295,15 +295,15 @@
│ │ │            
    
│ │ │                
    i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.23199s (cpu); 1.72322s (thread); 0s (gc)
│ │ │ + -- used 2.21243s (cpu); 1.82591s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -367,15 +367,15 @@
│ │ │              
    		
│ │ │                
    i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.12853s (cpu); 2.52738s (thread); 0s (gc)
│ │ │ + -- used 3.27801s (cpu); 2.80235s (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 0.238683s (cpu); 0.19446s (thread); 0s (gc)
│ │ │ │ + -- used 0.350617s (cpu); 0.253758s (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 0.185817s (cpu); 0.150042s (thread); 0s (gc)
│ │ │ │ + -- used 0.286801s (cpu); 0.184497s (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 2.23199s (cpu); 1.72322s (thread); 0s (gc)
│ │ │ │ + -- used 2.21243s (cpu); 1.82591s (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 3.12853s (cpu); 2.52738s (thread); 0s (gc)
│ │ │ │ + -- used 3.27801s (cpu); 2.80235s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = -- rational map --
│ │ │ │                                   ZZ
│ │ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │                {
│ │ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
    
│ │ │              
    
│ │ │                
    i5 : time degreeMap phi
│ │ │ - -- used 0.0449938s (cpu); 0.0449687s (thread); 0s (gc)
│ │ │ + -- used 0.0559744s (cpu); 0.0559739s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    
│ │ │              
    
│ │ │                
    i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │ @@ -113,15 +113,15 @@
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
    
│ │ │              
    
│ │ │                
    i7 : time degreeMap phi'
│ │ │ - -- used 1.1282s (cpu); 0.686245s (thread); 0s (gc)
│ │ │ + -- used 1.55683s (cpu); 0.882117s (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 0.0449938s (cpu); 0.0449687s (thread); 0s (gc)
│ │ │ │ + -- used 0.0559744s (cpu); 0.0559739s (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 1.1282s (cpu); 0.686245s (thread); 0s (gc)
│ │ │ │ + -- used 1.55683s (cpu); 0.882117s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = 14
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map
│ │ │ │      * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s -- projective degrees of a rational map between
│ │ │ │        projective varieties
│ │ │ │  ********** WWaayyss ttoo uussee ddeeggrreeeeMMaapp:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_force__Image.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.00068776s (cpu); 0.000679514s (thread); 0s (gc)
    │ │ │ + -- used 0.000884035s (cpu); 0.000876136s (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 0.00068776s (cpu); 0.000679514s (thread); 0s (gc) │ │ │ │ + -- used 0.000884035s (cpu); 0.000876136s (thread); 0s (gc) │ │ │ │ i5 : Phi; │ │ │ │ │ │ │ │ o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional │ │ │ │ subvariety of PP^9) │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ If the declaration is false, nonsensical answers may result. │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_graph.html │ │ │ @@ -113,15 +113,15 @@ │ │ │ │ │ │ o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0162802s (cpu); 0.0159994s (thread); 0s (gc)
    │ │ │ + -- used 0.0401547s (cpu); 0.0206611s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │ @@ -272,15 +272,15 @@
│ │ │            
│ │ │          
│ │ │          
    When the source of the rational map is a multi-projective variety, the method returns all the projections.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │  
│ │ │ │                        w w  - w w  + w w
│ │ │ │                         2 4    1 5    0 6
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │ │  i2 : time psi = inverseMap phi
│ │ │ │ - -- used 0.205898s (cpu); 0.128046s (thread); 0s (gc)
│ │ │ │ + -- used 0.240758s (cpu); 0.134812s (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 0.471188s (cpu); 0.313205s (thread); 0s (gc)
│ │ │ │ + -- used 0.462064s (cpu); 0.246758s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  ,
│ │ │ │  - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   -
│ │ │ │  w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   -
│ │ │ │  w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   +
│ │ │ │  w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   +
│ │ │ │  w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse_lp__Rational__Map_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time g = graph p2;
│ │ │ - -- used 0.0323697s (cpu); 0.0319186s (thread); 0s (gc)
    
│ │ │ + -- used 0.0659179s (cpu); 0.0437032s (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 0.0162802s (cpu); 0.0159994s (thread); 0s (gc)
│ │ │ │ + -- used 0.0401547s (cpu); 0.0206611s (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 0.0323697s (cpu); 0.0319186s (thread); 0s (gc)
│ │ │ │ + -- used 0.0659179s (cpu); 0.0437032s (thread); 0s (gc)
│ │ │ │  i10 : g_0;
│ │ │ │  
│ │ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ │ │  of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │ │  i11 : g_1;
│ │ │ │  
│ │ │ │  o11 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_ideal_lp__Rational__Map_rp.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │              
    
│ │ │                
    i3 : time ideal phi
│ │ │ - -- used 0.00381518s (cpu); 0.00381419s (thread); 0s (gc)
│ │ │ + -- used 0.00429425s (cpu); 0.00429116s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -195,15 +195,15 @@
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
    
│ │ │              
    
│ │ │                
    i6 : time ideal phi'
│ │ │ - -- used 0.0930291s (cpu); 0.0930347s (thread); 0s (gc)
│ │ │ + -- used 0.111615s (cpu); 0.111614s (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 0.00381518s (cpu); 0.00381419s (thread); 0s (gc)
│ │ │ │ + -- used 0.00429425s (cpu); 0.00429116s (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 0.0930291s (cpu); 0.0930347s (thread); 0s (gc)
│ │ │ │ + -- used 0.111615s (cpu); 0.111614s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal 1
│ │ │ │  
│ │ │ │  
│ │ │ │  QQ[x ..x , y ..y ]
│ │ │ │  
│ │ │ │  0   5   0   4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
    
│ │ │              
    
│ │ │                
    i2 : time psi = inverseMap phi
│ │ │ - -- used 0.205898s (cpu); 0.128046s (thread); 0s (gc)
│ │ │ + -- used 0.240758s (cpu); 0.134812s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -251,15 +251,15 @@
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i5 : time psi = inverseMap phi
│ │ │ - -- used 0.471188s (cpu); 0.313205s (thread); 0s (gc)
│ │ │ + -- used 0.462064s (cpu); 0.246758s (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 0.0580093s (cpu); 0.0580102s (thread); 0s (gc)
│ │ │ + -- used 0.0659368s (cpu); 0.0658969s (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 0.0580093s (cpu); 0.0580102s (thread); 0s (gc)
│ │ │ │ + -- used 0.0659368s (cpu); 0.0658969s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Birational.html
│ │ │ @@ -123,24 +123,24 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i3 : time isBirational phi
│ │ │ - -- used 0.017676s (cpu); 0.0176757s (thread); 0s (gc)
│ │ │ + -- used 0.022758s (cpu); 0.022757s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │                
    i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0144793s (cpu); 0.014089s (thread); 0s (gc)
│ │ │ + -- used 0.0295073s (cpu); 0.0170099s (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 0.017676s (cpu); 0.0176757s (thread); 0s (gc)
│ │ │ │ + -- used 0.022758s (cpu); 0.022757s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0144793s (cpu); 0.014089s (thread); 0s (gc)
│ │ │ │ + -- used 0.0295073s (cpu); 0.0170099s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_D_o_m_i_n_a_n_t -- whether a rational map is dominant
│ │ │ │  ********** WWaayyss ttoo uussee iissBBiirraattiioonnaall:: **********
│ │ │ │      * isBirational(RationalMap)
│ │ │ │      * isBirational(RingMap)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Dominant.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 2.42783s (cpu); 1.9977s (thread); 0s (gc)
│ │ │ + -- used 2.62625s (cpu); 2.23964s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : P7 = ZZ/101[x_0..x_7];
    │ │ │ @@ -115,15 +115,15 @@ │ │ │ o6 : RationalMap (cubic rational map from PP^7 to PP^7) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.61056s (cpu); 2.40993s (thread); 0s (gc)
│ │ │ + -- used 4.2377s (cpu); 2.9171s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ i1 : P8 = ZZ/101[x_0..x_8]; │ │ │ │ i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7}, │ │ │ │ {x_4..x_8}}; │ │ │ │ │ │ │ │ o2 : RationalMap (rational map from PP^8 to PP^8) │ │ │ │ i3 : time isDominant(phi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 2.42783s (cpu); 1.9977s (thread); 0s (gc) │ │ │ │ + -- used 2.62625s (cpu); 2.23964s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ i4 : P7 = ZZ/101[x_0..x_7]; │ │ │ │ i5 : -- hyperelliptic curve of genus 3 │ │ │ │ C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6- │ │ │ │ 26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5- │ │ │ │ 24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7- │ │ │ │ @@ -65,15 +65,15 @@ │ │ │ │ │ │ │ │ o5 : Ideal of P7 │ │ │ │ i6 : phi = rationalMap(C,3,2); │ │ │ │ │ │ │ │ o6 : RationalMap (cubic rational map from PP^7 to PP^7) │ │ │ │ i7 : time isDominant(phi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 3.61056s (cpu); 2.40993s (thread); 0s (gc) │ │ │ │ + -- used 4.2377s (cpu); 2.9171s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = false │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_B_i_r_a_t_i_o_n_a_l -- whether a rational map is birational │ │ │ │ ********** WWaayyss ttoo uussee iissDDoommiinnaanntt:: ********** │ │ │ │ * isDominant(RationalMap) │ │ │ │ * isDominant(RingMap) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_kernel_lp__Ring__Map_cm__Z__Z_rp.html │ │ │ @@ -90,26 +90,26 @@ │ │ │ o1 : RingMap QQ[x ..x ] <-- QQ[y ..y ] │ │ │ 0 8 0 11 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time kernel(phi,1)
│ │ │ - -- used 0.0179336s (cpu); 0.0179291s (thread); 0s (gc)
│ │ │ + -- used 0.0207501s (cpu); 0.0207511s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time kernel(phi,2)
│ │ │ - -- used 0.841697s (cpu); 0.443904s (thread); 0s (gc)
│ │ │ + -- used 1.25114s (cpu); 0.564235s (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 0.0179336s (cpu); 0.0179291s (thread); 0s (gc)
│ │ │ │ + -- used 0.0207501s (cpu); 0.0207511s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal ()
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │ │                    0   11
│ │ │ │  i3 : time kernel(phi,2)
│ │ │ │ - -- used 0.841697s (cpu); 0.443904s (thread); 0s (gc)
│ │ │ │ + -- used 1.25114s (cpu); 0.564235s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                             2
│ │ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │  
│ │ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_parametrize_lp__Ideal_rp.html
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time parametrize L
│ │ │ - -- used 0.00529365s (cpu); 0.00528742s (thread); 0s (gc)
│ │ │ + -- used 0.00671581s (cpu); 0.00671052s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -201,15 +201,15 @@
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time parametrize Q
│ │ │ - -- used 0.581724s (cpu); 0.39996s (thread); 0s (gc)
│ │ │ + -- used 0.608827s (cpu); 0.454077s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │       - 849671x  + 3034137x )
│ │ │ │                8           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o2 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i3 : time parametrize L
│ │ │ │ - -- used 0.00529365s (cpu); 0.00528742s (thread); 0s (gc)
│ │ │ │ + -- used 0.00671581s (cpu); 0.00671052s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -136,15 +136,15 @@
│ │ │ │       1211601x x  - 2168594x x  - 1801762x x  + 3022242x x  + 3618789x )
│ │ │ │               5 9           6 9           7 9           8 9           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o4 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i5 : time parametrize Q
│ │ │ │ - -- used 0.581724s (cpu); 0.39996s (thread); 0s (gc)
│ │ │ │ + -- used 0.608827s (cpu); 0.454077s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_point_lp__Quotient__Ring_rp.html
│ │ │ @@ -78,15 +78,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time p = point source f
│ │ │ - -- used 0.401446s (cpu); 0.196261s (thread); 0s (gc)
│ │ │ + -- used 0.542618s (cpu); 0.24148s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -100,15 +100,15 @@
│ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11   3 4    1 6    0 8   2 4    1 5    0 7
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time p == f^* f p
│ │ │ - -- used 0.206861s (cpu); 0.131113s (thread); 0s (gc)
│ │ │ + -- used 0.253346s (cpu); 0.145148s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ documentation) , see _p_o_i_n_t_(_M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_). │ │ │ │ Below we verify the birationality of a rational map. │ │ │ │ i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331); │ │ │ │ │ │ │ │ o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to │ │ │ │ PP^8) │ │ │ │ i2 : time p = point source f │ │ │ │ - -- used 0.401446s (cpu); 0.196261s (thread); 0s (gc) │ │ │ │ + -- used 0.542618s (cpu); 0.24148s (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 0.206861s (cpu); 0.131113s (thread); 0s (gc) │ │ │ │ + -- used 0.253346s (cpu); 0.145148s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t -- pick a random K rational point on the scheme X │ │ │ │ defined by I │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * point(PolynomialRing) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_projective__Degrees.html │ │ │ @@ -89,15 +89,15 @@ │ │ │ 0 4 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0160374s (cpu); 0.0155816s (thread); 0s (gc)
│ │ │ + -- used 0.033675s (cpu); 0.0182597s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -117,15 +117,15 @@ │ │ │ 2 3 1 4 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0118936s (cpu); 0.0115154s (thread); 0s (gc)
│ │ │ + -- used 0.0268169s (cpu); 0.0148575s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -143,25 +143,25 @@ │ │ │ o6 : RingMap ------[x ..x ] <-- ------[x ..x ] │ │ │ 300007 0 6 300007 0 6 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time projectiveDegrees phi
│ │ │ - -- used 4.799e-05s (cpu); 4.1858e-05s (thread); 0s (gc)
│ │ │ + -- used 7.56e-05s (cpu); 6.3659e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.7161e-05s (cpu); 2.693e-05s (thread); 0s (gc)
│ │ │ + -- used 2.1584e-05s (cpu); 2.1821e-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 0.0160374s (cpu); 0.0155816s (thread); 0s (gc) │ │ │ │ + -- used 0.033675s (cpu); 0.0182597s (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 0.0118936s (cpu); 0.0115154s (thread); 0s (gc) │ │ │ │ + -- used 0.0268169s (cpu); 0.0148575s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = {2, 4, 4, 2, 1} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : -- Cremona transformation of P^6 defined by the quadrics through a │ │ │ │ rational octic surface │ │ │ │ phi = map specialCremonaTransformation(7,ZZ/300007) │ │ │ │ @@ -119,21 +119,21 @@ │ │ │ │ 4 5 5 0 6 1 6 2 6 3 6 4 6 │ │ │ │ 5 6 │ │ │ │ │ │ │ │ ZZ ZZ │ │ │ │ o6 : RingMap ------[x ..x ] <-- ------[x ..x ] │ │ │ │ 300007 0 6 300007 0 6 │ │ │ │ i7 : time projectiveDegrees phi │ │ │ │ - -- used 4.799e-05s (cpu); 4.1858e-05s (thread); 0s (gc) │ │ │ │ + -- used 7.56e-05s (cpu); 6.3659e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {1, 2, 4, 8, 8, 4, 1} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : time projectiveDegrees(phi,NumDegrees=>1) │ │ │ │ - -- used 2.7161e-05s (cpu); 2.693e-05s (thread); 0s (gc) │ │ │ │ + -- used 2.1584e-05s (cpu); 2.1821e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = {4, 1} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ Another way to use this method is by passing an integer i as second argument. │ │ │ │ However, this is equivalent to first projectiveDegrees(phi,NumDegrees=>i) and │ │ │ │ generally it is not faster. │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.html │ │ │ @@ -88,15 +88,15 @@ │ │ │ o2 : Ideal of -----[x ..x ] │ │ │ 33331 0 6 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.102641s (cpu); 0.10261s (thread); 0s (gc)
│ │ │ + -- used 0.118146s (cpu); 0.118148s (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 0.102641s (cpu); 0.10261s (thread); 0s (gc)
│ │ │ │ + -- used 0.118146s (cpu); 0.118148s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                      ZZ
│ │ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │ │                    33331  0   1   2   3   4   5   6
│ │ │ │                      ZZ
│ │ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  ,
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ring_cm__Tally_rp.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │              
│ │ │                
    i5 : D = new Tally from {H => 2,C => 1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time phi = rationalMap D
│ │ │ - -- used 0.0313385s (cpu); 0.0313402s (thread); 0s (gc)
│ │ │ + -- used 0.0367759s (cpu); 0.0366658s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -219,15 +219,15 @@
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.09129s (cpu); 0.717208s (thread); 0s (gc)
│ │ │ + -- used 1.74204s (cpu); 0.718667s (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 0.0313385s (cpu); 0.0313402s (thread); 0s (gc)
│ │ │ │ + -- used 0.0367759s (cpu); 0.0366658s (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 1.09129s (cpu); 0.717208s (thread); 0s (gc)
│ │ │ │ + -- used 1.74204s (cpu); 0.718667s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │ │       hypersurfaces of degree 2
│ │ │ │  See also the package _W_e_i_l_D_i_v_i_s_o_r_s, which provides general tools for working
│ │ │ │  with divisors.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_t_i_o_n_a_l_M_a_p -- makes a rational map
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cremona__Transformation.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    A Cremona transformation is said to be special if the base locus scheme is smooth and irreducible. To ensure this condition, the field K must be large enough but no check is made.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.46815s (cpu); 1.12516s (thread); 0s (gc)
│ │ │ + -- used 1.73689s (cpu); 1.27091s (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 1.46815s (cpu); 1.12516s (thread); 0s (gc)
│ │ │ │ + -- used 1.73689s (cpu); 1.27091s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │ │        source variety: PP^3
│ │ │ │        target variety: PP^3
│ │ │ │        dominance: true
│ │ │ │        birationality: true
│ │ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cubic__Transformation.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.0991676s (cpu); 0.0991684s (thread); 0s (gc)
│ │ │ + -- used 0.0959956s (cpu); 0.0959964s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -138,15 +138,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.0190914s (cpu); 0.019092s (thread); 0s (gc)
│ │ │ + -- used 0.0195957s (cpu); 0.019597s (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 0.0991676s (cpu); 0.0991684s (thread); 0s (gc)
│ │ │ │ + -- used 0.0959956s (cpu); 0.0959964s (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 0.0190914s (cpu); 0.019092s (thread); 0s (gc)
│ │ │ │ + -- used 0.0195957s (cpu); 0.019597s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^6
│ │ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Quadratic__Transformation.html
│ │ │ @@ -70,15 +70,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0754958s (cpu); 0.0754978s (thread); 0s (gc)
│ │ │ + -- used 0.0852889s (cpu); 0.0852893s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.106283s (cpu); 0.0351122s (thread); 0s (gc)
│ │ │ + -- used 0.151762s (cpu); 0.033368s (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 0.0754958s (cpu); 0.0754978s (thread); 0s (gc)
│ │ │ │ + -- used 0.0852889s (cpu); 0.0852893s (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 0.106283s (cpu); 0.0351122s (thread); 0s (gc)
│ │ │ │ + -- used 0.151762s (cpu); 0.033368s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^8
│ │ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html
│ │ │ @@ -88,23 +88,23 @@
│ │ │  
│ │ │  o3 = 6927
    
│ │ │              
    
│ │ │                
    i4 : time phi' = value str;
│ │ │ - -- used 0.024537s (cpu); 0.0245365s (thread); 0s (gc)
│ │ │ + -- used 0.0256943s (cpu); 0.0256942s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
    
│ │ │              
    
│ │ │                
    i5 : time describe phi'
│ │ │ - -- used 0.00548258s (cpu); 0.00548337s (thread); 0s (gc)
│ │ │ + -- used 0.00669238s (cpu); 0.00670181s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ @@ -113,15 +113,15 @@
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
    
│ │ │              
    
│ │ │                
    i6 : time describe inverse phi'
│ │ │ - -- used 0.00454217s (cpu); 0.00454769s (thread); 0s (gc)
│ │ │ + -- used 0.00560702s (cpu); 0.00561764s (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 0.024537s (cpu); 0.0245365s (thread); 0s (gc)
│ │ │ │ + -- used 0.0256943s (cpu); 0.0256942s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i5 : time describe phi'
│ │ │ │ - -- used 0.00548258s (cpu); 0.00548337s (thread); 0s (gc)
│ │ │ │ + -- used 0.00669238s (cpu); 0.00670181s (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 0.00454217s (cpu); 0.00454769s (thread); 0s (gc)
│ │ │ │ + -- used 0.00560702s (cpu); 0.00561764s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = rational map defined by forms of degree 2
│ │ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │ │       target variety: PP^3
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/index.html
│ │ │ @@ -58,29 +58,29 @@
│ │ │              
    		
│ │ │                
    i1 : ZZ/300007[t_0..t_6];
    
│ │ │              
    
│ │ │                
    i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00420313s (cpu); 0.00419957s (thread); 0s (gc)
│ │ │ + -- used 0.00579333s (cpu); 0.00579207s (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 0.122781s (cpu); 0.065672s (thread); 0s (gc)
│ │ │ + -- used 0.183994s (cpu); 0.0865513s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │ @@ -88,43 +88,43 @@
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
    
│ │ │              
    
│ │ │                
    i4 : time degreeMap phi
│ │ │ - -- used 0.0292919s (cpu); 0.029296s (thread); 0s (gc)
│ │ │ + -- used 0.0361924s (cpu); 0.0362006s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.617907s (cpu); 0.436744s (thread); 0s (gc)
│ │ │ + -- used 0.742939s (cpu); 0.526145s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.061967s (cpu); 0.0619758s (thread); 0s (gc)
│ │ │ + -- used 0.0730622s (cpu); 0.0730746s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00229866s (cpu); 0.00229945s (thread); 0s (gc)
│ │ │ + -- used 0.00297404s (cpu); 0.00298069s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              ZZ                                                       300007  0   9                                                  3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o7 = map (------[t ..t ], ----------------------------------------------------------------------------------------------------, {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -136,15 +136,15 @@
│ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i8 : time psi = inverseMap phi
│ │ │ - -- used 0.461077s (cpu); 0.392442s (thread); 0s (gc)
│ │ │ + -- used 0.4282s (cpu); 0.428211s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       300007  0   9                                                ZZ              3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
│ │ │  o8 = map (----------------------------------------------------------------------------------------------------, ------[t ..t ], {x  - 2x x x  + x x  - x x x  + x x  + x x  + x x x  - x x x  + x x  - 2x x x  - x x x  - 2x x , x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x , x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x , x  - x x x  + x x x  + x x x  - 2x x x  - x x x , x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x , x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x , x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x })
│ │ │            (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )  300007  0   6     2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -156,44 +156,44 @@
│ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.00996664s (cpu); 0.00997121s (thread); 0s (gc)
│ │ │ + -- used 0.012497s (cpu); 0.0125013s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time degreeMap psi
│ │ │ - -- used 0.327999s (cpu); 0.203484s (thread); 0s (gc)
│ │ │ + -- used 0.618492s (cpu); 0.307599s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │              
    
│ │ │                
    i11 : time projectiveDegrees psi
│ │ │ - -- used 5.13464s (cpu); 4.44306s (thread); 0s (gc)
│ │ │ + -- used 6.6227s (cpu); 6.09155s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │          
    We repeat the example using the type RationalMap and using deterministic methods.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │        2
│ │ │ │  o6 = y T
│ │ │ │          2
│ │ │ │  
│ │ │ │  o6 : R[T ..T ]
│ │ │ │          1   3
│ │ │ │  i7 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.0160742s (cpu); 0.0146988s
│ │ │ │ +Finding easy relations           :  -- used 0.0627175s (cpu); 0.0219904s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = H
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Module.html
│ │ │ @@ -129,15 +129,15 @@
│ │ │  
│ │ │  o5 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.0022587s (cpu); 0.0022598s (thread); 0s (gc)
│ │ │ + -- used 0.00271443s (cpu); 0.00271919s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -242,15 +242,15 @@
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
    
│ │ │              
    
│ │ │                
    i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.153174s (cpu); 0.0865693s (thread); 0s (gc)
│ │ │ + -- used 0.212159s (cpu); 0.100868s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i14 : time phi^(-1)
│ │ │ - -- used 0.503127s (cpu); 0.422881s (thread); 0s (gc)
│ │ │ + -- used 0.490845s (cpu); 0.490851s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -376,49 +376,49 @@
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
    
│ │ │              
    
│ │ │                
    i15 : time degrees phi^(-1)
│ │ │ - -- used 0.313198s (cpu); 0.259718s (thread); 0s (gc)
│ │ │ + -- used 0.480484s (cpu); 0.36706s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │                
    i16 : time degrees phi
│ │ │ - -- used 0.0182114s (cpu); 0.0177219s (thread); 0s (gc)
│ │ │ + -- used 0.0323021s (cpu); 0.0202118s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
    
│ │ │              
    
│ │ │                
    i17 : time describe phi
│ │ │ - -- used 0.00315504s (cpu); 0.00315575s (thread); 0s (gc)
│ │ │ + -- used 0.00399366s (cpu); 0.00396264s (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 0.010093s (cpu); 0.0100937s (thread); 0s (gc)
│ │ │ + -- used 0.0121709s (cpu); 0.0121307s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ @@ -427,41 +427,41 @@
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.00979107s (cpu); 0.00979179s (thread); 0s (gc)
│ │ │ + -- used 0.0111708s (cpu); 0.0111818s (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 1.28655s (cpu); 0.923224s (thread); 0s (gc)
│ │ │ + -- used 1.28386s (cpu); 1.04453s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
    
│ │ │              
    
│ │ │                
    i22 : time degree f
│ │ │ - -- used 1.6732e-05s (cpu); 1.6331e-05s (thread); 0s (gc)
│ │ │ + -- used 2.8715e-05s (cpu); 2.459e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │              
    
│ │ │                
    i23 : time describe f
│ │ │ - -- used 0.00171978s (cpu); 0.00173089s (thread); 0s (gc)
│ │ │ + -- used 0.00173732s (cpu); 0.00174327s (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 0.00420313s (cpu); 0.00419957s (thread); 0s (gc)
│ │ │ │ + -- used 0.00579333s (cpu); 0.00579207s (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 0.122781s (cpu); 0.065672s (thread); 0s (gc)
│ │ │ │ + -- used 0.183994s (cpu); 0.0865513s (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 0.0292919s (cpu); 0.029296s (thread); 0s (gc)
│ │ │ │ + -- used 0.0361924s (cpu); 0.0362006s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.617907s (cpu); 0.436744s (thread); 0s (gc)
│ │ │ │ + -- used 0.742939s (cpu); 0.526145s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ │ - -- used 0.061967s (cpu); 0.0619758s (thread); 0s (gc)
│ │ │ │ + -- used 0.0730622s (cpu); 0.0730746s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.00229866s (cpu); 0.00229945s (thread); 0s (gc)
│ │ │ │ + -- used 0.00297404s (cpu); 0.00298069s (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 0.461077s (cpu); 0.392442s (thread); 0s (gc)
│ │ │ │ + -- used 0.4282s (cpu); 0.428211s (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 0.00996664s (cpu); 0.00997121s (thread); 0s (gc)
│ │ │ │ + -- used 0.012497s (cpu); 0.0125013s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.327999s (cpu); 0.203484s (thread); 0s (gc)
│ │ │ │ + -- used 0.618492s (cpu); 0.307599s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 5.13464s (cpu); 4.44306s (thread); 0s (gc)
│ │ │ │ + -- used 6.6227s (cpu); 6.09155s (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 0.0022587s (cpu); 0.0022598s (thread); 0s (gc)
│ │ │ │ + -- used 0.00271443s (cpu); 0.00271919s (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 0.153174s (cpu); 0.0865693s (thread); 0s (gc)
│ │ │ │ + -- used 0.212159s (cpu); 0.100868s (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 0.503127s (cpu); 0.422881s (thread); 0s (gc)
│ │ │ │ + -- used 0.490845s (cpu); 0.490851s (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 0.313198s (cpu); 0.259718s (thread); 0s (gc)
│ │ │ │ + -- used 0.480484s (cpu); 0.36706s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : time degrees phi
│ │ │ │ - -- used 0.0182114s (cpu); 0.0177219s (thread); 0s (gc)
│ │ │ │ + -- used 0.0323021s (cpu); 0.0202118s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : time describe phi
│ │ │ │ - -- used 0.00315504s (cpu); 0.00315575s (thread); 0s (gc)
│ │ │ │ + -- used 0.00399366s (cpu); 0.00396264s (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 0.010093s (cpu); 0.0100937s (thread); 0s (gc)
│ │ │ │ + -- used 0.0121709s (cpu); 0.0121307s (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 0.00979107s (cpu); 0.00979179s (thread); 0s (gc)
│ │ │ │ + -- used 0.0111708s (cpu); 0.0111818s (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 1.28655s (cpu); 0.923224s (thread); 0s (gc)
│ │ │ │ + -- used 1.28386s (cpu); 1.04453s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │ │  
│ │ │ │  o21 : List
│ │ │ │  i22 : time degree f
│ │ │ │ - -- used 1.6732e-05s (cpu); 1.6331e-05s (thread); 0s (gc)
│ │ │ │ + -- used 2.8715e-05s (cpu); 2.459e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 0.00171978s (cpu); 0.00173089s (thread); 0s (gc)
│ │ │ │ + -- used 0.00173732s (cpu); 0.00174327s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20
│ │ │ │  hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1,
│ │ │ │  1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2,
│ │ │ │  0},{2, 0})
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.out
│ │ │ @@ -155,15 +155,15 @@
│ │ │                                    2     2     2       2 2     2 2      2 2      2 2     2 2        2 2       2 2        2       2       2
│ │ │         Differential => {a, b, c, a T , b T , c T , a*b c T , b c T , -a b T , -a c T , b c T T , -a c T T , b c T T , -a T T , c T T , b T T }
│ │ │                                      1     2     3         1       4        6        5       3 4        3 5       2 4      1 7     3 7     2 7
│ │ │  
│ │ │  o16 : DGAlgebra
│ │ │  
│ │ │  i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0167421s (cpu); 0.0157323s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0401684s (cpu); 0.0194995s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │        Underlying algebra => R[S ..S ]
│ │ │                                 1   4
│ │ │        Differential => {a, b, c, d}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0208192s (cpu); 0.0197414s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0610767s (cpu); 0.027962s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i6 : describe HB
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │                                      2
│ │ │        Differential => {a, b, c, d, a T }
│ │ │                                        1
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │  
│ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.0198731s (cpu); 0.0185999s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0347516s (cpu); 0.0220718s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out
│ │ │ @@ -55,15 +55,15 @@
│ │ │                 {2} | 0 |
│ │ │                 {2} | 0 |
│ │ │                 {2} | 1 |
│ │ │  
│ │ │  o6 : ComplexMap
│ │ │  
│ │ │  i7 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0244752s (cpu); 0.022974s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0466916s (cpu); 0.0195647s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out
│ │ │ @@ -49,15 +49,15 @@
│ │ │                                1                                                             {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |     3
│ │ │                                                                                      
│ │ │                                                                                     2
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.117877s (cpu); 0.0777341s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.291163s (cpu); 0.0752399s (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 0.516265s (cpu); 0.47005s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.684938s (cpu); 0.669715s (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 0.0203978s (cpu); 0.0180659s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0368644s (cpu); 0.0239421s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.018462s (cpu); 0.0176619s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0753265s (cpu); 0.0302492s (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 0.0840126s (cpu); 0.0810399s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.451218s (cpu); 0.151345s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
│ │ │  
│ │ │  i9 : numgens HA
│ │ │  
│ │ │ @@ -114,15 +114,15 @@
│ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o15 = {1, 7, 7, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0524524s (cpu); 0.0506388s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.15493s (cpu); 0.0818177s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : R = ZZ/101[a,b,c,d]
│ │ │  
│ │ │ @@ -151,14 +151,14 @@
│ │ │         Underlying algebra => S[T ..T ]
│ │ │                                  1   4
│ │ │         Differential => {a, b, c, d}
│ │ │  
│ │ │  o20 : DGAlgebra
│ │ │  
│ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.0197566s (cpu); 0.018554s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.05998s (cpu); 0.0270712s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out
│ │ │ @@ -43,15 +43,15 @@
│ │ │  o6 = y T
│ │ │          2
│ │ │  
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
│ │ │  
│ │ │  i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0160742s (cpu); 0.0146988s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0627175s (cpu); 0.0219904s (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 0.174259s (cpu); 0.130484s (thread); 0s (gc)
│ │ │ + -- used 0.325197s (cpu); 0.217693s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
│ │ │  
│ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product.out
│ │ │ @@ -68,15 +68,15 @@
│ │ │                 2
│ │ │  o9 = (true, x y T T T  - x x y T T T )
│ │ │               2 2 1 2 3    1 2 2 2 3 4
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.532177s (cpu); 0.481269s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.807258s (cpu); 0.666616s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T T T T
│ │ │         1 2 2   2 3 4 5
│ │ │  
│ │ │  o10 : R[T ..T ]
│ │ │           1   5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -27,15 +27,15 @@
│ │ │                                 1   4
│ │ │        Differential => {t , t , t , t }
│ │ │                          1   2   3   4
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.144963s (cpu); 0.141058s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.236522s (cpu); 0.178883s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_tor__Algebra_lp__Ring_cm__Ring_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ - -- used 0.465825s (cpu); 0.404911s (thread); 0s (gc)
│ │ │ + -- used 0.713688s (cpu); 0.553672s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.html
│ │ │ @@ -289,15 +289,15 @@
│ │ │          
    
│ │ │            
    One can also obtain the map on homology induced by a DGAlgebra map.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0167421s (cpu); 0.0157323s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0401684s (cpu); 0.0194995s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -210,15 +210,15 @@
│ │ │ │  a c T , b c T T , -a c T T , b c T T , -a T T , c T T , b T T }
│ │ │ │                                      1     2     3         1       4        6
│ │ │ │  5       3 4        3 5       2 4      1 7     3 7     2 7
│ │ │ │  
│ │ │ │  o16 : DGAlgebra
│ │ │ │  One can also obtain the map on homology induced by a DGAlgebra map.
│ │ │ │  i17 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0167421s (cpu); 0.0157323s
│ │ │ │ +Finding easy relations           :  -- used 0.0401684s (cpu); 0.0194995s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                            ZZ
│ │ │ │                           ---[a..c]
│ │ │ │              ZZ           101
│ │ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │             101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │          
    
│ │ │            
    One can compute the homology algebra of a DGAlgebra using the homology (or HH) command.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -174,15 +174,15 @@
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │                                 1   4
│ │ │ │        Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  One can compute the homology algebra of a DGAlgebra using the homology (or HH)
│ │ │ │  command.
│ │ │ │  i5 : HB = HH B
│ │ │ │ -Finding easy relations           :  -- used 0.0208192s (cpu); 0.0197414s
│ │ │ │ +Finding easy relations           :  -- used 0.0610767s (cpu); 0.027962s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HB
│ │ │ │  
│ │ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i6 : describe HB
│ │ │ │  
│ │ │ │ @@ -87,15 +87,15 @@
│ │ │ │                                 1   5
│ │ │ │                                      2
│ │ │ │        Differential => {a, b, c, d, a T }
│ │ │ │                                        1
│ │ │ │  
│ │ │ │  o9 : DGAlgebra
│ │ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ │ -Finding easy relations           :  -- used 0.0198731s (cpu); 0.0185999s
│ │ │ │ +Finding easy relations           :  -- used 0.0347516s (cpu); 0.0220718s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │         ZZ
│ │ │ │  o10 = ---[X ..X ]
│ │ │ │        101  1   3
│ │ │ │  
│ │ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  
│ │ │  o6 : ComplexMap
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -242,15 +242,15 @@
│ │ │  
│ │ │  o15 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0208192s (cpu); 0.0197414s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0610767s (cpu); 0.027962s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.0198731s (cpu); 0.0185999s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0347516s (cpu); 0.0220718s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i7 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0244752s (cpu); 0.022974s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0466916s (cpu); 0.0195647s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,15 +62,15 @@
│ │ │ │       2 : R  <------------- R  : 2
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 0 |
│ │ │ │                 {2} | 1 |
│ │ │ │  
│ │ │ │  o6 : ComplexMap
│ │ │ │  i7 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0244752s (cpu); 0.022974s
│ │ │ │ +Finding easy relations           :  -- used 0.0466916s (cpu); 0.0195647s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                           ZZ
│ │ │ │                          ---[a..c]
│ │ │ │             ZZ           101
│ │ │ │  o7 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │            101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html
│ │ │ @@ -138,15 +138,15 @@
│ │ │          
    
│ │ │            
    Since the Koszul complex is a DG algebra, its homology is itself an algebra.  One can obtain this algebra using the command homology, homologyAlgebra, or HH (all commands work).  This algebra structure can detect whether or not the ring is a complete intersection or Gorenstein.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │  
│ │ │ │  o6 : Complex
│ │ │ │  Since the Koszul complex is a DG algebra, its homology is itself an algebra.
│ │ │ │  One can obtain this algebra using the command homology, homologyAlgebra, or HH
│ │ │ │  (all commands work). This algebra structure can detect whether or not the ring
│ │ │ │  is a complete intersection or Gorenstein.
│ │ │ │  i7 : HKR = HH KR
│ │ │ │ -Finding easy relations           :  -- used 0.117877s (cpu); 0.0777341s
│ │ │ │ +Finding easy relations           :  -- used 0.291163s (cpu); 0.0752399s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = HKR
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i8 : ideal HKR
│ │ │ │  
│ │ │ │ @@ -94,16 +94,16 @@
│ │ │ │  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 0.516265s (cpu); 0.47005s (thread);
│ │ │ │ -0s (gc)
│ │ │ │ +Finding easy relations           :  -- used 0.684938s (cpu); 0.669715s
│ │ │ │ +(thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = HKR'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : numgens HKR'
│ │ │ │  
│ │ │ │  o11 = 34
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_cycles.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0203978s (cpu); 0.0180659s
│ │ │ │ +Finding easy relations           :  -- used 0.0368644s (cpu); 0.0239421s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.117877s (cpu); 0.0777341s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.291163s (cpu); 0.0752399s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.516265s (cpu); 0.47005s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.684938s (cpu); 0.669715s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0203978s (cpu); 0.0180659s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0368644s (cpu); 0.0239421s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.018462s (cpu); 0.0176619s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0753265s (cpu); 0.0302492s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0840126s (cpu); 0.0810399s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.451218s (cpu); 0.151345s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
    
│ │ │              
    
│ │ │                
    i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0524524s (cpu); 0.0506388s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.15493s (cpu); 0.0818177s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
    
│ │ │ @@ -302,15 +302,15 @@
│ │ │  
│ │ │  o20 : DGAlgebra
    
│ │ │              
    
│ │ │                
    i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.0197566s (cpu); 0.018554s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.05998s (cpu); 0.0270712s (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 0.018462s (cpu); 0.0176619s
│ │ │ │ +Finding easy relations           :  -- used 0.0753265s (cpu); 0.0302492s
│ │ │ │  (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 0.0840126s (cpu); 0.0810399s
│ │ │ │ +Finding easy relations           :  -- used 0.451218s (cpu); 0.151345s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HA
│ │ │ │  
│ │ │ │  o8 : QuotientRing
│ │ │ │  i9 : numgens HA
│ │ │ │  
│ │ │ │ @@ -122,15 +122,15 @@
│ │ │ │  o14 : DGAlgebra
│ │ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o15 = {1, 7, 7, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0524524s (cpu); 0.0506388s
│ │ │ │ +Finding easy relations           :  -- used 0.15493s (cpu); 0.0818177s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = HA
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  One can check that HA has Poincare duality since R is Gorenstein.
│ │ │ │  If your DGAlgebra has generators in even degrees, then one must specify the
│ │ │ │ @@ -158,15 +158,15 @@
│ │ │ │  o20 = {Ring => S                      }
│ │ │ │         Underlying algebra => S[T ..T ]
│ │ │ │                                  1   4
│ │ │ │         Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o20 : DGAlgebra
│ │ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ │ -Finding easy relations           :  -- used 0.0197566s (cpu); 0.018554s
│ │ │ │ +Finding easy relations           :  -- used 0.05998s (cpu); 0.0270712s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = HB
│ │ │ │  
│ │ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  ********** WWaayyss ttoo uussee hhoommoollooggyyAAllggeebbrraa:: **********
│ │ │ │      * homologyAlgebra(DGAlgebra)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html
│ │ │ @@ -135,15 +135,15 @@
│ │ │  o6 : R[T ..T ]
│ │ │          1   3
    
│ │ │              
    
│ │ │                
    i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0160742s (cpu); 0.0146988s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0627175s (cpu); 0.0219904s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i6 : HKR = HH(KR)
│ │ │ - -- used 0.174259s (cpu); 0.130484s (thread); 0s (gc)
│ │ │ + -- used 0.325197s (cpu); 0.217693s (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 0.174259s (cpu); 0.130484s (thread); 0s (gc)
│ │ │ │ + -- used 0.325197s (cpu); 0.217693s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o6 = HKR
│ │ │ │  
│ │ │ │  o6 : QuotientRing
│ │ │ │  The following is the graded canonical module of R:
│ │ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product.html
│ │ │ @@ -192,15 +192,15 @@
│ │ │          
    
│ │ │            
    Given cycles z1,z2,z3 such that z1*z2 and z2*z3 are boundaries, the Massey triple product of the homology classes represented by z1,z2 and z3 is the homology class of lift12*z3 + z1*lift23.  To see this, we compute and check:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │        Underlying algebra => R[T ..T ]
│ │ │ │                                 1   4
│ │ │ │        Differential => {t , t , t , t }
│ │ │ │                          1   2   3   4
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.144963s (cpu); 0.141058s
│ │ │ │ +Finding easy relations           :  -- used 0.236522s (cpu); 0.178883s
│ │ │ │  (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
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  i2 : M = coker matrix {{a^3*b^3*c^3*d^3}};
│ │ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ │ - -- used 0.465825s (cpu); 0.404911s (thread); 0s (gc)
│ │ │ │ + -- used 0.713688s (cpu); 0.553672s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o4 = HB
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : numgens HB
│ │ │ │  
│ │ │ │  o5 = 35
│ │ ├── ./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     2   3     1   3   4   5
│ │ │ +                              2   3   4     2   5     1   3   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o3 : HyperGraph
│ │ │  
│ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              3   4   5     1   3     1   2   4   5
│ │ │ +                              2   4   5     1   5     1   2   3   4
│ │ │                  "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
│ │ │ @@ -97,15 +97,15 @@
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  i3 : T = {{a,b},{d,e}}
│ │ │ │  
│ │ │ │  o3 = {{a, b}, {d, e}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : gprime = deleteEdges (g,T)
│ │ │ │  
│ │ │ │ -o4 = HyperGraph{"edges" => {{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
│ │ │ @@ -88,28 +88,28 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,24 +25,24 @@
│ │ │ │  _T_i_m_e_L_i_m_i_t). The method will return null if it cannot find a hypergraph within
│ │ │ │  the branch and time limits.
│ │ │ │  i1 : R = QQ[x_1..x_5];
│ │ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │ │  
│ │ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ -                              2   4   5     2   3     1   3   4   5
│ │ │ │ +                              2   3   4     2   5     1   3   4   5
│ │ │ │                  "ring" => R
│ │ │ │                  "vertices" => {x , x , x , x , x }
│ │ │ │                                  1   2   3   4   5
│ │ │ │  
│ │ │ │  o3 : HyperGraph
│ │ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │ │  
│ │ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ -                              3   4   5     1   3     1   2   4   5
│ │ │ │ +                              2   4   5     1   5     1   2   3   4
│ │ │ │                  "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/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;
│ │ │ - -- .243242s elapsed
│ │ │ + -- .251877s elapsed
│ │ │  
│ │ │  i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/html/index.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,22 +29,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00269974s (cpu); 0.00269704s (thread); 0s (gc)
│ │ │ │ + -- used 0.00328374s (cpu); 0.00328124s (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 0.00166863s (cpu); 0.00166902s (thread); 0s (gc)
│ │ │ │ + -- used 0.00183563s (cpu); 0.00183797s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html
│ │ │ @@ -97,26 +97,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,22 +30,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00283303s (cpu); 0.00283058s (thread); 0s (gc)
│ │ │ │ + -- used 0.00330379s (cpu); 0.00329732s (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 0.00168679s (cpu); 0.00168707s (thread); 0s (gc)
│ │ │ │ + -- used 0.00185437s (cpu); 0.00185573s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -105,15 +105,15 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.532177s (cpu); 0.481269s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.807258s (cpu); 0.666616s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T T T T
│ │ │         1 2 2   2 3 4 5
│ │ │  
│ │ │  o10 : R[T ..T ]
│ │ │           1   5
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -90,15 +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 0.532177s (cpu); 0.481269s
│ │ │ │ +Finding easy relations           :  -- used 0.807258s (cpu); 0.666616s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2
│ │ │ │  o10 = x x y z T T T T
│ │ │ │         1 2 2   2 3 4 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
│ │ │ @@ -119,15 +119,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
    
│ │ │                
    i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.144963s (cpu); 0.141058s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.236522s (cpu); 0.178883s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ - -- used 0.465825s (cpu); 0.404911s (thread); 0s (gc)
│ │ │ + -- used 0.713688s (cpu); 0.553672s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
    
│ │ │                
    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     2   3     1   3   4   5
│ │ │ +                              2   3   4     2   5     1   3   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o3 : HyperGraph
    
│ │ │              
    
│ │ │                
    i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              3   4   5     1   3     1   2   4   5
│ │ │ +                              2   4   5     1   5     1   2   3   4
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .243242s elapsed
    
│ │ │ + -- .251877s 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;
│ │ │ │ - -- .243242s elapsed
│ │ │ │ + -- .251877s elapsed
│ │ │ │  i4 : #sols -- 156 solutions
│ │ │ │  
│ │ │ │  o4 = 156
│ │ │ │  The authors would like to acknowledge the June 2020 Macaulay2 workshop held
│ │ │ │  virtually at Warwick, where this package was first developed.
│ │ │ │  RReeffeerreenncceess:
│ │ │ │      * [1] Sturmfels, Bernd. Solving systems of polynomial equations. No. 97.
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -17,23 +17,23 @@
│ │ │  
│ │ │        2
│ │ │  o3 = x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00269974s (cpu); 0.00269704s (thread); 0s (gc)
│ │ │ + -- used 0.00328374s (cpu); 0.00328124s (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 0.00166863s (cpu); 0.00166902s (thread); 0s (gc)
│ │ │ + -- used 0.00183563s (cpu); 0.00183797s (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 0.00283303s (cpu); 0.00283058s (thread); 0s (gc)
│ │ │ + -- used 0.00330379s (cpu); 0.00329732s (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 0.00168679s (cpu); 0.00168707s (thread); 0s (gc)
│ │ │ + -- used 0.00185437s (cpu); 0.00185573s (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 1.635s (cpu); 1.364s (thread); 0s (gc)
│ │ │ + -- used 1.52981s (cpu); 1.33959s (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 0.0160364s (cpu); 0.0160384s (thread); 0s (gc)
│ │ │ + -- used 0.0175569s (cpu); 0.0175622s (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 1.60254s (cpu); 1.38345s (thread); 0s (gc)
│ │ │ + -- used 1.58104s (cpu); 1.4079s (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 0.015046s (cpu); 0.0150471s (thread); 0s (gc)
│ │ │ + -- used 0.0154346s (cpu); 0.0154363s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -103,26 +103,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00269974s (cpu); 0.00269704s (thread); 0s (gc)
│ │ │ + -- used 0.00328374s (cpu); 0.00328124s (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 0.00166863s (cpu); 0.00166902s (thread); 0s (gc)
│ │ │ + -- used 0.00183563s (cpu); 0.00183797s (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 0.00283303s (cpu); 0.00283058s (thread); 0s (gc)
│ │ │ + -- used 0.00330379s (cpu); 0.00329732s (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 0.00168679s (cpu); 0.00168707s (thread); 0s (gc)
│ │ │ + -- used 0.00185437s (cpu); 0.00185573s (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 1.635s (cpu); 1.364s (thread); 0s (gc)
│ │ │ + -- used 1.52981s (cpu); 1.33959s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -164,15 +164,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0160364s (cpu); 0.0160384s (thread); 0s (gc)
│ │ │ + -- used 0.0175569s (cpu); 0.0175622s (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 1.635s (cpu); 1.364s (thread); 0s (gc)
│ │ │ │ + -- used 1.52981s (cpu); 1.33959s (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 0.0160364s (cpu); 0.0160384s (thread); 0s (gc)
│ │ │ │ + -- used 0.0175569s (cpu); 0.0175622s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o4 : R
    
│ │ │              
    
│ │ │                
    i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.60254s (cpu); 1.38345s (thread); 0s (gc)
│ │ │ + -- used 1.58104s (cpu); 1.4079s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -163,15 +163,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.015046s (cpu); 0.0150471s (thread); 0s (gc)
│ │ │ + -- used 0.0154346s (cpu); 0.0154363s (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 1.60254s (cpu); 1.38345s (thread); 0s (gc)
│ │ │ │ + -- used 1.58104s (cpu); 1.4079s (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 0.015046s (cpu); 0.0150471s (thread); 0s (gc)
│ │ │ │ + -- used 0.0154346s (cpu); 0.0154363s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1331975673177
│ │ │  
│ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.0277008s (cpu); 0.0277012s (thread); 0s (gc)
│ │ │ + -- used 0.0326255s (cpu); 0.0326274s (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 0.335663s (cpu); 0.283945s (thread); 0s (gc)
│ │ │ + -- used 0.380507s (cpu); 0.31082s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time rationalCurve(3)
│ │ │ - -- used 0.213143s (cpu); 0.160223s (thread); 0s (gc)
│ │ │ + -- used 0.133618s (cpu); 0.133629s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
│ │ │  
│ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.41543s (cpu); 4.71039s (thread); 0s (gc)
│ │ │ + -- used 5.33961s (cpu); 4.55825s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.2255s (cpu); 0.170509s (thread); 0s (gc)
│ │ │ + -- used 0.137383s (cpu); 0.137392s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
│ │ │  
│ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 5.77039s (cpu); 5.00326s (thread); 0s (gc)
│ │ │ + -- used 5.56438s (cpu); 4.71817s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time rationalCurve(4)
│ │ │ - -- used 1.91616s (cpu); 1.70276s (thread); 0s (gc)
│ │ │ + -- used 1.55765s (cpu); 1.38141s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.19116s (cpu); 5.70109s (thread); 0s (gc)
│ │ │ + -- used 7.62831s (cpu); 6.12484s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.65688s (cpu); 1.4475s (thread); 0s (gc)
│ │ │ + -- used 1.72982s (cpu); 1.47984s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
│ │ │  
│ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 7.26424s (cpu); 5.93744s (thread); 0s (gc)
│ │ │ + -- used 7.42048s (cpu); 5.9041s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
│ │ │  
│ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.25779s (cpu); 5.7832s (thread); 0s (gc)
│ │ │ + -- used 7.56335s (cpu); 6.06623s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │            
    Computes the number of lines on a general hypersurface of degree 2n - 3 in \mathbb P^n.
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the number of lines on a general hypersurface of degree
│ │ │ │              2n - 3 in \mathbb P^n
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Computes the number of lines on a general hypersurface of degree 2n - 3 in
│ │ │ │  \mathbb P^n.
│ │ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ │ - -- used 0.0277008s (cpu); 0.0277012s (thread); 0s (gc)
│ │ │ │ + -- used 0.0326255s (cpu); 0.0326274s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       289139638632755625, 520764738758073845321}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  ********** WWaayyss ttoo uussee lliinneessHHyyppeerrssuurrffaaccee:: **********
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html
│ │ │ @@ -152,15 +152,15 @@
│ │ │            
    The numbers of conics on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.0277008s (cpu); 0.0277012s (thread); 0s (gc)
│ │ │ + -- used 0.0326255s (cpu); 0.0326274s (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 0.335663s (cpu); 0.283945s (thread); 0s (gc)
│ │ │ + -- used 0.380507s (cpu); 0.31082s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ @@ -168,27 +168,27 @@
│ │ │            
    For rational curves of degree 3:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -198,15 +198,15 @@
│ │ │            
    The number of rational curves of degree 3 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i8 : time rationalCurve(3)
│ │ │ - -- used 0.213143s (cpu); 0.160223s (thread); 0s (gc)
│ │ │ + -- used 0.133618s (cpu); 0.133629s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
    
│ │ │              
    
│ │ │                
    i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 5.41543s (cpu); 4.71039s (thread); 0s (gc)
│ │ │ + -- used 5.33961s (cpu); 4.55825s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.2255s (cpu); 0.170509s (thread); 0s (gc)
│ │ │ + -- used 0.137383s (cpu); 0.137392s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
    
│ │ │              
    
│ │ │ @@ -214,15 +214,15 @@
│ │ │            
    The numbers of rational curves of degree 3 on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 5.77039s (cpu); 5.00326s (thread); 0s (gc)
│ │ │ + -- used 5.56438s (cpu); 4.71817s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │ @@ -230,27 +230,27 @@
│ │ │            
    For rational curves of degree 4:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time rationalCurve(4)
│ │ │ - -- used 1.91616s (cpu); 1.70276s (thread); 0s (gc)
│ │ │ + -- used 1.55765s (cpu); 1.38141s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    
│ │ │              
    
│ │ │                
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 7.19116s (cpu); 5.70109s (thread); 0s (gc)
│ │ │ + -- used 7.62831s (cpu); 6.12484s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
    
│ │ │              
    
│ │ │ @@ -258,15 +258,15 @@
│ │ │            
    The number of rational curves of degree 4 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 1.65688s (cpu); 1.4475s (thread); 0s (gc)
│ │ │ + -- used 1.72982s (cpu); 1.47984s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
    
│ │ │              
    
│ │ │ @@ -274,25 +274,25 @@
│ │ │            
    The numbers of rational curves of degree 4 on general complete intersections of types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 7.26424s (cpu); 5.93744s (thread); 0s (gc)
│ │ │ + -- used 7.42048s (cpu); 5.9041s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
    
│ │ │              
    
│ │ │                
    i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 7.25779s (cpu); 5.7832s (thread); 0s (gc)
│ │ │ + -- used 7.56335s (cpu); 6.06623s (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 0.335663s (cpu); 0.283945s (thread); 0s (gc)
│ │ │ │ + -- used 0.380507s (cpu); 0.31082s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  For rational curves of degree 3:
│ │ │ │  i8 : time rationalCurve(3)
│ │ │ │ - -- used 0.213143s (cpu); 0.160223s (thread); 0s (gc)
│ │ │ │ + -- used 0.133618s (cpu); 0.133629s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       8564575000
│ │ │ │  o8 = ----------
│ │ │ │           27
│ │ │ │  
│ │ │ │  o8 : QQ
│ │ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ │ - -- used 5.41543s (cpu); 4.71039s (thread); 0s (gc)
│ │ │ │ + -- used 5.33961s (cpu); 4.55825s (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 0.2255s (cpu); 0.170509s (thread); 0s (gc)
│ │ │ │ + -- used 0.137383s (cpu); 0.137392s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 317206375
│ │ │ │  
│ │ │ │  o10 : QQ
│ │ │ │  The numbers of rational curves of degree 3 on general complete intersection
│ │ │ │  Calabi-Yau threefolds can be computed as follows:
│ │ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ │ - -- used 5.77039s (cpu); 5.00326s (thread); 0s (gc)
│ │ │ │ + -- used 5.56438s (cpu); 4.71817s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  For rational curves of degree 4:
│ │ │ │  i12 : time rationalCurve(4)
│ │ │ │ - -- used 1.91616s (cpu); 1.70276s (thread); 0s (gc)
│ │ │ │ + -- used 1.55765s (cpu); 1.38141s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        15517926796875
│ │ │ │  o12 = --------------
│ │ │ │              64
│ │ │ │  
│ │ │ │  o12 : QQ
│ │ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ │ - -- used 7.19116s (cpu); 5.70109s (thread); 0s (gc)
│ │ │ │ + -- used 7.62831s (cpu); 6.12484s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = 3883914084
│ │ │ │  
│ │ │ │  o13 : QQ
│ │ │ │  The number of rational curves of degree 4 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ │ - -- used 1.65688s (cpu); 1.4475s (thread); 0s (gc)
│ │ │ │ + -- used 1.72982s (cpu); 1.47984s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = 242467530000
│ │ │ │  
│ │ │ │  o14 : QQ
│ │ │ │  The numbers of rational curves of degree 4 on general complete intersections of
│ │ │ │  types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
│ │ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ │ - -- used 7.26424s (cpu); 5.93744s (thread); 0s (gc)
│ │ │ │ + -- used 7.42048s (cpu); 5.9041s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3883902528
│ │ │ │  
│ │ │ │  o15 : QQ
│ │ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ │ - -- used 7.25779s (cpu); 5.7832s (thread); 0s (gc)
│ │ │ │ + -- used 7.56335s (cpu); 6.06623s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 1139448384
│ │ │ │  
│ │ │ │  o16 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee rraattiioonnaallCCuurrvvee:: **********
│ │ │ │      * rationalCurve(ZZ)
│ │ │ │      * rationalCurve(ZZ,List)
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out
│ │ │ @@ -10,34 +10,34 @@
│ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │                    1   1 0   1 0   0
│ │ │  
│ │ │  o3 : RingMap R <-- S
│ │ │  
│ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .00212025 seconds
│ │ │ -     -- used .000581841 seconds
│ │ │ +     -- used .00262266 seconds
│ │ │ +     -- used .000693569 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .0034391 seconds
│ │ │ -     -- used .0043489 seconds
│ │ │ +     -- used .00380338 seconds
│ │ │ +     -- used .00505185 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00888019 seconds
│ │ │ -     -- used .0267665 seconds
│ │ │ +     -- used .00890672 seconds
│ │ │ +     -- used .0315301 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0185795 seconds
│ │ │ -     -- used .207487 seconds
│ │ │ +     -- used .0200075 seconds
│ │ │ +     -- used .255691 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .04064 seconds
│ │ │ -     -- used .812288 seconds
│ │ │ +     -- used .0439567 seconds
│ │ │ +     -- used .968007 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html
│ │ │ @@ -101,34 +101,34 @@
│ │ │  o3 : RingMap R <-- S
    
│ │ │              
    
│ │ │                
    i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .00212025 seconds
│ │ │ -     -- used .000581841 seconds
│ │ │ +     -- used .00262266 seconds
│ │ │ +     -- used .000693569 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .0034391 seconds
│ │ │ -     -- used .0043489 seconds
│ │ │ +     -- used .00380338 seconds
│ │ │ +     -- used .00505185 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00888019 seconds
│ │ │ -     -- used .0267665 seconds
│ │ │ +     -- used .00890672 seconds
│ │ │ +     -- used .0315301 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0185795 seconds
│ │ │ -     -- used .207487 seconds
│ │ │ +     -- used .0200075 seconds
│ │ │ +     -- used .255691 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .04064 seconds
│ │ │ -     -- used .812288 seconds
│ │ │ +     -- used .0439567 seconds
│ │ │ +     -- used .968007 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 .00212025 seconds
│ │ │ │ -     -- used .000581841 seconds
│ │ │ │ +     -- used .00262266 seconds
│ │ │ │ +     -- used .000693569 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .0034391 seconds
│ │ │ │ -     -- used .0043489 seconds
│ │ │ │ +     -- used .00380338 seconds
│ │ │ │ +     -- used .00505185 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00888019 seconds
│ │ │ │ -     -- used .0267665 seconds
│ │ │ │ +     -- used .00890672 seconds
│ │ │ │ +     -- used .0315301 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0185795 seconds
│ │ │ │ -     -- used .207487 seconds
│ │ │ │ +     -- used .0200075 seconds
│ │ │ │ +     -- used .255691 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .04064 seconds
│ │ │ │ -     -- used .812288 seconds
│ │ │ │ +     -- used .0439567 seconds
│ │ │ │ +     -- used .968007 seconds
│ │ │ │  (49, 217)
│ │ │ │  
│ │ │ │                                     2
│ │ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out
│ │ │ @@ -462,50 +462,50 @@
│ │ │                 3 2 4     3 6
│ │ │  o27 = ideal(12x x x  - 4x x )
│ │ │                 3 7 9     3 9
│ │ │  
│ │ │  o27 : Ideal of S
│ │ │  
│ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.176156s (cpu); 0.124654s (thread); 0s (gc)
│ │ │ + -- used 0.234994s (cpu); 0.156187s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
│ │ │  
│ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.305542s (cpu); 0.209068s (thread); 0s (gc)
│ │ │ + -- used 0.409906s (cpu); 0.237767s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
│ │ │  
│ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.444035s (cpu); 0.30863s (thread); 0s (gc)
│ │ │ + -- used 0.629586s (cpu); 0.371551s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
│ │ │  
│ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.192411s (cpu); 0.164502s (thread); 0s (gc)
│ │ │ + -- used 0.295396s (cpu); 0.214497s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
│ │ │  
│ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.345276s (cpu); 0.206372s (thread); 0s (gc)
│ │ │ + -- used 0.495143s (cpu); 0.245083s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
│ │ │  
│ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.33846s (cpu); 0.224657s (thread); 0s (gc)
│ │ │ + -- used 0.4256s (cpu); 0.268384s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
│ │ │  
│ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.272654s (cpu); 0.18907s (thread); 0s (gc)
│ │ │ + -- used 0.395062s (cpu); 0.236132s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
│ │ │  
│ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 15.6451s (cpu); 11.048s (thread); 0s (gc)
│ │ │ + -- used 20.1925s (cpu); 12.4113s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
│ │ │  
│ │ │  i36 : peek StrategyDefault
│ │ │  
│ │ │  o36 = OptionTable{GRevLexLargest => 0      }
│ │ │                    GRevLexSmallest => 16
│ │ │ @@ -514,15 +514,15 @@
│ │ │                    LexSmallest => 16
│ │ │                    LexSmallestTerm => 16
│ │ │                    Points => 0
│ │ │                    Random => 16
│ │ │                    RandomNonzero => 16
│ │ │  
│ │ │  i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.36572s (cpu); 0.330934s (thread); 0s (gc)
│ │ │ + -- used 0.47935s (cpu); 0.391832s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -582,15 +582,15 @@
│ │ │  i41 : ptsStratGeometric = new OptionTable from (options chooseGoodMinors)#PointOptions;
│ │ │  
│ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │  
│ │ │  o42 = true
│ │ │  
│ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.499388s (cpu); 0.444367s (thread); 0s (gc)
│ │ │ + -- used 0.733144s (cpu); 0.650783s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
│ │ │  
│ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │  
│ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │                    DimensionFunction => dim
│ │ │ @@ -605,47 +605,47 @@
│ │ │  o44 : OptionTable
│ │ │  
│ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │  
│ │ │  o45 = false
│ │ │  
│ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.456658s (cpu); 0.361956s (thread); 0s (gc)
│ │ │ + -- used 0.673375s (cpu); 0.489117s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
│ │ │  
│ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 3.39152s (cpu); 3.05769s (thread); 0s (gc)
│ │ │ + -- used 4.40716s (cpu); 3.86968s (thread); 0s (gc)
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 0.771095s (cpu); 0.681377s (thread); 0s (gc)
│ │ │ + -- used 0.989466s (cpu); 0.814996s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
│ │ │  
│ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.84296s (cpu); 2.62077s (thread); 0s (gc)
│ │ │ + -- used 3.48102s (cpu); 3.20736s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.29048s (cpu); 1.9516s (thread); 0s (gc)
│ │ │ + -- used 3.06145s (cpu); 2.42175s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.869374s (cpu); 0.790991s (thread); 0s (gc)
│ │ │ + -- used 0.965885s (cpu); 0.880369s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
│ │ │  
│ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.73026s (cpu); 2.30339s (thread); 0s (gc)
│ │ │ + -- used 3.46726s (cpu); 2.77333s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.16893s (cpu); 2.7704s (thread); 0s (gc)
│ │ │ + -- used 4.12878s (cpu); 3.39966s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 9.08616s (cpu); 7.66782s (thread); 0s (gc)
│ │ │ + -- used 12.3758s (cpu); 9.91985s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
│ │ │  
│ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 7.23875s (cpu); 6.02045s (thread); 0s (gc)
│ │ │ + -- used 9.54822s (cpu); 7.61112s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
│ │ │  
│ │ │  i56 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  o2 : Ideal of S
│ │ │  
│ │ │  i3 : dim (S/J)
│ │ │  
│ │ │  o3 = 4
│ │ │  
│ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 0.864231s (cpu); 0.611071s (thread); 0s (gc)
│ │ │ + -- used 1.32195s (cpu); 0.842862s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 11.0835s (cpu); 8.33071s (thread); 0s (gc)
│ │ │ + -- used 14.149s (cpu); 9.44792s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time regularInCodimension(1, S/J, Verbose=>true)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 452.908 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ @@ -87,21 +87,21 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 49, and computed = 39
│ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ -regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.3211s (cpu); 0.975451s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.82316s (cpu); 1.18293s (thread); 0s (gc)
│ │ │  d = 39.  singular locus dimension appears to be = 2
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ - -- used 0.182671s (cpu); 0.133167s (thread); 0s (gc)
│ │ │ + -- used 0.221987s (cpu); 0.147568s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -115,15 +115,15 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom, Verbose=>true)
│ │ │ - -- used 0.135093s (cpu); 0.115447s (thread); 0s (gc)
│ │ │ + -- used 0.309692s (cpu); 0.236845s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -137,15 +137,15 @@
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t->3, Verbose=>true)
│ │ │ - -- used 0.586765s (cpu); 0.44115s (thread); 0s (gc)
│ │ │ + -- used 0.766929s (cpu); 0.525284s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -165,15 +165,15 @@
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 10, and computed = 10.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t->t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ - -- used 0.650722s (cpu); 0.488427s (thread); 0s (gc)
│ │ │ + -- used 0.905427s (cpu); 0.596929s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ @@ -214,15 +214,15 @@
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 25, and computed = 23
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 25, and computed = 23.  singular locus dimension appears to be = 3
│ │ │  
│ │ │  i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim => true, Verbose=>true)
│ │ │ - -- used 0.441839s (cpu); 0.319692s (thread); 0s (gc)
│ │ │ + -- used 0.583593s (cpu); 0.361698s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5509279875405941999
│ │ │  
│ │ │  i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
│ │ │  
│ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 1.59228s elapsed
│ │ │ + -- 1.56285s 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)
│ │ │ - -- 1.02929s elapsed
│ │ │ + -- .952057s 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 0.000336902s (cpu); 0.00243334s (thread); 0s (gc)
│ │ │ + -- used 0.00402664s (cpu); 0.00336435s (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 0.000522079s (cpu); 0.00238391s (thread); 0s (gc)
│ │ │ + -- used 0.00331098s (cpu); 0.00300599s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 0.00114405s (cpu); 0.00222255s (thread); 0s (gc)
│ │ │ + -- used 0.00223138s (cpu); 0.00292183s (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 0.221015s (cpu); 0.155202s (thread); 0s (gc)
│ │ │ + -- used 0.389228s (cpu); 0.231021s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0111545s (cpu); 0.0139544s (thread); 0s (gc)
│ │ │ + -- used 0.0132883s (cpu); 0.0154692s (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 0.793132s (cpu); 0.746832s (thread); 0s (gc)
│ │ │ + -- used 0.555063s (cpu); 0.480218s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.47691s (cpu); 1.27617s (thread); 0s (gc)
│ │ │ + -- used 1.4366s (cpu); 1.27504s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : I1 == I2
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out
│ │ │ @@ -17,44 +17,44 @@
│ │ │  i6 : S = T/I;
│ │ │  
│ │ │  i7 : dim S
│ │ │  
│ │ │  o7 = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.694973s (cpu); 0.541805s (thread); 0s (gc)
│ │ │ + -- used 0.834255s (cpu); 0.616431s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 6.89948s (cpu); 5.27952s (thread); 0s (gc)
│ │ │ + -- used 8.16578s (cpu); 5.66619s (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 0.0210586s (cpu); 0.0202034s (thread); 0s (gc)
│ │ │ + -- used 0.0200381s (cpu); 0.0217863s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.182896s (cpu); 0.137715s (thread); 0s (gc)
│ │ │ + -- used 0.235151s (cpu); 0.157628s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.982565s (cpu); 0.690681s (thread); 0s (gc)
│ │ │ + -- used 1.25257s (cpu); 0.740226s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 1.38818s (cpu); 0.988524s (thread); 0s (gc)
│ │ │ + -- used 1.71559s (cpu); 1.05506s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
│ │ │  
│ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 327.599 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -386,15 +386,15 @@
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ -internalChooseMinor: Ch -- used 6.66371s (cpu); 5.28322s (thread); 0s (gc)
│ │ │ +internalChooseMinor: Ch -- used 8.60343s (cpu); 6.06955s (thread); 0s (gc)
│ │ │  oosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -430,15 +430,15 @@
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 328, and computed = 180
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 328, and computed = 180.  singular locus dimension appears to be = 1
│ │ │  
│ │ │  i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ - -- used 1.27921s (cpu); 1.02648s (thread); 0s (gc)
│ │ │ + -- used 1.72484s (cpu); 1.24048s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ @@ -490,59 +490,59 @@
│ │ │  i18 : StrategyCurrent#Random = 0;
│ │ │  
│ │ │  i19 : StrategyCurrent#LexSmallest = 100;
│ │ │  
│ │ │  i20 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │  
│ │ │  i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.308707s (cpu); 0.21708s (thread); 0s (gc)
│ │ │ + -- used 0.44499s (cpu); 0.275425s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.122371s (cpu); 0.0749652s (thread); 0s (gc)
│ │ │ + -- used 0.176976s (cpu); 0.0962802s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.373838s (cpu); 0.285558s (thread); 0s (gc)
│ │ │ + -- used 0.536569s (cpu); 0.36789s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.66577s (cpu); 1.25232s (thread); 0s (gc)
│ │ │ + -- used 2.34674s (cpu); 1.59893s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │  
│ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │  
│ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.2665s (cpu); 1.66206s (thread); 0s (gc)
│ │ │ + -- used 3.0589s (cpu); 2.00445s (thread); 0s (gc)
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.34659s (cpu); 1.70143s (thread); 0s (gc)
│ │ │ + -- used 3.0428s (cpu); 1.93984s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
│ │ │  
│ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.408217s (cpu); 0.307796s (thread); 0s (gc)
│ │ │ + -- used 0.53559s (cpu); 0.387166s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.711845s (cpu); 0.560503s (thread); 0s (gc)
│ │ │ + -- used 0.934341s (cpu); 0.689238s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 0.994761s (cpu); 0.81258s (thread); 0s (gc)
│ │ │ + -- used 1.32284s (cpu); 1.00247s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.66547s (cpu); 1.33943s (thread); 0s (gc)
│ │ │ + -- used 2.15611s (cpu); 1.60242s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
│ │ │  
│ │ │  i33 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html
│ │ │ @@ -620,71 +620,71 @@
│ │ │          
    
│ │ │            
    Here the $1$ passed to the function says how many minors to compute.  For instance, let's compute 8 minors for each of these strategies and see if that was enough to verify that the ring is regular in codimension 1.  In other words, if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/J$ has dimension 3).
    
│ │ │          
    
│ │ │          
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│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.176156s (cpu); 0.124654s (thread); 0s (gc)
│ │ │ + -- used 0.234994s (cpu); 0.156187s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
    
│ │ │              
    
│ │ │                
    i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.305542s (cpu); 0.209068s (thread); 0s (gc)
│ │ │ + -- used 0.409906s (cpu); 0.237767s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
    
│ │ │              
    
│ │ │                
    i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.444035s (cpu); 0.30863s (thread); 0s (gc)
│ │ │ + -- used 0.629586s (cpu); 0.371551s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
    
│ │ │              
    
│ │ │                
    i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.192411s (cpu); 0.164502s (thread); 0s (gc)
│ │ │ + -- used 0.295396s (cpu); 0.214497s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
    
│ │ │              
    
│ │ │                
    i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.345276s (cpu); 0.206372s (thread); 0s (gc)
│ │ │ + -- used 0.495143s (cpu); 0.245083s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
    
│ │ │              
    
│ │ │                
    i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.33846s (cpu); 0.224657s (thread); 0s (gc)
│ │ │ + -- used 0.4256s (cpu); 0.268384s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
    
│ │ │              
    
│ │ │                
    i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.272654s (cpu); 0.18907s (thread); 0s (gc)
│ │ │ + -- used 0.395062s (cpu); 0.236132s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
    
│ │ │              
    
│ │ │                
    i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 15.6451s (cpu); 11.048s (thread); 0s (gc)
│ │ │ + -- used 20.1925s (cpu); 12.4113s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Indeed, in this example, even computing determinants of 1,000 random submatrices is not typically enough to verify that $V(J)$ is regular in codimension 1.  On the other hand, Points is almost always quite effective at finding valuable submatrices, but can be quite slow.  In this particular example, we can see that LexSmallestTerm also performs very well (and does it quickly). Since different strategies work better or worse on different examples, the default strategy actually mixes and matches various strategies.  The default strategy, which we now elucidate,
    
│ │ │ @@ -709,15 +709,15 @@
│ │ │          
    
│ │ │            
    says that we should use GRevLexSmallest, GRevLexSmallestTerm, LexSmallest, LexSmallestTerm, Random, RandomNonzero all with equal probability (note RandomNonzero, which we have not yet discussed chooses random submatrices where no row or column is zero, which is good for working in sparse matrices).  For instance, if we run:
    
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│ │ │          
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│ │ │              
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│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault, Verbose=>true);
│ │ │ - -- used 0.36572s (cpu); 0.330934s (thread); 0s (gc)
│ │ │ + -- used 0.47935s (cpu); 0.391832s (thread); 0s (gc)
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -820,15 +820,15 @@
│ │ │  
│ │ │  o42 = true
    
│ │ │              
    
│ │ │                
    i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.499388s (cpu); 0.444367s (thread); 0s (gc)
│ │ │ + -- used 0.733144s (cpu); 0.650783s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
    
│ │ │              
    
│ │ │                
    i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │ @@ -852,15 +852,15 @@
│ │ │  
│ │ │  o45 = false
    
│ │ │              
    
│ │ │                
    i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.456658s (cpu); 0.361956s (thread); 0s (gc)
│ │ │ + -- used 0.673375s (cpu); 0.489117s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Other options may also be passed to the RandomPoints package via the PointOptions option.
    
│ │ │ @@ -868,69 +868,69 @@
│ │ │          
    
│ │ │            
    regularInCodimension:  It is reasonable to think that you should find a few minors (with one strategy or another), and see if perhaps the minors you have computed so far are enough to verify our ring is regular in codimension 1.  This is exactly what regularInCodimension does.  One can control at a fine level how frequently new minors are computed, and how frequently the dimension of what we have computed so far is checked, by the option codimCheckFunction.  For more on that, see RegularInCodimensionTutorial and regularInCodimension.  Let us finish running regularInCodimension on our example with several different strategies.
    
│ │ │          
    
│ │ │          
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│ │ │              
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│ │ │              
│ │ │            
│ │ │            
│ │ │              
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│ │ │              
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│ │ │              
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│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 3.39152s (cpu); 3.05769s (thread); 0s (gc)
    
│ │ │ + -- used 4.40716s (cpu); 3.86968s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 0.771095s (cpu); 0.681377s (thread); 0s (gc)
│ │ │ + -- used 0.989466s (cpu); 0.814996s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
    
│ │ │              
    
│ │ │                
    i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.84296s (cpu); 2.62077s (thread); 0s (gc)
    
│ │ │ + -- used 3.48102s (cpu); 3.20736s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.29048s (cpu); 1.9516s (thread); 0s (gc)
    
│ │ │ + -- used 3.06145s (cpu); 2.42175s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.869374s (cpu); 0.790991s (thread); 0s (gc)
│ │ │ + -- used 0.965885s (cpu); 0.880369s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
    
│ │ │              
    
│ │ │                
    i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.73026s (cpu); 2.30339s (thread); 0s (gc)
    
│ │ │ + -- used 3.46726s (cpu); 2.77333s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.16893s (cpu); 2.7704s (thread); 0s (gc)
    
│ │ │ + -- used 4.12878s (cpu); 3.39966s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 9.08616s (cpu); 7.66782s (thread); 0s (gc)
│ │ │ + -- used 12.3758s (cpu); 9.91985s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
    
│ │ │              
    
│ │ │                
    i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 7.23875s (cpu); 6.02045s (thread); 0s (gc)
│ │ │ + -- used 9.54822s (cpu); 7.61112s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If regularInCodimension outputs nothing, then it couldn't verify that the ring was regular in that codimension.  We set MaxMinors => 100 to keep it from running too long with an ineffective strategy.  Again, even though GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular example, in others they perform better than other strategies.  Note similar considerations also apply to projDim.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -486,44 +486,44 @@
│ │ │ │  o27 : Ideal of S
│ │ │ │  Here the $1$ passed to the function says how many minors to compute. For
│ │ │ │  instance, let's compute 8 minors for each of these strategies and see if that
│ │ │ │  was enough to verify that the ring is regular in codimension 1. In other words,
│ │ │ │  if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/
│ │ │ │  J$ has dimension 3).
│ │ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ │ - -- used 0.176156s (cpu); 0.124654s (thread); 0s (gc)
│ │ │ │ + -- used 0.234994s (cpu); 0.156187s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 2
│ │ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ │ - -- used 0.305542s (cpu); 0.209068s (thread); 0s (gc)
│ │ │ │ + -- used 0.409906s (cpu); 0.237767s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = 3
│ │ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ │ - -- used 0.444035s (cpu); 0.30863s (thread); 0s (gc)
│ │ │ │ + -- used 0.629586s (cpu); 0.371551s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 1
│ │ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ │ - -- used 0.192411s (cpu); 0.164502s (thread); 0s (gc)
│ │ │ │ + -- used 0.295396s (cpu); 0.214497s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 2
│ │ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ │ - -- used 0.345276s (cpu); 0.206372s (thread); 0s (gc)
│ │ │ │ + -- used 0.495143s (cpu); 0.245083s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = 3
│ │ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J,
│ │ │ │  Strategy=>GRevLexSmallestTerm))
│ │ │ │ - -- used 0.33846s (cpu); 0.224657s (thread); 0s (gc)
│ │ │ │ + -- used 0.4256s (cpu); 0.268384s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o33 = 3
│ │ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ │ - -- used 0.272654s (cpu); 0.18907s (thread); 0s (gc)
│ │ │ │ + -- used 0.395062s (cpu); 0.236132s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o34 = 3
│ │ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ │ - -- used 15.6451s (cpu); 11.048s (thread); 0s (gc)
│ │ │ │ + -- used 20.1925s (cpu); 12.4113s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o35 = 1
│ │ │ │  Indeed, in this example, even computing determinants of 1,000 random
│ │ │ │  submatrices is not typically enough to verify that $V(J)$ is regular in
│ │ │ │  codimension 1. On the other hand, Points is almost always quite effective at
│ │ │ │  finding valuable submatrices, but can be quite slow. In this particular
│ │ │ │  example, we can see that LexSmallestTerm also performs very well (and does it
│ │ │ │ @@ -544,15 +544,15 @@
│ │ │ │  says that we should use GRevLexSmallest, GRevLexSmallestTerm, LexSmallest,
│ │ │ │  LexSmallestTerm, Random, RandomNonzero all with equal probability (note
│ │ │ │  RandomNonzero, which we have not yet discussed chooses random submatrices where
│ │ │ │  no row or column is zero, which is good for working in sparse matrices). For
│ │ │ │  instance, if we run:
│ │ │ │  i37 : time chooseGoodMinors(20, 6, M, J, Strategy=>StrategyDefault,
│ │ │ │  Verbose=>true);
│ │ │ │ - -- used 0.36572s (cpu); 0.330934s (thread); 0s (gc)
│ │ │ │ + -- used 0.47935s (cpu); 0.391832s (thread); 0s (gc)
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │ @@ -633,15 +633,15 @@
│ │ │ │  i41 : ptsStratGeometric = new OptionTable from (options
│ │ │ │  chooseGoodMinors)#PointOptions;
│ │ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │ │  
│ │ │ │  o42 = true
│ │ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratGeometric))
│ │ │ │ - -- used 0.499388s (cpu); 0.444367s (thread); 0s (gc)
│ │ │ │ + -- used 0.733144s (cpu); 0.650783s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o43 = 2
│ │ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that
│ │ │ │  value
│ │ │ │  
│ │ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │ │                    DimensionFunction => dim
│ │ │ │ @@ -655,58 +655,58 @@
│ │ │ │  
│ │ │ │  o44 : OptionTable
│ │ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │ │  
│ │ │ │  o45 = false
│ │ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratRational))
│ │ │ │ - -- used 0.456658s (cpu); 0.361956s (thread); 0s (gc)
│ │ │ │ + -- used 0.673375s (cpu); 0.489117s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = 2
│ │ │ │  Other options may also be passed to the _R_a_n_d_o_m_P_o_i_n_t_s package via the
│ │ │ │  _P_o_i_n_t_O_p_t_i_o_n_s option.
│ │ │ │  rreegguullaarrIInnCCooddiimmeennssiioonn:: It is reasonable to think that you should find a few
│ │ │ │  minors (with one strategy or another), and see if perhaps the minors you have
│ │ │ │  computed so far are enough to verify our ring is regular in codimension 1. This
│ │ │ │  is exactly what regularInCodimension does. One can control at a fine level how
│ │ │ │  frequently new minors are computed, and how frequently the dimension of what we
│ │ │ │  have computed so far is checked, by the option codimCheckFunction. For more on
│ │ │ │  that, see _R_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n_T_u_t_o_r_i_a_l and _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n. Let us finish
│ │ │ │  running regularInCodimension on our example with several different strategies.
│ │ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefault)
│ │ │ │ - -- used 3.39152s (cpu); 3.05769s (thread); 0s (gc)
│ │ │ │ + -- used 4.40716s (cpu); 3.86968s (thread); 0s (gc)
│ │ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultNonRandom)
│ │ │ │ - -- used 0.771095s (cpu); 0.681377s (thread); 0s (gc)
│ │ │ │ + -- used 0.989466s (cpu); 0.814996s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o48 = true
│ │ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ │ - -- used 2.84296s (cpu); 2.62077s (thread); 0s (gc)
│ │ │ │ + -- used 3.48102s (cpu); 3.20736s (thread); 0s (gc)
│ │ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallest)
│ │ │ │ - -- used 2.29048s (cpu); 1.9516s (thread); 0s (gc)
│ │ │ │ + -- used 3.06145s (cpu); 2.42175s (thread); 0s (gc)
│ │ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallestTerm)
│ │ │ │ - -- used 0.869374s (cpu); 0.790991s (thread); 0s (gc)
│ │ │ │ + -- used 0.965885s (cpu); 0.880369s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = true
│ │ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallest)
│ │ │ │ - -- used 2.73026s (cpu); 2.30339s (thread); 0s (gc)
│ │ │ │ + -- used 3.46726s (cpu); 2.77333s (thread); 0s (gc)
│ │ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallestTerm)
│ │ │ │ - -- used 3.16893s (cpu); 2.7704s (thread); 0s (gc)
│ │ │ │ + -- used 4.12878s (cpu); 3.39966s (thread); 0s (gc)
│ │ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ │ - -- used 9.08616s (cpu); 7.66782s (thread); 0s (gc)
│ │ │ │ + -- used 12.3758s (cpu); 9.91985s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o54 = true
│ │ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultWithPoints)
│ │ │ │ - -- used 7.23875s (cpu); 6.02045s (thread); 0s (gc)
│ │ │ │ + -- used 9.54822s (cpu); 7.61112s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o55 = true
│ │ │ │  If regularInCodimension outputs nothing, then it couldn't verify that the ring
│ │ │ │  was regular in that codimension. We set MaxMinors => 100 to keep it from
│ │ │ │  running too long with an ineffective strategy. Again, even though
│ │ │ │  GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular
│ │ │ │  example, in others they perform better than other strategies. Note similar
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html
│ │ │ @@ -81,23 +81,23 @@
│ │ │          
    
│ │ │            
    It is the cone over $P^2 \times E$ where $E$ is an elliptic curve.  We have embedded it with a Segre embedding inside $P^8$.  In particular, this example is even regular in codimension 3.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 0.864231s (cpu); 0.611071s (thread); 0s (gc)
│ │ │ + -- used 1.32195s (cpu); 0.842862s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │                
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 11.0835s (cpu); 8.33071s (thread); 0s (gc)
    
│ │ │ + -- used 14.149s (cpu); 9.44792s (thread); 0s (gc)
│ │ │              
    
│ │ │          
    
│ │ │            
    We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the ideal made up of a small number of minors of the Jacobian matrix. In this example, instead of computing all relevant 1465128 minors to compute the singular locus, and then trying to compute the dimension of the ideal they generate, we instead compute a few of them.  regularInCodimension returns true if it verified that the ring is regular in codim 1 or 2 (respectively) and null if not.  Because of the randomness that exists in terms of selecting minors, the execution time can actually vary quite a bit.   Let's take a look at what is occurring by using the Verbose option.  We go through the output and explain what each line is telling us.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -172,29 +172,29 @@
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 49, and computed = 39
│ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ -regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.3211s (cpu); 0.975451s (thread); 0s (gc)
│ │ │ +regularInCodimension:  Loop completed, submatrices considered = 49, and compute -- used 1.82316s (cpu); 1.18293s (thread); 0s (gc)
│ │ │  d = 39.  singular locus dimension appears to be = 2
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │          
    
│ │ │          
    
│ │ │            
    MaxMinors.  The first output says that we will compute up to 452.9 minors before giving up.  We can control that by setting the option MaxMinors.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ - -- used 0.182671s (cpu); 0.133167s (thread); 0s (gc)
│ │ │ + -- used 0.221987s (cpu); 0.147568s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -219,15 +219,15 @@
│ │ │          
    
│ │ │            
    Selecting submatrices of the Jacobian.  We also see output like: ``Choosing LexSmallest'' or ``Choosing Random''.  This is saying how we are selecting a given submatrix.  For instance, we can run:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom, Verbose=>true)
│ │ │ - -- used 0.135093s (cpu); 0.115447s (thread); 0s (gc)
│ │ │ + -- used 0.309692s (cpu); 0.236845s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │ @@ -252,15 +252,15 @@
│ │ │          
    
│ │ │            
    Computing minors vs considering the dimension of what has been computed.  Periodically we compute the codimension of the partial ideal of minors we have computed so far.  There are two options to control this.  First, we can tell the function when to first compute the dimension of the working partial ideal of minors.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t->3, Verbose=>true)
│ │ │ - -- used 0.586765s (cpu); 0.44115s (thread); 0s (gc)
│ │ │ + -- used 0.766929s (cpu); 0.525284s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -291,15 +291,15 @@
│ │ │          
    
│ │ │            
    CodimCheckFunction. The option CodimCheckFunction controls how frequently the dimension of the partial ideal of minors is computed.  For instance, setting CodimCheckFunction => t -> t/5 will say it should compute dimension after every 5 minors are examined.  In general, after the output of the CodimCheckFunction increases by an integer we compute the codimension again.  The default function has the space between computations grow exponentially.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t->t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ - -- used 0.650722s (cpu); 0.488427s (thread); 0s (gc)
│ │ │ + -- used 0.905427s (cpu); 0.596929s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ @@ -348,15 +348,15 @@
│ │ │          
    
│ │ │            
    isCodimAtLeast and dim.  We see the lines about the ``isCodimAtLeast failed''.  This means that isCodimAtLeast was not enough on its own to verify that our ring is regular in codimension 1.  After this, ``partial singular locus dimension computed'' indicates we did a complete dimension computation of the partial ideal defining the singular locus.  How isCodimAtLeast is called can be controlled via the options SPairsFunction and PairLimit, which are simply passed to isCodimAtLeast.  You can force the function to only use isCodimAtLeast and not call dimension by setting UseOnlyFastCodim => true.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim => true, Verbose=>true)
│ │ │ - -- used 0.441839s (cpu); 0.319692s (thread); 0s (gc)
│ │ │ + -- used 0.583593s (cpu); 0.361698s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  i3 : dim (S/J)
│ │ │ │  
│ │ │ │  o3 = 4
│ │ │ │  It is the cone over $P^2 \times E$ where $E$ is an elliptic curve. We have
│ │ │ │  embedded it with a Segre embedding inside $P^8$. In particular, this example is
│ │ │ │  even regular in codimension 3.
│ │ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ │ - -- used 0.864231s (cpu); 0.611071s (thread); 0s (gc)
│ │ │ │ + -- used 1.32195s (cpu); 0.842862s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 11.0835s (cpu); 8.33071s (thread); 0s (gc)
│ │ │ │ + -- used 14.149s (cpu); 9.44792s (thread); 0s (gc)
│ │ │ │  We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the
│ │ │ │  ideal made up of a small number of minors of the Jacobian matrix. In this
│ │ │ │  example, instead of computing all relevant 1465128 minors to compute the
│ │ │ │  singular locus, and then trying to compute the dimension of the ideal they
│ │ │ │  generate, we instead compute a few of them. regularInCodimension returns true
│ │ │ │  if it verified that the ring is regular in codim 1 or 2 (respectively) and null
│ │ │ │  if not. Because of the randomness that exists in terms of selecting minors, the
│ │ │ │ @@ -121,22 +121,22 @@
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 49, and computed = 39
│ │ │ │  regularInCodimension:  singularLocus dimension verified by isCodimAtLeast
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 2
│ │ │ │  regularInCodimension:  Loop completed, submatrices considered = 49, and compute
│ │ │ │ --- used 1.3211s (cpu); 0.975451s (thread); 0s (gc)
│ │ │ │ +-- used 1.82316s (cpu); 1.18293s (thread); 0s (gc)
│ │ │ │  d = 39.  singular locus dimension appears to be = 2
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  MMaaxxMMiinnoorrss.. The first output says that we will compute up to 452.9 minors before
│ │ │ │  giving up. We can control that by setting the option MaxMinors.
│ │ │ │  i7 : time regularInCodimension(1, S/J, MaxMinors=>10, Verbose=>true)
│ │ │ │ - -- used 0.182671s (cpu); 0.133167s (thread); 0s (gc)
│ │ │ │ + -- used 0.221987s (cpu); 0.147568s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -159,15 +159,15 @@
│ │ │ │  There are other finer ways to control the MaxMinors option, but they will not
│ │ │ │  be discussed in this tutorial. See _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n.
│ │ │ │  SSeelleeccttiinngg ssuubbmmaattrriicceess ooff tthhee JJaaccoobbiiaann.. We also see output like: ``Choosing
│ │ │ │  LexSmallest'' or ``Choosing Random''. This is saying how we are selecting a
│ │ │ │  given submatrix. For instance, we can run:
│ │ │ │  i8 : time regularInCodimension(1, S/J, MaxMinors=>10, Strategy=>StrategyRandom,
│ │ │ │  Verbose=>true)
│ │ │ │ - -- used 0.135093s (cpu); 0.115447s (thread); 0s (gc)
│ │ │ │ + -- used 0.309692s (cpu); 0.236845s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │ @@ -197,15 +197,15 @@
│ │ │ │  CCoommppuuttiinngg mmiinnoorrss vvss ccoonnssiiddeerriinngg tthhee ddiimmeennssiioonn ooff wwhhaatt hhaass bbeeeenn ccoommppuutteedd..
│ │ │ │  Periodically we compute the codimension of the partial ideal of minors we have
│ │ │ │  computed so far. There are two options to control this. First, we can tell the
│ │ │ │  function when to first compute the dimension of the working partial ideal of
│ │ │ │  minors.
│ │ │ │  i9 : time regularInCodimension(1, S/J, MaxMinors=>10, MinMinorsFunction => t-
│ │ │ │  >3, Verbose=>true)
│ │ │ │ - -- used 0.586765s (cpu); 0.44115s (thread); 0s (gc)
│ │ │ │ + -- used 0.766929s (cpu); 0.525284s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │ @@ -243,15 +243,15 @@
│ │ │ │  dimension of the partial ideal of minors is computed. For instance, setting
│ │ │ │  CodimCheckFunction => t -> t/5 will say it should compute dimension after every
│ │ │ │  5 minors are examined. In general, after the output of the CodimCheckFunction
│ │ │ │  increases by an integer we compute the codimension again. The default function
│ │ │ │  has the space between computations grow exponentially.
│ │ │ │  i10 : time regularInCodimension(1, S/J, MaxMinors=>25, CodimCheckFunction => t-
│ │ │ │  >t/5, MinMinorsFunction => t->2, Verbose=>true)
│ │ │ │ - -- used 0.650722s (cpu); 0.488427s (thread); 0s (gc)
│ │ │ │ + -- used 0.905427s (cpu); 0.596929s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 2, and computed = 2
│ │ │ │ @@ -308,15 +308,15 @@
│ │ │ │  dimension computed'' indicates we did a complete dimension computation of the
│ │ │ │  partial ideal defining the singular locus. How isCodimAtLeast is called can be
│ │ │ │  controlled via the options SPairsFunction and PairLimit, which are simply
│ │ │ │  passed to _i_s_C_o_d_i_m_A_t_L_e_a_s_t. You can force the function to only use isCodimAtLeast
│ │ │ │  and not call dimension by setting UseOnlyFastCodim => true.
│ │ │ │  i11 : time regularInCodimension(1, S/J, MaxMinors=>25, UseOnlyFastCodim =>
│ │ │ │  true, Verbose=>true)
│ │ │ │ - -- used 0.441839s (cpu); 0.319692s (thread); 0s (gc)
│ │ │ │ + -- used 0.583593s (cpu); 0.361698s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html
│ │ │ @@ -68,15 +68,15 @@
│ │ │              
    		
│ │ │                
    i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 1.59228s elapsed
│ │ │ + -- 1.56285s elapsed
│ │ │  
│ │ │  o2 = true
    
│ │ │              
    
│ │ │  In this particular example, on one machine, we list average time to completion of each of the above strategies after 100 runs.        
      
│ │ │            
    • StrategyDefault: 1.65 seconds
      • 
│ │ │ @@ -122,15 +122,15 @@
│ │ │            
    • StrategyPoints: choose all submatrices via Points.
      • 
│ │ │            
    • StrategyDefaultWithPoints: like StrategyDefault but replaces the Random and RandomNonZero submatrices as with matrices chosen as in Points.
      • 
│ │ │          
    
│ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported.  It begins as the default strategy, but the user can modify it.

    Using a single heuristic  Alternatively, if the user only wants to use say LexSmallestTerm they can set, Strategy to point to that symbol, instead of a creating a custom strategy HashTable.  For example:         
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.02929s elapsed
│ │ │ + -- .952057s elapsed
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-
│ │ │ │  b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-
│ │ │ │  e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-
│ │ │ │  a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-
│ │ │ │  g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-
│ │ │ │  h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
│ │ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ │ - -- 1.59228s elapsed
│ │ │ │ + -- 1.56285s 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)
│ │ │ │ - -- 1.02929s elapsed
│ │ │ │ + -- .952057s elapsed
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _S_t_r_a_t_e_g_y_D_e_f_a_u_l_t is an _o_p_t_i_o_n_ _t_a_b_l_e.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/FastMinors.m2:1993:0.
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.000336902s (cpu); 0.00243334s (thread); 0s (gc)
│ │ │ + -- used 0.00402664s (cpu); 0.00336435s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The function works by computing gb(I, PairLimit=>f(i)) for successive values of i.  Here f(i) is a function that takes t, some approximation of the base degree value of the polynomial ring (for example, in a standard graded polynomial ring, this is probably expected to be \{1\}).  And i is a counting variable. You can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=>( (i) -> f(i) ), the default function is SPairsFunction=>i->ceiling(1.5^i)   Perhaps more commonly however, the user may want to instead tell the function to compute for larger values of i.  This is done via the option PairLimit.  This is the max value of i to be plugged into SPairsFunction before the function gives up.  In other words, PairLimit=>5 will tell the function to check codimension 5 times.
    
│ │ │ @@ -136,24 +136,24 @@
│ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
    
│ │ │              
    
│ │ │                
    i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.000522079s (cpu); 0.00238391s (thread); 0s (gc)
│ │ │ + -- used 0.00331098s (cpu); 0.00300599s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 0.00114405s (cpu); 0.00222255s (thread); 0s (gc)
│ │ │ + -- used 0.00223138s (cpu); 0.00292183s (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 0.000336902s (cpu); 0.00243334s (thread); 0s (gc)
│ │ │ │ + -- used 0.00402664s (cpu); 0.00336435s (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 0.000522079s (cpu); 0.00238391s (thread); 0s (gc)
│ │ │ │ + -- used 0.00331098s (cpu); 0.00300599s (thread); 0s (gc)
│ │ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ │ - -- used 0.00114405s (cpu); 0.00222255s (thread); 0s (gc)
│ │ │ │ + -- used 0.00223138s (cpu); 0.00292183s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  Notice in the first case the function returned null, because the depth of
│ │ │ │  search was not high enough. It only computed codim 5 times. The second returned
│ │ │ │  true, but it did so as soon as the answer was found (and before we hit the
│ │ │ │  PairLimit limit).
│ │ │ │  ********** WWaayyss ttoo uussee iissCCooddiimmAAttLLeeaasstt:: **********
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html
│ │ │ @@ -99,23 +99,23 @@
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.221015s (cpu); 0.155202s (thread); 0s (gc)
│ │ │ + -- used 0.389228s (cpu); 0.231021s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.0111545s (cpu); 0.0139544s (thread); 0s (gc)
│ │ │ + -- used 0.0132883s (cpu); 0.0154692s (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 0.221015s (cpu); 0.155202s (thread); 0s (gc)
│ │ │ │ + -- used 0.389228s (cpu); 0.231021s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ │ - -- used 0.0111545s (cpu); 0.0139544s (thread); 0s (gc)
│ │ │ │ + -- used 0.0132883s (cpu); 0.0154692s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 1
│ │ │ │  The option MaxMinors can be used to control how many minors are computed at
│ │ │ │  each step. If this is not specified, the number of minors is a function of the
│ │ │ │  dimension $d$ of the polynomial ring and the possible minors $c$. Specifically
│ │ │ │  it is 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors
│ │ │ │  => ZZ to specify that a fixed integer is used for each step. Alternatively, the
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html
│ │ │ @@ -92,23 +92,23 @@
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.793132s (cpu); 0.746832s (thread); 0s (gc)
│ │ │ + -- used 0.555063s (cpu); 0.480218s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.47691s (cpu); 1.27617s (thread); 0s (gc)
│ │ │ + -- used 1.4366s (cpu); 1.27504s (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 0.793132s (cpu); 0.746832s (thread); 0s (gc)
│ │ │ │ + -- used 0.555063s (cpu); 0.480218s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ │ - -- used 1.47691s (cpu); 1.27617s (thread); 0s (gc)
│ │ │ │ + -- used 1.4366s (cpu); 1.27504s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : I1 == I2
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r_s -- ideal generated by minors
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html
│ │ │ @@ -131,23 +131,23 @@
│ │ │  
│ │ │  o7 = 3
    
│ │ │              
    
│ │ │                
    i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.694973s (cpu); 0.541805s (thread); 0s (gc)
│ │ │ + -- used 0.834255s (cpu); 0.616431s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 6.89948s (cpu); 5.27952s (thread); 0s (gc)
    
│ │ │ + -- used 8.16578s (cpu); 5.66619s (thread); 0s (gc)
│ │ │              
    
│ │ │          
    
│ │ │            
    There are numerous examples where regularInCodimension is several orders of magnitude faster that calls of dim singularLocus.
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -165,39 +165,39 @@
│ │ │  
│ │ │  o11 = 2
    
│ │ │              
    
│ │ │                
    i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0210586s (cpu); 0.0202034s (thread); 0s (gc)
│ │ │ + -- used 0.0200381s (cpu); 0.0217863s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    
│ │ │              
    
│ │ │                
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.182896s (cpu); 0.137715s (thread); 0s (gc)
│ │ │ + -- used 0.235151s (cpu); 0.157628s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    
│ │ │              
    
│ │ │                
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.982565s (cpu); 0.690681s (thread); 0s (gc)
│ │ │ + -- used 1.25257s (cpu); 0.740226s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    
│ │ │              
    
│ │ │                
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 1.38818s (cpu); 0.988524s (thread); 0s (gc)
│ │ │ + -- used 1.71559s (cpu); 1.05506s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The function works by choosing interesting looking submatrices, computing their determinants, and periodically (based on a logarithmic growth setting), computing the dimension of a subideal of the Jacobian. The option Verbose can be used to see this in action.
    
│ │ │ @@ -537,15 +537,15 @@
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ -internalChooseMinor: Ch -- used 6.66371s (cpu); 5.28322s (thread); 0s (gc)
│ │ │ +internalChooseMinor: Ch -- used 8.60343s (cpu); 6.06955s (thread); 0s (gc)
│ │ │  oosing GRevLexSmallestTerm
│ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing Random
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ @@ -589,15 +589,15 @@
│ │ │          
    
│ │ │            
    The maximum number of minors considered can be controlled by the option MaxMinors.  Alternatively, it can be controlled in a more precise way by passing a function to the option MaxMinors.  This function should have two inputs; the first is minimum number of minors needed to determine whether the ring is regular in codimension n, and the second is the total number of minors available in the Jacobian. The function regularInCodimension does not recompute determinants, so MaxMinors or is only an upper bound on the number of minors computed.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ - -- used 1.27921s (cpu); 1.02648s (thread); 0s (gc)
│ │ │ + -- used 1.72484s (cpu); 1.24048s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ @@ -666,39 +666,39 @@
│ │ │              
    		
│ │ │                
    i20 : StrategyCurrent#LexSmallestTerm = 0;
    
│ │ │              
    
│ │ │                
    i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.308707s (cpu); 0.21708s (thread); 0s (gc)
│ │ │ + -- used 0.44499s (cpu); 0.275425s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │                
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.122371s (cpu); 0.0749652s (thread); 0s (gc)
│ │ │ + -- used 0.176976s (cpu); 0.0962802s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    
│ │ │              
    
│ │ │                
    i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.373838s (cpu); 0.285558s (thread); 0s (gc)
│ │ │ + -- used 0.536569s (cpu); 0.36789s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 1.66577s (cpu); 1.25232s (thread); 0s (gc)
│ │ │ + -- used 2.34674s (cpu); 1.59893s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │                
    i25 : StrategyCurrent#LexSmallest = 0;
    
│ │ │ @@ -708,53 +708,53 @@
│ │ │                  		
│ │ │                
    i26 : StrategyCurrent#LexSmallestTerm = 100;
    
│ │ │              
    
│ │ │                
    i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.2665s (cpu); 1.66206s (thread); 0s (gc)
    
│ │ │ + -- used 3.0589s (cpu); 2.00445s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 2.34659s (cpu); 1.70143s (thread); 0s (gc)
│ │ │ + -- used 3.0428s (cpu); 1.93984s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
    
│ │ │              
    
│ │ │                
    i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.408217s (cpu); 0.307796s (thread); 0s (gc)
│ │ │ + -- used 0.53559s (cpu); 0.387166s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = true
    
│ │ │              
    
│ │ │                
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.711845s (cpu); 0.560503s (thread); 0s (gc)
│ │ │ + -- used 0.934341s (cpu); 0.689238s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    
│ │ │              
    
│ │ │                
    i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 0.994761s (cpu); 0.81258s (thread); 0s (gc)
│ │ │ + -- used 1.32284s (cpu); 1.00247s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    
│ │ │              
    
│ │ │                
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.66547s (cpu); 1.33943s (thread); 0s (gc)
│ │ │ + -- used 2.15611s (cpu); 1.60242s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The minimum number of minors computed before checking the codimension can also be controlled by an option MinMinorsFunction.  This is should be a function of a single variable, the number of minors computed.  Finally, via the option CodimCheckFunction, you can pass the regularInCodimension a function which controls how frequently the codimension of the partial Jacobian ideal is computed.  By default this is the floor of 1.3^k. Finally, passing the option Modulus => p will do the computation after changing the coefficient ring to ZZ/p.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -72,19 +72,19 @@
│ │ │ │  
│ │ │ │  o5 : Ideal of T
│ │ │ │  i6 : S = T/I;
│ │ │ │  i7 : dim S
│ │ │ │  
│ │ │ │  o7 = 3
│ │ │ │  i8 : time regularInCodimension(1, S)
│ │ │ │ - -- used 0.694973s (cpu); 0.541805s (thread); 0s (gc)
│ │ │ │ + -- used 0.834255s (cpu); 0.616431s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : time regularInCodimension(2, S)
│ │ │ │ - -- used 6.89948s (cpu); 5.27952s (thread); 0s (gc)
│ │ │ │ + -- used 8.16578s (cpu); 5.66619s (thread); 0s (gc)
│ │ │ │  There are numerous examples where regularInCodimension is several orders of
│ │ │ │  magnitude faster that calls of dim singularLocus.
│ │ │ │  The following is a (pruned) affine chart on an Abelian surface obtained as a
│ │ │ │  product of two elliptic curves. It is nonsingular, as our function verifies. If
│ │ │ │  one does not prune it, then the dim singularLocus call takes an enormous amount
│ │ │ │  of time, otherwise the running times of dim singularLocus and our function are
│ │ │ │  frequently about the same.
│ │ │ │ @@ -92,27 +92,27 @@
│ │ │ │  (g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-
│ │ │ │  f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-
│ │ │ │  c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);
│ │ │ │  i11 : dim(R)
│ │ │ │  
│ │ │ │  o11 = 2
│ │ │ │  i12 : time (dim singularLocus (R))
│ │ │ │ - -- used 0.0210586s (cpu); 0.0202034s (thread); 0s (gc)
│ │ │ │ + -- used 0.0200381s (cpu); 0.0217863s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -1
│ │ │ │  i13 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.182896s (cpu); 0.137715s (thread); 0s (gc)
│ │ │ │ + -- used 0.235151s (cpu); 0.157628s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = true
│ │ │ │  i14 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.982565s (cpu); 0.690681s (thread); 0s (gc)
│ │ │ │ + -- used 1.25257s (cpu); 0.740226s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : time regularInCodimension(2, R)
│ │ │ │ - -- used 1.38818s (cpu); 0.988524s (thread); 0s (gc)
│ │ │ │ + -- used 1.71559s (cpu); 1.05506s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  The function works by choosing interesting looking submatrices, computing their
│ │ │ │  determinants, and periodically (based on a logarithmic growth setting),
│ │ │ │  computing the dimension of a subideal of the Jacobian. The option Verbose can
│ │ │ │  be used to see this in action.
│ │ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │ │ @@ -461,15 +461,15 @@
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │ -internalChooseMinor: Ch -- used 6.66371s (cpu); 5.28322s (thread); 0s (gc)
│ │ │ │ +internalChooseMinor: Ch -- used 8.60343s (cpu); 6.06955s (thread); 0s (gc)
│ │ │ │  oosing GRevLexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing RandomNonZero
│ │ │ │  internalChooseMinor: Choosing LexSmallest
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing Random
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │ @@ -515,15 +515,15 @@
│ │ │ │  a function to the option MaxMinors. This function should have two inputs; the
│ │ │ │  first is minimum number of minors needed to determine whether the ring is
│ │ │ │  regular in codimension n, and the second is the total number of minors
│ │ │ │  available in the Jacobian. The function regularInCodimension does not recompute
│ │ │ │  determinants, so MaxMinors or is only an upper bound on the number of minors
│ │ │ │  computed.
│ │ │ │  i17 : time regularInCodimension(2, S, Verbose=>true, MaxMinors=>30)
│ │ │ │ - -- used 1.27921s (cpu); 1.02648s (thread); 0s (gc)
│ │ │ │ + -- used 1.72484s (cpu); 1.24048s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4
│ │ │ │  minors, we will compute up to 30 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │  internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │ @@ -590,51 +590,51 @@
│ │ │ │  because there are a small number of entries with nonzero constant terms, which
│ │ │ │  are selected repeatedly). However, in our first example, the LexSmallestTerm is
│ │ │ │  much faster, and Random does not perform well at all.
│ │ │ │  i18 : StrategyCurrent#Random = 0;
│ │ │ │  i19 : StrategyCurrent#LexSmallest = 100;
│ │ │ │  i20 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │ │  i21 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.308707s (cpu); 0.21708s (thread); 0s (gc)
│ │ │ │ + -- used 0.44499s (cpu); 0.275425s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.122371s (cpu); 0.0749652s (thread); 0s (gc)
│ │ │ │ + -- used 0.176976s (cpu); 0.0962802s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.373838s (cpu); 0.285558s (thread); 0s (gc)
│ │ │ │ + -- used 0.536569s (cpu); 0.36789s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 1.66577s (cpu); 1.25232s (thread); 0s (gc)
│ │ │ │ + -- used 2.34674s (cpu); 1.59893s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  i25 : StrategyCurrent#LexSmallest = 0;
│ │ │ │  i26 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │ │  i27 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 2.2665s (cpu); 1.66206s (thread); 0s (gc)
│ │ │ │ + -- used 3.0589s (cpu); 2.00445s (thread); 0s (gc)
│ │ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 2.34659s (cpu); 1.70143s (thread); 0s (gc)
│ │ │ │ + -- used 3.0428s (cpu); 1.93984s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = true
│ │ │ │  i29 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.408217s (cpu); 0.307796s (thread); 0s (gc)
│ │ │ │ + -- used 0.53559s (cpu); 0.387166s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = true
│ │ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.711845s (cpu); 0.560503s (thread); 0s (gc)
│ │ │ │ + -- used 0.934341s (cpu); 0.689238s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 0.994761s (cpu); 0.81258s (thread); 0s (gc)
│ │ │ │ + -- used 1.32284s (cpu); 1.00247s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 1.66547s (cpu); 1.33943s (thread); 0s (gc)
│ │ │ │ + -- used 2.15611s (cpu); 1.60242s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = true
│ │ │ │  The minimum number of minors computed before checking the codimension can also
│ │ │ │  be controlled by an option MinMinorsFunction. This is should be a function of a
│ │ │ │  single variable, the number of minors computed. Finally, via the option
│ │ │ │  CodimCheckFunction, you can pass the regularInCodimension a function which
│ │ │ │  controls how frequently the codimension of the partial Jacobian ideal is
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out
│ │ │ @@ -81,23 +81,23 @@
│ │ │  
│ │ │  i14 : K3=nextDegree(gens ker Q2,2,S);
│ │ │  
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
│ │ │  
│ │ │  i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00488759s (cpu); 0.00488444s (thread); 0s (gc)
│ │ │ + -- used 0.00314991s (cpu); 0.00314654s (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 0.0110816s (cpu); 0.0110876s (thread); 0s (gc)
│ │ │ + -- used 0.00661407s (cpu); 0.00661586s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : I==J
│ │ │  
│ │ │  o17 = true
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html
│ │ │ @@ -202,26 +202,26 @@
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00488759s (cpu); 0.00488444s (thread); 0s (gc)
│ │ │ + -- used 0.00314991s (cpu); 0.00314654s (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 0.0110816s (cpu); 0.0110876s (thread); 0s (gc)
│ │ │ + -- used 0.00661407s (cpu); 0.00661586s (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 0.00488759s (cpu); 0.00488444s (thread); 0s (gc)
│ │ │ │ + -- used 0.00314991s (cpu); 0.00314654s (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 0.0110816s (cpu); 0.0110876s (thread); 0s (gc)
│ │ │ │ + -- used 0.00661407s (cpu); 0.00661586s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : I==J
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  Note that our method is a bit faster for this small example, and for rank 2
│ │ │ │  quotients of S^3=\mathbb{Z}[x,y]^3 the time difference is massive.
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Object.out
│ │ │ @@ -4,19 +4,19 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7fc5abdd74a0}
│ │ │ +o2 = int32{Address => 0x7f209494ac80}
│ │ │  
│ │ │  i3 : address x
│ │ │  
│ │ │ -o3 = 0x7fc5abdd74a0
│ │ │ +o3 = 0x7f209494ac80
│ │ │  
│ │ │  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 = {0x7fc5abe22090, 0x7fc5abe22080, 0x7fc5abe22070}
│ │ │ +o3 = {0x7f2094996480, 0x7f2094996470, 0x7f2094996460}
│ │ │  
│ │ │  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 = {0x7fc5abe6ced0, 0x7fc5abe6cec0, 0x7fc5abe6ceb0}
│ │ │ +o2 = {0x7f2094996270, 0x7f2094996260, 0x7f2094996250}
│ │ │  
│ │ │  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 = 0x7fc5ae702770
│ │ │ +o1 = 0x7f20972dce00
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7fc5ae702770
│ │ │ +o2 = 0x7f20972dce00
│ │ │  
│ │ │  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 = 0x7fc5abdd7640
│ │ │ +o2 = 0x7f209494a440
│ │ │  
│ │ │  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 = 0x7fc5abe22b40
│ │ │ +o1 = 0x7f209494a180
│ │ │  
│ │ │  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" => 6.94084e-310}
│ │ │ +o2 = HashTable{"bar" => 6.90582e-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 => 0x7fc5abdd7460}
│ │ │ +o2 = int32{Address => 0x7f20949072d0}
│ │ │  
│ │ │  i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7fc5abdd7460
│ │ │ +o3 = 0x7f20949072d0
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7fc5abdd7465
│ │ │ +o4 = 0x7f20949072d5
│ │ │  
│ │ │  o4 : Pointer
│ │ │  
│ │ │  i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7fc5abdd745d
│ │ │ +o5 = 0x7f20949072cd
│ │ │  
│ │ │  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{0x7fc5ba51d550, mpfr}
│ │ │ +o2 = SharedLibrary{0x7f20a88c4550, 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 = 0x7fc5abe22de0
│ │ │ +o2 = 0x7f209494a420
│ │ │  
│ │ │  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 = 0x564762a03b40
│ │ │ +o1 = 0x5607a8edfb40
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7fc5abdd7920
│ │ │ +o2 = 0x7f2094907450
│ │ │  
│ │ │  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 = 0x7fa47006a4f0
│ │ │ +o17 = 0x7f256806a4f0
│ │ │  
│ │ │  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 = 0x7fc5b5f45200
│ │ │ +o1 = 0x7f20a3b61490
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7fc5abdd71a0
│ │ │ +o2 = 0x7f209494afe0
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7fc5abdd70a0
│ │ │ +o3 = 0x7f209494aed0
│ │ │  
│ │ │  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 0x7fc59c07f9b0freeing memory at 0x7fc59c07f990
│ │ │ -freeing memory at 0x7fc59c07f930
│ │ │ -freeing memory at 0x7fc59c07f950
│ │ │ -freeing memory at 0x7fc59c07f8f0
│ │ │ -freeing memory at 0x7fc59c07f250
│ │ │ -freeing memory at 0x7fc59c07f230
│ │ │ -freeing memory at 0x7fc59c07f910
│ │ │ -freeing memory at 0x7fc59c07f970
│ │ │ -freeing memory at 0x7fc59c07f9b0
│ │ │ +freeing memory at 0x7f207c07f950
│ │ │ +freeing memory at 0x7f207c07f910
│ │ │ +freeing memory at 0x7f207c07f230
│ │ │ +freeing memory at 0x7f207c07f9b0
│ │ │ +freeing memory at 0x7f207c07f970
│ │ │ +freeing memory at 0x7f207c07f990
│ │ │ +freeing memory at 0x7f207c07f250
│ │ │ +freeing memory at 0x7f207c07f8f0
│ │ │ +freeing memory at 0x7f207c07f930
│ │ │  
│ │ │  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 = 0x7fc5abe22b10
│ │ │ +o5 = 0x7f209494a6d0
│ │ │  
│ │ │  o5 : ForeignObject of type void*
│ │ │  
│ │ │  i6 : value x
│ │ │  
│ │ │ -o6 = 0x7fc5abe22b10
│ │ │ +o6 = 0x7f209494a6d0
│ │ │  
│ │ │  o6 : Pointer
│ │ │  
│ │ │  i7 : x = charstar "Hello, world!"
│ │ │  
│ │ │  o7 = Hello, world!
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Object.html
│ │ │ @@ -64,27 +64,27 @@
│ │ │  o1 : ForeignObject of type int32
    
│ │ │              
    
│ │ │                
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7fc5abdd74a0}
    
│ │ │ +o2 = int32{Address => 0x7f209494ac80}
│ │ │              
    
│ │ │          
    
│ │ │            
    To get this, use address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7fc5abdd74a0
│ │ │ +o3 = 0x7f209494ac80
│ │ │  
│ │ │  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 => 0x7fc5abdd74a0}
│ │ │ │ +o2 = int32{Address => 0x7f209494ac80}
│ │ │ │  To get this, use _a_d_d_r_e_s_s.
│ │ │ │  i3 : address x
│ │ │ │  
│ │ │ │ -o3 = 0x7fc5abdd74a0
│ │ │ │ +o3 = 0x7f209494ac80
│ │ │ │  
│ │ │ │  o3 : Pointer
│ │ │ │  Use _c_l_a_s_s to determine the type of the object.
│ │ │ │  i4 : class x
│ │ │ │  
│ │ │ │  o4 = int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │  o2 : ForeignObject of type char**
    
│ │ │              
    
│ │ │                
    i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7fc5abe22090, 0x7fc5abe22080, 0x7fc5abe22070}
│ │ │ +o3 = {0x7f2094996480, 0x7f2094996470, 0x7f2094996460}
│ │ │  
│ │ │  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 = {0x7fc5abe22090, 0x7fc5abe22080, 0x7fc5abe22070}
│ │ │ │ +o3 = {0x7f2094996480, 0x7f2094996470, 0x7f2094996460}
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type void**
│ │ │ │  Foreign pointer arrays may be subscripted using __.
│ │ │ │  i4 : x = charstarstar {"foo", "bar", "baz"}
│ │ │ │  
│ │ │ │  o4 = {foo, bar, baz}
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type_sp__Visible__List.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignObject of type char**
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7fc5abe6ced0, 0x7fc5abe6cec0, 0x7fc5abe6ceb0}
│ │ │ +o2 = {0x7f2094996270, 0x7f2094996260, 0x7f2094996250}
│ │ │  
│ │ │  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 = {0x7fc5abe6ced0, 0x7fc5abe6cec0, 0x7fc5abe6ceb0}
│ │ │ │ +o2 = {0x7f2094996270, 0x7f2094996260, 0x7f2094996250}
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void**
│ │ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ │  
│ │ │ │  o3 = int32[2]*
│ │ │ │  
│ │ │ │  o3 : ForeignPointerArrayType
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Type_sp__Pointer.html
│ │ │ @@ -73,24 +73,24 @@
│ │ │            
    To cast a Macaulay2 pointer to a foreign object with a pointer type, give the type followed by the pointer.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7fc5ae702770
│ │ │ +o1 = 0x7f20972dce00
│ │ │  
│ │ │  o1 : Pointer
    │ │ │
    │ │ │ 
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7fc5ae702770
│ │ │ +o2 = 0x7f20972dce00
│ │ │  
│ │ │  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 = 0x7fc5ae702770 │ │ │ │ +o1 = 0x7f20972dce00 │ │ │ │ │ │ │ │ o1 : Pointer │ │ │ │ i2 : voidstar ptr │ │ │ │ │ │ │ │ -o2 = 0x7fc5ae702770 │ │ │ │ +o2 = 0x7f20972dce00 │ │ │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _F_o_r_e_i_g_n_P_o_i_n_t_e_r_T_y_p_e_ _P_o_i_n_t_e_r -- cast a Macaulay2 pointer to a foreign │ │ │ │ pointer │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp__Pointer.html │ │ │ @@ -82,15 +82,15 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7fc5abdd7640
│ │ │ +o2 = 0x7f209494a440
│ │ │  
│ │ │  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 = 0x7fc5abdd7640
│ │ │ │ +o2 = 0x7f209494a440
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : int ptr
│ │ │ │  
│ │ │ │  o3 = 5
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp_st_spvoidstar.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │            
    This is syntactic sugar for T value ptr (see ForeignType Pointer) for dereferencing pointers.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7fc5abe22b40
│ │ │ +o1 = 0x7f209494a180
│ │ │  
│ │ │  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 = 0x7fc5abe22b40
│ │ │ │ +o1 = 0x7f209494a180
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type void*
│ │ │ │  i2 : int * ptr
│ │ │ │  
│ │ │ │  o2 = 5
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  o1 : ForeignUnionType
    │ │ │
    │ │ │ 
    i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.94084e-310}
│ │ │ +o2 = HashTable{"bar" => 6.90582e-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" => 6.94084e-310} │ │ │ │ +o2 = HashTable{"bar" => 6.90582e-310} │ │ │ │ "foo" => 27 │ │ │ │ │ │ │ │ o2 : ForeignObject of type myunion │ │ │ │ i3 : myunion pi │ │ │ │ │ │ │ │ o3 = HashTable{"bar" => 3.14159 } │ │ │ │ "foo" => 1413754136 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Pointer.html │ │ │ @@ -64,50 +64,50 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │
    │ │ │ 
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7fc5abdd7460}
    │ │ │ +o2 = int32{Address => 0x7f20949072d0} │ │ │
    │ │ │
    │ │ │

    These pointers can be accessed using address.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7fc5abdd7460
│ │ │ +o3 = 0x7f20949072d0
│ │ │  
│ │ │  o3 : Pointer
    │ │ │
    │ │ │
    │ │ │

    Simple arithmetic can be performed on pointers.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7fc5abdd7465
│ │ │ +o4 = 0x7f20949072d5
│ │ │  
│ │ │  o4 : Pointer
    │ │ │
    │ │ │ 
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7fc5abdd745d
│ │ │ +o5 = 0x7f20949072cd
│ │ │  
│ │ │  o5 : Pointer
    │ │ │
    │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -10,30 +10,30 @@ │ │ │ │ i1 : x = int 20 │ │ │ │ │ │ │ │ o1 = 20 │ │ │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ i2 : peek x │ │ │ │ │ │ │ │ -o2 = int32{Address => 0x7fc5abdd7460} │ │ │ │ +o2 = int32{Address => 0x7f20949072d0} │ │ │ │ These pointers can be accessed using _a_d_d_r_e_s_s. │ │ │ │ i3 : ptr = address x │ │ │ │ │ │ │ │ -o3 = 0x7fc5abdd7460 │ │ │ │ +o3 = 0x7f20949072d0 │ │ │ │ │ │ │ │ o3 : Pointer │ │ │ │ Simple arithmetic can be performed on pointers. │ │ │ │ i4 : ptr + 5 │ │ │ │ │ │ │ │ -o4 = 0x7fc5abdd7465 │ │ │ │ +o4 = 0x7f20949072d5 │ │ │ │ │ │ │ │ o4 : Pointer │ │ │ │ i5 : ptr - 3 │ │ │ │ │ │ │ │ -o5 = 0x7fc5abdd745d │ │ │ │ +o5 = 0x7f20949072cd │ │ │ │ │ │ │ │ o5 : Pointer │ │ │ │ ******** MMeennuu ******** │ │ │ │ * _n_u_l_l_P_o_i_n_t_e_r -- the null pointer │ │ │ │ * _a_d_d_r_e_s_s -- pointer to type or object │ │ │ │ * _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r -- dereference a pointer │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa ppooiinntteerr:: ********** │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Shared__Library.html │ │ │ @@ -64,15 +64,15 @@ │ │ │ o1 : SharedLibrary │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7fc5ba51d550, mpfr}
    │ │ │ +o2 = SharedLibrary{0x7f20a88c4550, mpfr} │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Menu

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : address int
│ │ │  
│ │ │ -o1 = 0x564762a03b40
│ │ │ +o1 = 0x5607a8edfb40
│ │ │  
│ │ │  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 = 0x7fc5abdd7920
│ │ │ +o2 = 0x7f2094907450
│ │ │  
│ │ │  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 = 0x564762a03b40 │ │ │ │ +o1 = 0x5607a8edfb40 │ │ │ │ │ │ │ │ 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 = 0x7fc5abdd7920 │ │ │ │ +o2 = 0x7f2094907450 │ │ │ │ │ │ │ │ o2 : Pointer │ │ │ │ ********** WWaayyss ttoo uussee aaddddrreessss:: ********** │ │ │ │ * address(ForeignObject) │ │ │ │ * address(ForeignType) │ │ │ │ * address(Nothing) (missing documentation) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_foreign__Function.html │ │ │ @@ -232,15 +232,15 @@ │ │ │ o16 : ForeignFunction │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7fa47006a4f0
│ │ │ +o17 = 0x7f256806a4f0
│ │ │  
│ │ │  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 = 0x7fa47006a4f0 │ │ │ │ +o17 = 0x7f256806a4f0 │ │ │ │ │ │ │ │ o17 : ForeignObject of type void* │ │ │ │ i18 : registerFinalizer(x, free) │ │ │ │ ********** WWaayyss ttoo uussee ffoorreeiiggnnFFuunnccttiioonn:: ********** │ │ │ │ * foreignFunction(Pointer,String,ForeignType,VisibleList) │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,ForeignType) │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,VisibleList) │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_get__Memory.html │ │ │ @@ -77,43 +77,43 @@ │ │ │

    Allocate n bytes of memory using the GC garbage collector.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7fc5b5f45200
│ │ │ +o1 = 0x7f20a3b61490
│ │ │  
│ │ │  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 = 0x7fc5abdd71a0
│ │ │ +o2 = 0x7f209494afe0
│ │ │  
│ │ │  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 = 0x7fc5abdd70a0
│ │ │ +o3 = 0x7f209494aed0
│ │ │  
│ │ │  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 = 0x7fc5b5f45200 │ │ │ │ +o1 = 0x7f20a3b61490 │ │ │ │ │ │ │ │ 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 = 0x7fc5abdd71a0 │ │ │ │ +o2 = 0x7f209494afe0 │ │ │ │ │ │ │ │ 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 = 0x7fc5abdd70a0 │ │ │ │ +o3 = 0x7f209494aed0 │ │ │ │ │ │ │ │ o3 : ForeignObject of type void* │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_g_i_s_t_e_r_F_i_n_a_l_i_z_e_r_(_F_o_r_e_i_g_n_O_b_j_e_c_t_,_F_u_n_c_t_i_o_n_) -- register a finalizer for a │ │ │ │ foreign object │ │ │ │ ********** WWaayyss ttoo uussee ggeettMMeemmoorryy:: ********** │ │ │ │ * getMemory(ForeignType) │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.html │ │ │ @@ -100,23 +100,23 @@ │ │ │ │ │ │ 
    i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : collectGarbage()
│ │ │ -freeing memory at 0x7fc59c07f9b0freeing memory at 0x7fc59c07f990
│ │ │ -freeing memory at 0x7fc59c07f930
│ │ │ -freeing memory at 0x7fc59c07f950
│ │ │ -freeing memory at 0x7fc59c07f8f0
│ │ │ -freeing memory at 0x7fc59c07f250
│ │ │ -freeing memory at 0x7fc59c07f230
│ │ │ -freeing memory at 0x7fc59c07f910
│ │ │ -freeing memory at 0x7fc59c07f970
│ │ │ -freeing memory at 0x7fc59c07f9b0
    │ │ │ +freeing memory at 0x7f207c07f950 │ │ │ +freeing memory at 0x7f207c07f910 │ │ │ +freeing memory at 0x7f207c07f230 │ │ │ +freeing memory at 0x7f207c07f9b0 │ │ │ +freeing memory at 0x7f207c07f970 │ │ │ +freeing memory at 0x7f207c07f990 │ │ │ +freeing memory at 0x7f207c07f250 │ │ │ +freeing memory at 0x7f207c07f8f0 │ │ │ +freeing memory at 0x7f207c07f930 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7fc5abe22b10
│ │ │ +o5 = 0x7f209494a6d0
│ │ │  
│ │ │  o5 : ForeignObject of type void*
    │ │ │
    │ │ │ 
    i6 : value x
│ │ │  
│ │ │ -o6 = 0x7fc5abe22b10
│ │ │ +o6 = 0x7f209494a6d0
│ │ │  
│ │ │  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 = 0x7fc5abe22b10 │ │ │ │ +o5 = 0x7f209494a6d0 │ │ │ │ │ │ │ │ o5 : ForeignObject of type void* │ │ │ │ i6 : value x │ │ │ │ │ │ │ │ -o6 = 0x7fc5abe22b10 │ │ │ │ +o6 = 0x7f209494a6d0 │ │ │ │ │ │ │ │ o6 : Pointer │ │ │ │ Foreign string objects are converted to strings. │ │ │ │ i7 : x = charstar "Hello, world!" │ │ │ │ │ │ │ │ o7 = Hello, world! │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/example-output/_put__Matrix.out │ │ │ @@ -6,27 +6,27 @@ │ │ │ | 1 2 3 4 | │ │ │ │ │ │ 2 4 │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i2 : s = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-15970-0/0 │ │ │ +o2 = /tmp/M2-21095-0/0 │ │ │ │ │ │ i3 : F = openOut(s) │ │ │ │ │ │ -o3 = /tmp/M2-15970-0/0 │ │ │ +o3 = /tmp/M2-21095-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : putMatrix(F,A) │ │ │ │ │ │ i5 : close(F) │ │ │ │ │ │ -o5 = /tmp/M2-15970-0/0 │ │ │ +o5 = /tmp/M2-21095-0/0 │ │ │ │ │ │ o5 : File │ │ │ │ │ │ i6 : getMatrix(s) │ │ │ │ │ │ o6 = | 1 1 1 1 | │ │ │ | 1 2 3 4 | │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/html/_put__Matrix.html │ │ │ @@ -79,36 +79,36 @@ │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-15970-0/0
    │ │ │ +o2 = /tmp/M2-21095-0/0 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15970-0/0
│ │ │ +o3 = /tmp/M2-21095-0/0
│ │ │  
│ │ │  o3 : File
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : putMatrix(F,A)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15970-0/0
│ │ │ +o5 = /tmp/M2-21095-0/0
│ │ │  
│ │ │  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/M2-15970-0/0
│ │ │ │ +o2 = /tmp/M2-21095-0/0
│ │ │ │  i3 : F = openOut(s)
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-15970-0/0
│ │ │ │ +o3 = /tmp/M2-21095-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : putMatrix(F,A)
│ │ │ │  i5 : close(F)
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-15970-0/0
│ │ │ │ +o5 = /tmp/M2-21095-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : getMatrix(s)
│ │ │ │  
│ │ │ │  o6 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out
│ │ │ @@ -155,31 +155,31 @@
│ │ │  i26 : numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true) -- FinalAttempt improves the estimate slightly
│ │ │  
│ │ │  o26 = {.142067, .144}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 2.34145s (cpu); 1.52418s (thread); 0s (gc)
│ │ │ + -- used 2.93425s (cpu); 1.5746s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.37654s (cpu); 0.917052s (thread); 0s (gc)
│ │ │ + -- used 1.72202s (cpu); 0.952119s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
│ │ │  
│ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.15354s (cpu); 0.709506s (thread); 0s (gc)
│ │ │ + -- used 1.35233s (cpu); 0.781143s (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 0.00222093s (cpu); 0.00433484s (thread); 0s (gc)
│ │ │ + -- used 0.00805633s (cpu); 0.00597248s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
│ │ │  
│ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.398199s (cpu); 0.259534s (thread); 0s (gc)
│ │ │ + -- used 0.436302s (cpu); 0.298274s (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 0.288129s (cpu); 0.213097s (thread); 0s (gc)
│ │ │ + -- used 0.553702s (cpu); 0.323267s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
│ │ │  
│ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.5014s (cpu); 1.17856s (thread); 0s (gc)
│ │ │ + -- used 1.48704s (cpu); 1.25232s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
│ │ │  
│ │ │  i21 : R = ZZ/3[x,y];
│ │ │  
│ │ │  i22 : M = ideal(x, y);
│ │ │  
│ │ │ @@ -85,34 +85,34 @@
│ │ │  o24 = 8
│ │ │  
│ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │  
│ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 1.01826s (cpu); 0.670069s (thread); 0s (gc)
│ │ │ + -- used 1.27503s (cpu); 0.773707s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
│ │ │  
│ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.5111s (cpu); 0.975742s (thread); 0s (gc)
│ │ │ + -- used 1.87579s (cpu); 1.1398s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
│ │ │  
│ │ │  i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 1.71561s (cpu); 1.41782s (thread); 0s (gc)
│ │ │ + -- used 1.87909s (cpu); 1.54041s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
│ │ │  
│ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.622933s (cpu); 0.512418s (thread); 0s (gc)
│ │ │ + -- used 0.609691s (cpu); 0.544908s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
│ │ │  
│ │ │  i32 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html
│ │ │ @@ -363,37 +363,37 @@
│ │ │          
    
│ │ │            
    The computations performed when FinalAttempt is set to true are often slow, and often fail to improve the estimate, and for this reason, this option should be used sparingly. It is often more effective to increase the values of Attempts or DepthOfSearch, instead.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -228,29 +228,29 @@
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  The computations performed when FinalAttempt is set to true are often slow, and
│ │ │ │  often fail to improve the estimate, and for this reason, this option should be
│ │ │ │  used sparingly. It is often more effective to increase the values of Attempts
│ │ │ │  or DepthOfSearch, instead.
│ │ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ │ - -- used 2.34145s (cpu); 1.52418s (thread); 0s (gc)
│ │ │ │ + -- used 2.93425s (cpu); 1.5746s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {.142067, .144}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ │ - -- used 1.37654s (cpu); 0.917052s (thread); 0s (gc)
│ │ │ │ + -- used 1.72202s (cpu); 0.952119s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o28 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o28 : QQ
│ │ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ │ - -- used 1.15354s (cpu); 0.709506s (thread); 0s (gc)
│ │ │ │ + -- used 1.35233s (cpu); 0.781143s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o29 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o29 : QQ
│ │ │ │  As seen in several examples above, when the exact answer is not found, a list
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html
│ │ │ @@ -192,23 +192,23 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 2.34145s (cpu); 1.52418s (thread); 0s (gc)
│ │ │ + -- used 2.93425s (cpu); 1.5746s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.37654s (cpu); 0.917052s (thread); 0s (gc)
│ │ │ + -- used 1.72202s (cpu); 0.952119s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
    
│ │ │              
    
│ │ │                
    i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 1.15354s (cpu); 0.709506s (thread); 0s (gc)
│ │ │ + -- used 1.35233s (cpu); 0.781143s (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 0.00222093s (cpu); 0.00433484s (thread); 0s (gc)
│ │ │ + -- used 0.00805633s (cpu); 0.00597248s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
    
│ │ │              
    
│ │ │                
    i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.398199s (cpu); 0.259534s (thread); 0s (gc)
│ │ │ + -- used 0.436302s (cpu); 0.298274s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The valid values for the option ContainmentTest are FrobeniusPower, FrobeniusRoot, and StandardPower. The default value of this option depends on what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to StandardPower if frobeniusNu is passed an ideal $I$. We describe the consequences of setting ContainmentTest to each of these values below.
    
│ │ │ @@ -225,23 +225,23 @@
│ │ │              
│ │ │                
    i18 : f = x^3 + y^3 + z^3 + x*y*z;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.288129s (cpu); 0.213097s (thread); 0s (gc)
│ │ │ + -- used 0.553702s (cpu); 0.323267s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.5014s (cpu); 1.17856s (thread); 0s (gc)
│ │ │ + -- used 1.48704s (cpu); 1.25232s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Finally, when ContainmentTest is set to FrobeniusPower, then instead of producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as defined in the paper Frobenius Powers by Hernández, Teixeira, and Witt, which can be computed with the function frobeniusPower, from the TestIdeals package. In particular, frobeniusNu(e,I,J) and frobeniusNu(e,I,J,ContainmentTest=>FrobeniusPower) need not agree. However, they will agree when $I$ is a principal ideal.
    
│ │ │ @@ -287,46 +287,46 @@
│ │ │              
│ │ │                
    i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 1.01826s (cpu); 0.670069s (thread); 0s (gc)
│ │ │ + -- used 1.27503s (cpu); 0.773707s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.5111s (cpu); 0.975742s (thread); 0s (gc)
│ │ │ + -- used 1.87579s (cpu); 1.1398s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 1.71561s (cpu); 1.41782s (thread); 0s (gc)
│ │ │ + -- used 1.87909s (cpu); 1.54041s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.622933s (cpu); 0.512418s (thread); 0s (gc)
│ │ │ + -- used 0.609691s (cpu); 0.544908s (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 0.00222093s (cpu); 0.00433484s (thread); 0s (gc)
│ │ │ │ + -- used 0.00805633s (cpu); 0.00597248s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3756
│ │ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ │ - -- used 0.398199s (cpu); 0.259534s (thread); 0s (gc)
│ │ │ │ + -- used 0.436302s (cpu); 0.298274s (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 0.288129s (cpu); 0.213097s (thread); 0s (gc)
│ │ │ │ + -- used 0.553702s (cpu); 0.323267s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = 499
│ │ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ │ - -- used 1.5014s (cpu); 1.17856s (thread); 0s (gc)
│ │ │ │ + -- used 1.48704s (cpu); 1.25232s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = 499
│ │ │ │  Finally, when ContainmentTest is set to FrobeniusPower, then instead of
│ │ │ │  producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead
│ │ │ │  outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius
│ │ │ │  power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the
│ │ │ │  $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as
│ │ │ │ @@ -167,31 +167,31 @@
│ │ │ │  The function frobeniusNu works by searching through the list of potential
│ │ │ │  integers $n$ and checking containments of $I^n$ in the specified Frobenius
│ │ │ │  power of $J$. The way this search is approached is specified by the option
│ │ │ │  Search, which can be set to Binary (the default value) or Linear.
│ │ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ │ - -- used 1.01826s (cpu); 0.670069s (thread); 0s (gc)
│ │ │ │ + -- used 1.27503s (cpu); 0.773707s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = 1124
│ │ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ │ - -- used 1.5111s (cpu); 0.975742s (thread); 0s (gc)
│ │ │ │ + -- used 1.87579s (cpu); 1.1398s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 1124
│ │ │ │  i29 : M = ideal(x, y, z);
│ │ │ │  
│ │ │ │  o29 : Ideal of R
│ │ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ │ - -- used 1.71561s (cpu); 1.41782s (thread); 0s (gc)
│ │ │ │ + -- used 1.87909s (cpu); 1.54041s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 97
│ │ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets
│ │ │ │  luckier
│ │ │ │ - -- used 0.622933s (cpu); 0.512418s (thread); 0s (gc)
│ │ │ │ + -- used 0.609691s (cpu); 0.544908s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 97
│ │ │ │  The option AtOrigin (default value true) can be turned off to tell frobeniusNu
│ │ │ │  to effectively do the computation over all possible maximal ideals $J$ and take
│ │ │ │  the minimum.
│ │ │ │  i32 : R = ZZ/7[x,y];
│ │ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out
│ │ │ @@ -208,21 +208,21 @@
│ │ │        | 3/7 5/4 3/7  10   |
│ │ │        | 6/7 2/9 5    3/2  |
│ │ │  
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
│ │ │  
│ │ │  i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.609678s (cpu); 0.386401s (thread); 0s (gc)
│ │ │ + -- used 1.78911s (cpu); 0.510325s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
│ │ │  
│ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.84273s (cpu); 1.07771s (thread); 0s (gc)
│ │ │ + -- used 3.55807s (cpu); 1.12459s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
│ │ │  
│ │ │  i29 :
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html
│ │ │ @@ -386,25 +386,25 @@
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.609678s (cpu); 0.386401s (thread); 0s (gc)
│ │ │ + -- used 1.78911s (cpu); 0.510325s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 1.84273s (cpu); 1.07771s (thread); 0s (gc)
│ │ │ + -- used 3.55807s (cpu); 1.12459s (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 0.609678s (cpu); 0.386401s (thread); 0s (gc) │ │ │ │ + -- used 1.78911s (cpu); 0.510325s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ │ │ o27 : KClass │ │ │ │ i28 : time C = orbitClosure(X,Mat, RREFMethod => true) │ │ │ │ - -- used 1.84273s (cpu); 1.07771s (thread); 0s (gc) │ │ │ │ + -- used 3.55807s (cpu); 1.12459s (thread); 0s (gc) │ │ │ │ │ │ │ │ o28 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ │ │ o28 : KClass │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_a_l_i_z_e_d_F_l_a_g_V_a_r_i_e_t_y -- makes a generalized flag variety as a GKM │ │ │ │ variety │ │ ├── ./usr/share/doc/Macaulay2/Graphs/example-output/_new__Digraph.out │ │ │ @@ -32,12 +32,12 @@ │ │ │ 5 => {6} │ │ │ 6 => {} │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ │ │ i3 : keys H │ │ │ │ │ │ -o3 = {map, digraph, newDigraph} │ │ │ +o3 = {newDigraph, digraph, map} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Graphs/html/_new__Digraph.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ o2 : SortedDigraph │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : keys H
│ │ │  
│ │ │ -o3 = {map, digraph, newDigraph}
│ │ │ +o3 = {newDigraph, digraph, map}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ 4 => {} │ │ │ │ 5 => {6} │ │ │ │ 6 => {} │ │ │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ i3 : keys H │ │ │ │ │ │ │ │ -o3 = {map, digraph, 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/example-output/_nonminimal__Maps.out │ │ │ @@ -103,46 +103,46 @@ │ │ │ │ │ │ i13 : #compsJ │ │ │ │ │ │ o13 = 2 │ │ │ │ │ │ i14 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o14 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21 │ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20 │ │ │ ----------------------------------------------------------------------- │ │ │ - -36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 | │ │ │ + 40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 | │ │ │ │ │ │ 1 36 │ │ │ o14 : Matrix kk <-- kk │ │ │ │ │ │ i15 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16 │ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47 │ │ │ ----------------------------------------------------------------------- │ │ │ - -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 | │ │ │ + 28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 | │ │ │ │ │ │ 1 36 │ │ │ o15 : Matrix kk <-- kk │ │ │ │ │ │ i16 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o16 = ideal (a - 44b*c - 35c + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c + │ │ │ + 2 2 2 │ │ │ +o16 = ideal (a + 25b*c - 43c + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 15c - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c - 16a*d + 2b*d + │ │ │ + 2 2 2 │ │ │ + 46c + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c + 39a*d - 20b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 2 │ │ │ - 19c*d + 34d , b - 30b*c + 19c - 22a*d + 21b*d + 50c*d - 12d , b*c - │ │ │ + 2 2 2 2 2 │ │ │ + - 22c*d - 2d , b + 24b*c + 19c - 10a*d + 21b*d - 16c*d - 35d , b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 3 3 2 │ │ │ - 29b*c*d - 36c d + 24a*d + 2b*d + 50c*d + 9d , c - 24b*c*d + 39c d - │ │ │ + 2 2 2 2 3 3 2 │ │ │ + 29b*c*d - 30c d - 36a*d + 34b*d + 48c*d - 23d , c - 24b*c*d - 16c d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 3 │ │ │ - 8a*d - 29b*d - 29c*d - 17d ) │ │ │ + 2 2 2 3 │ │ │ + + 19a*d - 38b*d - 29c*d - 26d ) │ │ │ │ │ │ o16 : Ideal of S │ │ │ │ │ │ i17 : betti res F1 │ │ │ │ │ │ 0 1 2 3 │ │ │ o17 = total: 1 6 8 3 │ │ │ @@ -150,28 +150,28 @@ │ │ │ 1: . 4 4 1 │ │ │ 2: . 2 4 2 │ │ │ │ │ │ o17 : BettiTally │ │ │ │ │ │ i18 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 │ │ │ -o18 = ideal (a - 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c - │ │ │ + 2 2 2 │ │ │ +o18 = ideal (a - 9b*c - 18c - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 45c - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c + 33a*d + 16b*d │ │ │ + 2 2 2 │ │ │ + 6c + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c + 27a*d - 47b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 2 │ │ │ - - 18c*d - 39d , b - 47b*c - 28c - 5b*d - c*d - 30d , b*c - 43b*c*d + │ │ │ + 2 2 2 2 2 │ │ │ + - 28c*d - d , b + 19b*c - 13c - 37b*d + 32c*d + 15d , b*c - 43b*c*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 3 3 2 2 │ │ │ - 19c d + 34a*d + 21b*d + 46c*d + 20d , c + 38b*c*d + 22c d - 15a*d │ │ │ + 2 2 2 2 3 3 2 2 │ │ │ + - 47c d + 34a*d - 22b*d - 6c*d + 17d , c + 2b*c*d + 22c d - 18a*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 3 │ │ │ - - 47b*d - 39c*d + 40d ) │ │ │ + 2 2 3 │ │ │ + + 38b*d - 39c*d - d ) │ │ │ │ │ │ o18 : Ideal of S │ │ │ │ │ │ i19 : betti res F2 │ │ │ │ │ │ 0 1 2 3 │ │ │ o19 = total: 1 6 8 3 │ │ │ @@ -179,27 +179,30 @@ │ │ │ 1: . 4 4 1 │ │ │ 2: . 2 4 2 │ │ │ │ │ │ o19 : BettiTally │ │ │ │ │ │ i20 : netList decompose F1 │ │ │ │ │ │ - +------------------------------------------------------+ │ │ │ -o20 = |ideal (c - 13d, b + 32d, a + 36d) | │ │ │ - +------------------------------------------------------+ │ │ │ - |ideal (c - 16d, b + d, a + 16d) | │ │ │ - +------------------------------------------------------+ │ │ │ - | 2 2 | │ │ │ - |ideal (b - 6c + 33d, a - 36c + 2d, c + 43c*d - d ) | │ │ │ - +------------------------------------------------------+ │ │ │ - | 2 2 | │ │ │ - |ideal (b + 29c + 7d, a - 19c + 24d, c - 20c*d - 30d )| │ │ │ - +------------------------------------------------------+ │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d) | │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 3 2 2 2 3 2 2 2 3 | │ │ │ + |ideal (a - 29b - 8c - 13d, b + 24b*c + 19c + 34b*d + 5c*d + 37d , c - 24b*c*d - 16c d + 8b*d + 22c*d + 19d , b*c - 29b*c*d - 30c d - 38c*d + 14d )| │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i21 : netList decompose F2 │ │ │ │ │ │ - +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ - | 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 | │ │ │ -o21 = |ideal (a*c + 2b*c - 13c + 33a*d + 16b*d - 18c*d - 39d , b - 47b*c - 28c - 5b*d - c*d - 30d , a*b - 20b*c - 45c - 35a*d + 21b*d - 34c*d + 10d , a - 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , c + 38b*c*d + 22c d - 15a*d - 47b*d - 39c*d + 40d , b*c - 43b*c*d + 19c d + 34a*d + 21b*d + 46c*d + 20d )| │ │ │ - +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + +-------------------------------------------------------+ │ │ │ +o21 = |ideal (c - 32d, b - 5d, a - 29d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 43d, b - 47d, a - 27d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 24d, b - 49d, a) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 14d, b + 31d, a - 16d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b + 11c + 22d, a + 11c + 42d, c - 43c*d + 31d )| │ │ │ + +-------------------------------------------------------+ │ │ │ │ │ │ i22 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ @@ -200,67 +200,72 @@ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o13 = | 50 15 46 -33 2 -43 -46 8 33 19 -2 -18 -8 -22 43 -29 19 3 -16 -29 -38 │ │ │ +o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8 │ │ │ ----------------------------------------------------------------------- │ │ │ - -24 -10 -29 | │ │ │ + -29 -22 -29 -24 | │ │ │ │ │ │ 1 24 │ │ │ o13 : Matrix kk <-- kk │ │ │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o14 = ideal (a + 33b*c - 33c + 19a*d + 2b*d + 15c*d + 50d , a*b + 43b*c - │ │ │ + 2 2 2 │ │ │ +o14 = ideal (a - 30b*c + 23c - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 2c - 29a*d - 18b*d - 43c*d + 46d , a*c - 38b*c + 19c - 24a*d + 3b*d - │ │ │ + 2 2 2 │ │ │ + 33c - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c - 22a*d + 32b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - 8c*d - 46d , b - 10b*c - 16c - 29a*d - 29b*d - 22c*d + 8d ) │ │ │ + 2 2 2 2 │ │ │ + + 19c*d - 4d , b - 29b*c - 29c - 24a*d - 8b*d + 19c*d - 12d ) │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ │ │ i15 : decompose F1 │ │ │ │ │ │ - 2 2 2 │ │ │ -o15 = {ideal (a - 38b + 19c + 44d, b - 10b*c - 16c - 20b*d + 24c*d - 29d ), │ │ │ + 2 2 2 │ │ │ +o15 = {ideal (a - 29b - 38c - 9d, b - 29b*c - 29c + 3b*d + 16c*d - 26d ), │ │ │ ----------------------------------------------------------------------- │ │ │ - ideal (c - 24d, b - 38d, a + 15d)} │ │ │ + ideal (c - 22d, b - 21d, a + 8d)} │ │ │ │ │ │ o15 : List │ │ │ │ │ │ i16 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19 │ │ │ +o16 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47 │ │ │ ----------------------------------------------------------------------- │ │ │ - -47 -39 34 0 | │ │ │ + 34 19 21 39 0 | │ │ │ │ │ │ 1 24 │ │ │ o16 : Matrix kk <-- kk │ │ │ │ │ │ i17 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 │ │ │ -o17 = ideal (a - 8b*c + 26c - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c - │ │ │ + 2 2 2 2 │ │ │ +o17 = ideal (a + 17b*c - 20c - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 38c + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c - 39a*d + 21b*d │ │ │ + 2 2 │ │ │ + - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c + 21a*d - 13b*d - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - - 13c*d - 35d , b + 34b*c - 18c + 19b*d + 29c*d + 41d ) │ │ │ + 2 2 2 2 │ │ │ + 39c*d - 43d , b + 39b*c - 47c + 34b*d + 40c*d - 41d ) │ │ │ │ │ │ o17 : Ideal of S │ │ │ │ │ │ i18 : decompose F2 │ │ │ │ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c + │ │ │ + 2 2 2 │ │ │ +o18 = {ideal (a*c + 19b*c - 18c + 21a*d - 13b*d - 39c*d - 43d , b + 39b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 26d)} │ │ │ + 2 2 2 │ │ │ + 47c + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c - 39a*d - 29b*d - 30c*d │ │ │ + ----------------------------------------------------------------------- │ │ │ + 2 2 2 2 │ │ │ + + 16d , a + 17b*c - 20c - 4a*d - b*d - 2c*d + 46d )} │ │ │ │ │ │ o18 : List │ │ │ │ │ │ i19 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.out │ │ │ @@ -68,54 +68,54 @@ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : netList randomPointsOnRationalVariety(compsJ_0, 10) │ │ │ │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ -o13 = || 29 -40 15 -49 3 -13 -6 -39 2 39 47 15 19 -47 -46 -39 -16 32 -43 34 -13 -18 21 -38 | | │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ - || 37 -7 -24 8 -26 38 9 -31 24 -47 -34 12 16 22 -22 45 -28 16 -47 2 -48 -34 38 -15 | | │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ - || 6 1 -31 -7 44 8 -50 24 -48 -16 23 23 -23 39 -5 43 19 -15 48 15 -11 -17 7 47 | | │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ - || -41 -49 6 -16 -12 31 23 6 -7 11 3 -42 40 11 -28 46 35 -28 -3 33 1 -28 -38 36 | | │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ - || -11 -27 -4 40 -34 6 44 -2 19 -23 -29 21 29 -47 -37 15 -47 -24 -10 2 -13 -37 -7 22 | | │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ - || -50 42 20 -30 -46 -48 -5 40 -47 39 13 47 32 -9 41 -32 -18 25 -30 -22 24 -20 27 30 | | │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ - || 50 22 -30 3 -43 -29 -33 -18 6 39 -29 24 -49 -33 -15 -19 -15 -37 44 33 -20 17 0 -48 || │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ - || -9 31 -37 -42 -7 -8 -11 -21 12 9 13 -9 13 -26 11 22 36 34 -8 4 -11 -49 -39 -39 | | │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ - || 47 14 -11 -16 -20 -40 42 5 -2 36 8 -45 -30 41 -26 16 -8 -34 35 -22 -6 -28 -3 43 | | │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ - || 23 -8 -3 -17 38 0 11 -33 -7 6 -31 -4 -31 25 6 -2 -35 -11 -13 3 -49 -41 40 -9 | | │ │ │ - +--------------------------------------------------------------------------------------+ │ │ │ - │ │ │ -i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) │ │ │ - │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 | | │ │ │ +o13 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -18 -43 21 -38 | | │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ - || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 || │ │ │ + || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -34 -47 38 -15 || │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ - || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 | | │ │ │ + || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17 48 7 47 | | │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ - || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 | | │ │ │ + || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 -3 -38 36 | | │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ - || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 | | │ │ │ + || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -37 -10 -7 22 | | │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ - || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 | | │ │ │ + || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -20 -30 27 30 | | │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ - || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 | | │ │ │ + || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 17 44 0 -48 | | │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ - || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 | | │ │ │ + || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -49 -8 -39 -39 | | │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ - || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 | | │ │ │ + || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 35 -3 43 | | │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ - || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 | | │ │ │ + || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -41 -13 40 -9 | | │ │ │ +-------------------------------------------------------------------------------------+ │ │ │ │ │ │ +i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) │ │ │ + │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 -35 -47 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 -48 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 -22 -18 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42 30 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27 -18 18 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 -39 19 20 -37 0 || │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33 -9 -28 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42 -28 26 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 -13 4 0 | | │ │ │ + +---------------------------------------------------------------------------------------+ │ │ │ + │ │ │ i15 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ @@ -232,52 +232,52 @@ │ │ │ o13 = 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21
│ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 |
│ │ │ +      40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │ +      28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                            2               
│ │ │ -o16 = ideal (a  - 44b*c - 35c  + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c +
│ │ │ +              2              2                             2               
│ │ │ +o16 = ideal (a  + 25b*c - 43c  + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                 
│ │ │ -      15c  - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c  - 16a*d + 2b*d +
│ │ │ +         2                              2                  2                
│ │ │ +      46c  + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c  + 39a*d - 20b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                              2     2  
│ │ │ -      19c*d + 34d , b  - 30b*c + 19c  - 22a*d + 21b*d + 50c*d - 12d , b*c  -
│ │ │ +                  2   2              2                              2     2  
│ │ │ +      - 22c*d - 2d , b  + 24b*c + 19c  - 10a*d + 21b*d - 16c*d - 35d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2       2        2     3   3                2   
│ │ │ -      29b*c*d - 36c d + 24a*d  + 2b*d  + 50c*d  + 9d , c  - 24b*c*d + 39c d -
│ │ │ +                   2         2        2        2      3   3                2 
│ │ │ +      29b*c*d - 30c d - 36a*d  + 34b*d  + 48c*d  - 23d , c  - 24b*c*d - 16c d
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2        2        2      3
│ │ │ -      8a*d  - 29b*d  - 29c*d  - 17d )
│ │ │ +             2        2        2      3
│ │ │ +      + 19a*d  - 38b*d  - 29c*d  - 26d )
│ │ │  
│ │ │  o16 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : betti res F1
│ │ │ @@ -291,28 +291,28 @@
│ │ │  o17 : BettiTally
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │ +              2             2                              2               
│ │ │ +o18 = ideal (a  - 9b*c - 18c  - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                  2                
│ │ │ -      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │ +        2                              2                   2                
│ │ │ +      6c  + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c  + 27a*d - 47b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                   2     2            
│ │ │ -      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*d +
│ │ │ +                 2   2              2                      2     2          
│ │ │ +      - 28c*d - d , b  + 19b*c - 13c  - 37b*d + 32c*d + 15d , b*c  - 43b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2         2        2        2      3   3                2         2
│ │ │ -      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*d 
│ │ │ +           2         2        2       2      3   3               2         2
│ │ │ +      - 47c d + 34a*d  - 22b*d  - 6c*d  + 17d , c  + 2b*c*d + 22c d - 18a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2      3
│ │ │ -      - 47b*d  - 39c*d  + 40d )
│ │ │ +             2        2    3
│ │ │ +      + 38b*d  - 39c*d  - d )
│ │ │  
│ │ │  o18 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i19 : betti res F2
│ │ │ @@ -331,35 +331,38 @@
│ │ │            
    What are the ideals F1 and F2?
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i20 : netList decompose F1
│ │ │  
│ │ │ -      +------------------------------------------------------+
│ │ │ -o20 = |ideal (c - 13d, b + 32d, a + 36d)                     |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |ideal (c - 16d, b + d, a + 16d)                       |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                    2            2    |
│ │ │ -      |ideal (b - 6c + 33d, a - 36c + 2d, c  + 43c*d - d )   |
│ │ │ -      +------------------------------------------------------+
│ │ │ -      |                                     2              2 |
│ │ │ -      |ideal (b + 29c + 7d, a - 19c + 24d, c  - 20c*d - 30d )|
│ │ │ -      +------------------------------------------------------+
    │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d) | │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 3 2 2 2 3 2 2 2 3 | │ │ │ + |ideal (a - 29b - 8c - 13d, b + 24b*c + 19c + 34b*d + 5c*d + 37d , c - 24b*c*d - 16c d + 8b*d + 22c*d + 19d , b*c - 29b*c*d - 30c d - 38c*d + 14d )| │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ 
    i21 : netList decompose F2
│ │ │  
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                       2                              2   2              2                   2                   2                              2   2              2                              2   3                2         2        2        2      3     2                2         2        2        2      3 |
│ │ │ -o21 = |ideal (a*c + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  - 47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d )|
│ │ │ -      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    │ │ │ + +-------------------------------------------------------+ │ │ │ +o21 = |ideal (c - 32d, b - 5d, a - 29d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 43d, b - 47d, a - 27d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 24d, b - 49d, a) | │ │ │ + +-------------------------------------------------------+ │ │ │ + |ideal (c + 14d, b + 31d, a - 16d) | │ │ │ + +-------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b + 11c + 22d, a + 11c + 42d, c - 43c*d + 31d )| │ │ │ + +-------------------------------------------------------+ │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -170,117 +170,115 @@ │ │ │ │ 32 13 21 33 19 31 │ │ │ │ i12 : compsJ = decompose J; │ │ │ │ i13 : #compsJ │ │ │ │ │ │ │ │ o13 = 2 │ │ │ │ i14 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o14 = | 42 9 39 9 34 7 -12 -17 -29 -35 50 2 13 19 -44 50 2 -29 15 2 -27 21 │ │ │ │ +o14 = | -6 48 44 -23 -2 -11 -35 -26 27 -43 48 27 15 -22 25 -16 34 -29 46 -20 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -36 -29 -39 -10 24 -16 19 -29 39 -38 -22 -8 -30 -24 | │ │ │ │ + 40 21 -30 -38 -19 -8 -36 39 19 -29 -16 -29 -10 19 24 -24 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o14 : Matrix kk <-- kk │ │ │ │ i15 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16 │ │ │ │ +o15 = | -48 -46 16 17 -1 -43 15 -1 12 -18 -6 -28 14 -28 -9 32 -22 -39 6 -47 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 | │ │ │ │ + 28 -37 -47 38 -16 -15 34 27 -13 -43 22 16 0 -18 19 2 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o15 : Matrix kk <-- kk │ │ │ │ i16 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o16 = ideal (a - 44b*c - 35c + 2a*d + 7b*d + 39c*d + 42d , a*b - 39b*c + │ │ │ │ + 2 2 2 │ │ │ │ +o16 = ideal (a + 25b*c - 43c + 27a*d - 11b*d + 44c*d - 6d , a*b - 19b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 15c - 27a*d + 13b*d - 29c*d + 9d , a*c - 38b*c - 10c - 16a*d + 2b*d + │ │ │ │ + 2 2 2 │ │ │ │ + 46c + 40a*d + 15b*d + 27c*d + 48d , a*c - 29b*c - 8c + 39a*d - 20b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 2 │ │ │ │ - 19c*d + 34d , b - 30b*c + 19c - 22a*d + 21b*d + 50c*d - 12d , b*c - │ │ │ │ + 2 2 2 2 2 │ │ │ │ + - 22c*d - 2d , b + 24b*c + 19c - 10a*d + 21b*d - 16c*d - 35d , b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 │ │ │ │ - 29b*c*d - 36c d + 24a*d + 2b*d + 50c*d + 9d , c - 24b*c*d + 39c d - │ │ │ │ + 2 2 2 2 3 3 2 │ │ │ │ + 29b*c*d - 30c d - 36a*d + 34b*d + 48c*d - 23d , c - 24b*c*d - 16c d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 3 │ │ │ │ - 8a*d - 29b*d - 29c*d - 17d ) │ │ │ │ + 2 2 2 3 │ │ │ │ + + 19a*d - 38b*d - 29c*d - 26d ) │ │ │ │ │ │ │ │ o16 : Ideal of S │ │ │ │ i17 : betti res F1 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o17 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ 1: . 4 4 1 │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o17 : BettiTally │ │ │ │ i18 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o18 = ideal (a - 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o18 = ideal (a - 9b*c - 18c - 28a*d - 43b*d + 16c*d - 48d , a*b - 16b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 45c - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c + 33a*d + 16b*d │ │ │ │ + 2 2 2 │ │ │ │ + 6c + 28a*d + 14b*d + 12c*d - 46d , a*c + 16b*c - 15c + 27a*d - 47b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 2 │ │ │ │ - - 18c*d - 39d , b - 47b*c - 28c - 5b*d - c*d - 30d , b*c - 43b*c*d + │ │ │ │ + 2 2 2 2 2 │ │ │ │ + - 28c*d - d , b + 19b*c - 13c - 37b*d + 32c*d + 15d , b*c - 43b*c*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 2 │ │ │ │ - 19c d + 34a*d + 21b*d + 46c*d + 20d , c + 38b*c*d + 22c d - 15a*d │ │ │ │ + 2 2 2 2 3 3 2 2 │ │ │ │ + - 47c d + 34a*d - 22b*d - 6c*d + 17d , c + 2b*c*d + 22c d - 18a*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 3 │ │ │ │ - - 47b*d - 39c*d + 40d ) │ │ │ │ + 2 2 3 │ │ │ │ + + 38b*d - 39c*d - d ) │ │ │ │ │ │ │ │ o18 : Ideal of S │ │ │ │ i19 : betti res F2 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ o19 = total: 1 6 8 3 │ │ │ │ 0: 1 . . . │ │ │ │ 1: . 4 4 1 │ │ │ │ 2: . 2 4 2 │ │ │ │ │ │ │ │ o19 : BettiTally │ │ │ │ What are the ideals F1 and F2? │ │ │ │ i20 : netList decompose F1 │ │ │ │ │ │ │ │ - +------------------------------------------------------+ │ │ │ │ -o20 = |ideal (c - 13d, b + 32d, a + 36d) | │ │ │ │ - +------------------------------------------------------+ │ │ │ │ - |ideal (c - 16d, b + d, a + 16d) | │ │ │ │ - +------------------------------------------------------+ │ │ │ │ - | 2 2 | │ │ │ │ - |ideal (b - 6c + 33d, a - 36c + 2d, c + 43c*d - d ) | │ │ │ │ - +------------------------------------------------------+ │ │ │ │ - | 2 2 | │ │ │ │ - |ideal (b + 29c + 7d, a - 19c + 24d, c - 20c*d - 30d )| │ │ │ │ - +------------------------------------------------------+ │ │ │ │ -i21 : netList decompose F2 │ │ │ │ - │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ +--+ │ │ │ │ +o20 = |ideal (c + 39d, b + 27d, a - 18d) │ │ │ │ +| │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ --+ │ │ │ │ - | 2 2 2 │ │ │ │ -2 2 2 2 2 │ │ │ │ -2 2 3 2 2 2 │ │ │ │ -2 3 2 2 2 2 2 3 | │ │ │ │ -o21 = |ideal (a*c + 2b*c - 13c + 33a*d + 16b*d - 18c*d - 39d , b - 47b*c - │ │ │ │ -28c - 5b*d - c*d - 30d , a*b - 20b*c - 45c - 35a*d + 21b*d - 34c*d + 10d , a │ │ │ │ -- 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , c + 38b*c*d + 22c d - 15a*d - │ │ │ │ -47b*d - 39c*d + 40d , b*c - 43b*c*d + 19c d + 34a*d + 21b*d + 46c*d + 20d │ │ │ │ +--+ │ │ │ │ + | 2 2 2 3 │ │ │ │ +2 2 2 3 2 2 2 3 | │ │ │ │ + |ideal (a - 29b - 8c - 13d, b + 24b*c + 19c + 34b*d + 5c*d + 37d , c - │ │ │ │ +24b*c*d - 16c d + 8b*d + 22c*d + 19d , b*c - 29b*c*d - 30c d - 38c*d + 14d │ │ │ │ )| │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ -------------------------------------------------------------------------------- │ │ │ │ --+ │ │ │ │ +--+ │ │ │ │ +i21 : netList decompose F2 │ │ │ │ + │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ +o21 = |ideal (c - 32d, b - 5d, a - 29d) | │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ + |ideal (c + 43d, b - 47d, a - 27d) | │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ + |ideal (c + 24d, b - 49d, a) | │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ + |ideal (c + 14d, b + 31d, a - 16d) | │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ + | 2 2 | │ │ │ │ + |ideal (b + 11c + 22d, a + 11c + 42d, c - 43c*d + 31d )| │ │ │ │ + +-------------------------------------------------------+ │ │ │ │ We can determine what these represent. One should be a set of 6 points, where 5 │ │ │ │ lie on a plane. The other should be 6 points with 3 points on one line, and the │ │ │ │ other 3 points on a skew line. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y -- find a random point on a variety that can │ │ │ │ be detected to be rational │ │ │ │ ********** WWaayyss ttoo uussee nnoonnmmiinniimmaallMMaappss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ @@ -318,90 +318,95 @@ │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | 50 15 46 -33 2 -43 -46 8 33 19 -2 -18 -8 -22 43 -29 19 3 -16 -29 -38
│ │ │ +o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -24 -10 -29 |
│ │ │ +      -29 -22 -29 -24 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o14 = ideal (a  + 33b*c - 33c  + 19a*d + 2b*d + 15c*d + 50d , a*b + 43b*c -
│ │ │ +              2              2                              2               
│ │ │ +o14 = ideal (a  - 30b*c + 23c  - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2                   2                 
│ │ │ -      2c  - 29a*d - 18b*d - 43c*d + 46d , a*c - 38b*c + 19c  - 24a*d + 3b*d -
│ │ │ +         2                              2                   2                
│ │ │ +      33c  - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c  - 22a*d + 32b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                2   2              2                             2
│ │ │ -      8c*d - 46d , b  - 10b*c - 16c  - 29a*d - 29b*d - 22c*d + 8d )
│ │ │ +                  2   2              2                             2
│ │ │ +      + 19c*d - 4d , b  - 29b*c - 29c  - 24a*d - 8b*d + 19c*d - 12d )
│ │ │  
│ │ │  o14 : Ideal of S
    │ │ │
    │ │ │ 
    i15 : decompose F1
│ │ │  
│ │ │ -                                    2              2                      2
│ │ │ -o15 = {ideal (a - 38b + 19c + 44d, b  - 10b*c - 16c  - 20b*d + 24c*d - 29d ),
│ │ │ +                                   2              2                     2
│ │ │ +o15 = {ideal (a - 29b - 38c - 9d, b  - 29b*c - 29c  + 3b*d + 16c*d - 26d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 24d, b - 38d, a + 15d)}
│ │ │ +      ideal (c - 22d, b - 21d, a + 8d)}
│ │ │  
│ │ │  o15 : List
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i16 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19
│ │ │ +o16 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -47 -39 34 0 |
│ │ │ +      34 19 21 39 0 |
│ │ │  
│ │ │                 1       24
│ │ │  o16 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i17 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2             2                              2               
│ │ │ -o17 = ideal (a  - 8b*c + 26c  - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c -
│ │ │ +              2              2                          2                   2
│ │ │ +o17 = ideal (a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                
│ │ │ -      38c  + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c  - 39a*d + 21b*d
│ │ │ +                                   2                   2                  
│ │ │ +      - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c  + 21a*d - 13b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2
│ │ │ -      - 13c*d - 35d , b  + 34b*c - 18c  + 19b*d + 29c*d + 41d )
│ │ │ +                 2   2              2                      2
│ │ │ +      39c*d - 43d , b  + 39b*c - 47c  + 34b*d + 40c*d - 41d )
│ │ │  
│ │ │  o17 : Ideal of S
    │ │ │
    │ │ │ 
    i18 : decompose F2
│ │ │  
│ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c +
│ │ │ +                               2                              2   2          
│ │ │ +o18 = {ideal (a*c + 19b*c - 18c  + 21a*d - 13b*d - 39c*d - 43d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      26d)}
│ │ │ +         2                      2                   2                        
│ │ │ +      47c  + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c  - 39a*d - 29b*d - 30c*d
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +           2   2              2                          2
│ │ │ +      + 16d , a  + 17b*c - 20c  - 4a*d - b*d - 2c*d + 46d )}
│ │ │  
│ │ │  o18 : List
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │ ├── html2text {} │ │ │ │ @@ -212,67 +212,72 @@ │ │ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ There are 2 components. We attempt to find a point on the first component │ │ │ │ i13 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o13 = | 50 15 46 -33 2 -43 -46 8 33 19 -2 -18 -8 -22 43 -29 19 3 -16 -29 -38 │ │ │ │ +o13 = | 13 48 43 23 41 36 -4 -12 -30 -16 -33 -36 19 19 30 -10 -38 32 -29 -8 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -24 -10 -29 | │ │ │ │ + -29 -22 -29 -24 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o13 : Matrix kk <-- kk │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o14 = ideal (a + 33b*c - 33c + 19a*d + 2b*d + 15c*d + 50d , a*b + 43b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o14 = ideal (a - 30b*c + 23c - 16a*d + 41b*d + 48c*d + 13d , a*b + 30b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 2c - 29a*d - 18b*d - 43c*d + 46d , a*c - 38b*c + 19c - 24a*d + 3b*d - │ │ │ │ + 2 2 2 │ │ │ │ + 33c - 10a*d - 36b*d + 36c*d + 43d , a*c - 29b*c - 38c - 22a*d + 32b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 8c*d - 46d , b - 10b*c - 16c - 29a*d - 29b*d - 22c*d + 8d ) │ │ │ │ + 2 2 2 2 │ │ │ │ + + 19c*d - 4d , b - 29b*c - 29c - 24a*d - 8b*d + 19c*d - 12d ) │ │ │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ i15 : decompose F1 │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o15 = {ideal (a - 38b + 19c + 44d, b - 10b*c - 16c - 20b*d + 24c*d - 29d ), │ │ │ │ + 2 2 2 │ │ │ │ +o15 = {ideal (a - 29b - 38c - 9d, b - 29b*c - 29c + 3b*d + 16c*d - 26d ), │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - ideal (c - 24d, b - 38d, a + 15d)} │ │ │ │ + ideal (c - 22d, b - 21d, a + 8d)} │ │ │ │ │ │ │ │ o15 : List │ │ │ │ We attempt to find a point on the second component in parameter space, and its │ │ │ │ corresponding ideal. │ │ │ │ i16 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o16 = | -14 40 -5 26 -48 -26 -35 41 -8 -15 -38 31 -13 29 21 16 39 21 -18 19 │ │ │ │ +o16 = | 46 -2 16 -20 -1 -30 -43 -41 17 -4 -16 -29 -39 40 49 -39 -18 -13 -47 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -47 -39 34 0 | │ │ │ │ + 34 19 21 39 0 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o16 : Matrix kk <-- kk │ │ │ │ i17 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o17 = ideal (a - 8b*c + 26c - 15a*d - 48b*d + 40c*d - 14d , a*b + 21b*c - │ │ │ │ + 2 2 2 2 │ │ │ │ +o17 = ideal (a + 17b*c - 20c - 4a*d - b*d - 2c*d + 46d , a*b + 49b*c - 16c │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 38c + 16a*d + 31b*d - 26c*d - 5d , a*c - 47b*c + 39c - 39a*d + 21b*d │ │ │ │ + 2 2 │ │ │ │ + - 39a*d - 29b*d - 30c*d + 16d , a*c + 19b*c - 18c + 21a*d - 13b*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - - 13c*d - 35d , b + 34b*c - 18c + 19b*d + 29c*d + 41d ) │ │ │ │ + 2 2 2 2 │ │ │ │ + 39c*d - 43d , b + 39b*c - 47c + 34b*d + 40c*d - 41d ) │ │ │ │ │ │ │ │ o17 : Ideal of S │ │ │ │ i18 : decompose F2 │ │ │ │ │ │ │ │ -o18 = {ideal (b + 19c - 18d, a + 23c + 43d), ideal (b + 15c + 37d, a + 37c + │ │ │ │ + 2 2 2 │ │ │ │ +o18 = {ideal (a*c + 19b*c - 18c + 21a*d - 13b*d - 39c*d - 43d , b + 39b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 26d)} │ │ │ │ + 2 2 2 │ │ │ │ + 47c + 34b*d + 40c*d - 41d , a*b + 49b*c - 16c - 39a*d - 29b*d - 30c*d │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 2 2 2 2 │ │ │ │ + + 16d , a + 17b*c - 20c - 4a*d - b*d - 2c*d + 46d )} │ │ │ │ │ │ │ │ o18 : List │ │ │ │ It turns out that this is the ideal of 2 skew lines, just not defined over this │ │ │ │ field. │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ This routine expects the input to represent an irreducible variety │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html │ │ │ @@ -186,64 +186,64 @@ │ │ │

    There are 2 components. We attempt to find points on each of these two components. We are successful. This indicates that the corresponding varieties are both rational. Also, if we can find one point, we can find as many as we want.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ - │ │ │ - │ │ │ - │ │ │ │ │ │ + │ │ │ + │ │ │ + │ │ │
    │ │ │ 
    i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │  
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -o13 = || 29 -40 15 -49 3 -13 -6 -39 2 39 47 15 19 -47 -46 -39 -16 32 -43 34 -13 -18 21 -38 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 37 -7 -24 8 -26 38 9 -31 24 -47 -34 12 16 22 -22 45 -28 16 -47 2 -48 -34 38 -15 |   |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 6 1 -31 -7 44 8 -50 24 -48 -16 23 23 -23 39 -5 43 19 -15 48 15 -11 -17 7 47 |       |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -41 -49 6 -16 -12 31 23 6 -7 11 3 -42 40 11 -28 46 35 -28 -3 33 1 -28 -38 36 |      |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -11 -27 -4 40 -34 6 44 -2 19 -23 -29 21 29 -47 -37 15 -47 -24 -10 2 -13 -37 -7 22 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -50 42 20 -30 -46 -48 -5 40 -47 39 13 47 32 -9 41 -32 -18 25 -30 -22 24 -20 27 30 | |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 50 22 -30 3 -43 -29 -33 -18 6 39 -29 24 -49 -33 -15 -19 -15 -37 44 33 -20 17 0 -48 ||
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || -9 31 -37 -42 -7 -8 -11 -21 12 9 13 -9 13 -26 11 22 36 34 -8 4 -11 -49 -39 -39 |    |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 47 14 -11 -16 -20 -40 42 5 -2 36 8 -45 -30 41 -26 16 -8 -34 35 -22 -6 -28 -3 43 |   |
│ │ │ -      +--------------------------------------------------------------------------------------+
│ │ │ -      || 23 -8 -3 -17 38 0 11 -33 -7 6 -31 -4 -31 25 6 -2 -35 -11 -13 3 -49 -41 40 -9 |      |
│ │ │ -      +--------------------------------------------------------------------------------------+
    │ │ │ -
    │ │ │ - 
    i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ -
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 |   |
│ │ │ +o13 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -18 -43 21 -38 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 ||
│ │ │ +      || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -34 -47 38 -15 ||
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 |    |
│ │ │ +      || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17 48 7 47 |     |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 |     |
│ │ │ +      || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 -3 -38 36 |     |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 |    |
│ │ │ +      || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -37 -10 -7 22 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 |     |
│ │ │ +      || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -20 -30 27 30 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 |      |
│ │ │ +      || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 17 44 0 -48 |  |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 |    |
│ │ │ +      || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -49 -8 -39 -39 | |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 |    |
│ │ │ +      || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 35 -3 43 |      |
│ │ │        +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 |   |
│ │ │ +      || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -41 -13 40 -9 | |
│ │ │        +-------------------------------------------------------------------------------------+
    │ │ │
    │ │ │ + 
    i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ +
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 -35 -47 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 -48 0 |          |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 -22 -18 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13 0 |            |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42 30 0 |         |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27 -18 18 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 -39 19 20 -37 0 ||
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33 -9 -28 0 |     |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42 -28 26 0 |    |
│ │ │ +      +---------------------------------------------------------------------------------------+
│ │ │ +      || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 -13 4 0 |        |
│ │ │ +      +---------------------------------------------------------------------------------------+
    │ │ │ +
    │ │ │ │ │ │
    │ │ │

    Caveat

    │ │ │
    │ │ │

    This routine expects the input to represent an irreducible variety

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -84,99 +84,99 @@ │ │ │ │ There are 2 components. We attempt to find points on each of these two │ │ │ │ components. We are successful. This indicates that the corresponding varieties │ │ │ │ are both rational. Also, if we can find one point, we can find as many as we │ │ │ │ want. │ │ │ │ i13 : netList randomPointsOnRationalVariety(compsJ_0, 10) │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ -o13 = || 29 -40 15 -49 3 -13 -6 -39 2 39 47 15 19 -47 -46 -39 -16 32 -43 34 -13 │ │ │ │ --18 21 -38 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ - || 37 -7 -24 8 -26 38 9 -31 24 -47 -34 12 16 22 -22 45 -28 16 -47 2 -48 - │ │ │ │ -34 38 -15 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ - || 6 1 -31 -7 44 8 -50 24 -48 -16 23 23 -23 39 -5 43 19 -15 48 15 -11 -17 │ │ │ │ -7 47 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ - || -41 -49 6 -16 -12 31 23 6 -7 11 3 -42 40 11 -28 46 35 -28 -3 33 1 -28 │ │ │ │ --38 36 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ - || -11 -27 -4 40 -34 6 44 -2 19 -23 -29 21 29 -47 -37 15 -47 -24 -10 2 - │ │ │ │ -13 -37 -7 22 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ - || -50 42 20 -30 -46 -48 -5 40 -47 39 13 47 32 -9 41 -32 -18 25 -30 -22 │ │ │ │ -24 -20 27 30 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ - || 50 22 -30 3 -43 -29 -33 -18 6 39 -29 24 -49 -33 -15 -19 -15 -37 44 33 │ │ │ │ --20 17 0 -48 || │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ - || -9 31 -37 -42 -7 -8 -11 -21 12 9 13 -9 13 -26 11 22 36 34 -8 4 -11 -49 │ │ │ │ --39 -39 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ - || 47 14 -11 -16 -20 -40 42 5 -2 36 8 -45 -30 41 -26 16 -8 -34 35 -22 - │ │ │ │ -6 -28 -3 43 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ - || 23 -8 -3 -17 38 0 11 -33 -7 6 -31 -4 -31 25 6 -2 -35 -11 -13 3 -49 -41 │ │ │ │ -40 -9 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ ---------------+ │ │ │ │ -i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) │ │ │ │ - │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 │ │ │ │ -37 -47 0 | | │ │ │ │ +o13 = || 13 15 3 36 2 48 44 -35 -34 39 5 -32 34 19 -42 -47 -16 -34 -39 -13 -18 │ │ │ │ +-43 21 -38 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ - || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 - │ │ │ │ -48 30 -48 0 || │ │ │ │ + || -43 48 14 29 -47 -10 47 22 8 -47 15 -26 2 16 -49 22 -28 -18 45 -48 -34 │ │ │ │ +-47 38 -15 || │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ - || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 │ │ │ │ -40 -18 0 | | │ │ │ │ + || -3 45 42 47 -50 16 -30 28 43 -16 24 19 15 -23 37 39 19 -8 43 -11 -17 │ │ │ │ +48 7 47 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ - || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 │ │ │ │ -3 13 0 | | │ │ │ │ + || -49 7 32 -6 -30 -41 -10 2 44 11 -25 4 33 40 -19 11 35 -17 46 1 -28 - │ │ │ │ +3 -38 36 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ - || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 - │ │ │ │ -18 30 0 | | │ │ │ │ + || 35 -48 -2 45 -35 29 34 12 -32 -23 50 2 2 29 -3 -47 -47 -34 15 -13 -37 │ │ │ │ +-10 -7 22 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ - || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 │ │ │ │ -12 18 0 | | │ │ │ │ + || 47 8 -14 6 -1 -13 -7 16 -20 39 -34 -22 -22 32 17 -9 -18 -6 -32 24 -20 │ │ │ │ +-30 27 30 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ - || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 │ │ │ │ --37 0 | | │ │ │ │ + || -2 -36 -39 41 -6 34 -10 42 5 39 20 33 33 -49 -15 -33 -15 41 -19 -20 17 │ │ │ │ +44 0 -48 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ - || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 │ │ │ │ -6 -28 0 | | │ │ │ │ + || -30 37 -9 16 -36 19 -13 -14 -19 9 -33 5 4 13 44 -26 36 -12 22 -11 -49 │ │ │ │ +-8 -39 -39 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ - || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 - │ │ │ │ -33 26 0 | | │ │ │ │ + || 27 41 32 -44 40 -20 41 33 28 36 44 31 -22 -30 9 41 -8 30 16 -6 -28 35 │ │ │ │ +-3 43 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ - || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 │ │ │ │ --20 4 0 | | │ │ │ │ + || 37 -2 17 -42 -42 -12 18 -31 33 6 19 -31 3 -31 -11 25 -35 28 -2 -49 -41 │ │ │ │ +-13 40 -9 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -------------+ │ │ │ │ +i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) │ │ │ │ + │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ +o14 = || -41 -1 -48 25 40 4 35 16 26 -41 -28 -16 27 -14 -39 4 4 30 -40 37 -31 - │ │ │ │ +35 -47 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -1 19 -3 12 50 3 4 25 48 50 34 -6 -29 6 -5 36 -39 -31 -48 30 47 -37 - │ │ │ │ +48 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -27 -3 -40 22 27 3 -28 -41 -12 -34 -10 40 46 29 30 24 -49 28 1 40 10 - │ │ │ │ +22 -18 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -26 -6 24 28 -27 26 34 47 13 50 3 -42 -17 5 4 -35 7 30 -13 3 8 -41 13 │ │ │ │ +0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || 49 -7 48 1 48 25 25 -10 49 36 -16 35 -46 -5 25 -33 8 -29 49 -18 23 42 │ │ │ │ +30 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -35 28 -6 22 50 -49 2 -5 -11 -39 30 27 -16 34 -9 -34 -28 15 -46 12 27 │ │ │ │ +-18 18 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -49 -44 -16 -10 48 18 22 33 -35 -48 -28 -8 -23 -48 -25 -3 -21 23 44 - │ │ │ │ +39 19 20 -37 0 || │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || -33 -14 -18 10 2 -43 -26 45 10 19 -15 25 47 9 -15 -22 0 -47 -28 6 -33 │ │ │ │ +-9 -28 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || 20 -27 -17 2 -47 -23 13 40 -19 -13 39 -23 5 -3 47 -6 28 -29 -37 -33 42 │ │ │ │ +-28 26 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ + || 19 10 -10 47 41 20 -43 -34 -43 2 44 29 22 35 -42 16 44 30 5 -20 -29 - │ │ │ │ +13 4 0 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +---------------+ │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ This routine expects the input to represent an irreducible variety │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y_(_I_d_e_a_l_) -- find a random point on a variety │ │ │ │ that can be detected to be rational │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _r_a_n_d_o_m_P_o_i_n_t_s_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y_(_I_d_e_a_l_,_Z_Z_) -- find random points on a │ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out │ │ │ @@ -11,21 +11,21 @@ │ │ │ i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}]; │ │ │ │ │ │ i4 : I2 = sub(I1, R2); │ │ │ │ │ │ o4 : Ideal of R2 │ │ │ │ │ │ i5 : elapsedTime gb I2 │ │ │ - -- 3.14967s elapsed │ │ │ + -- 2.25281s elapsed │ │ │ │ │ │ o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16] │ │ │ │ │ │ o5 : GroebnerBasis │ │ │ │ │ │ i6 : elapsedTime groebnerWalk(gb I1, R2) │ │ │ - -- 2.15229s elapsed │ │ │ + -- 1.70012s elapsed │ │ │ │ │ │ o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0] │ │ │ │ │ │ o6 : GroebnerBasis │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/html/index.html │ │ │ @@ -92,30 +92,30 @@ │ │ │ │ │ │ o4 : Ideal of R2     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime gb I2
│ │ │ - -- 3.14967s elapsed
│ │ │ + -- 2.25281s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 2.15229s elapsed
│ │ │ + -- 1.70012s 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 │ │ │ │ - -- 3.14967s elapsed │ │ │ │ + -- 2.25281s elapsed │ │ │ │ │ │ │ │ o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16] │ │ │ │ │ │ │ │ o5 : GroebnerBasis │ │ │ │ but it is faster to compute directly in the first order and then use the │ │ │ │ Groebner walk. │ │ │ │ i6 : elapsedTime groebnerWalk(gb I1, R2) │ │ │ │ - -- 2.15229s elapsed │ │ │ │ + -- 1.70012s elapsed │ │ │ │ │ │ │ │ o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0] │ │ │ │ │ │ │ │ o6 : GroebnerBasis │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ The target ring must be the same ring as the ring of the starting ideal, except │ │ │ │ with different monomial order. The ring must be a polynomial ring over a field. │ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out │ │ │ @@ -44,19 +44,19 @@ │ │ │ o5 = {9, 1, 99999, 9999999, 3, 999} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : netList cssLeadTerm(Hbeta, w) │ │ │ Warning: F4 Algorithm not available over current coefficient ring or inhomogeneous ideal. │ │ │ Converting to Naive algorithm. │ │ │ - -- .00000508s elapsed │ │ │ - -- .000003055s elapsed │ │ │ - -- .00000561s elapsed │ │ │ - -- .000003727s elapsed │ │ │ - -- .000003476s elapsed │ │ │ + -- .000005598s elapsed │ │ │ + -- .000007122s elapsed │ │ │ + -- .000006739s elapsed │ │ │ + -- .00000619s elapsed │ │ │ + -- .000008272s elapsed │ │ │ │ │ │ +----------------------------------------------------+ │ │ │ | 1 5 5 5 | │ │ │ | - - - - - - | │ │ │ | 2 2 2 2 | │ │ │ o6 = |x x x x | │ │ │ | 1 2 4 5 | │ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out │ │ │ @@ -3,15 +3,15 @@ │ │ │ i1 : R = QQ[t_1..t_5]; │ │ │ │ │ │ i2 : I = ideal(t_1+t_2+t_3+t_4+t_5, t_1+t_2-t_4, t_2+t_3-t_4, t_1*t_3, t_2*t_4); │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ i3 : solveFrobeniusIdeal I │ │ │ - -- .000004529s elapsed │ │ │ + -- .000006559s elapsed │ │ │ Warning: F4 Algorithm not available over current coefficient ring or inhomogeneous ideal. │ │ │ Converting to Naive algorithm. │ │ │ │ │ │ │ │ │ o3 = {1, - 2logX + 3logX - 2logX + logX , - logX + logX - logX + logX , │ │ │ 0 1 2 3 0 1 2 4 │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -24,15 +24,15 @@ │ │ │ 2 4 0 4 4 1 2 4 2 4 4 3 4 │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : W = makeWeylAlgebra(QQ[x_1..x_5]); │ │ │ │ │ │ i5 : solveFrobeniusIdeal(I, W) │ │ │ - -- .000004469s elapsed │ │ │ + -- .000006912s elapsed │ │ │ Warning: F4 Algorithm not available over current coefficient ring or inhomogeneous ideal. │ │ │ Converting to Naive algorithm. │ │ │ │ │ │ │ │ │ o5 = {1, - 2logX + 3logX - 2logX + logX , - logX + logX - logX + logX , │ │ │ 0 1 2 3 0 1 2 4 │ │ │ ------------------------------------------------------------------------ │ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html │ │ │ @@ -134,19 +134,19 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .00000508s elapsed
│ │ │ - -- .000003055s elapsed
│ │ │ - -- .00000561s elapsed
│ │ │ - -- .000003727s elapsed
│ │ │ - -- .000003476s elapsed
│ │ │ + -- .000005598s elapsed
│ │ │ + -- .000007122s elapsed
│ │ │ + -- .000006739s elapsed
│ │ │ + -- .00000619s elapsed
│ │ │ + -- .000008272s 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.
│ │ │ │ - -- .00000508s elapsed
│ │ │ │ - -- .000003055s elapsed
│ │ │ │ - -- .00000561s elapsed
│ │ │ │ - -- .000003727s elapsed
│ │ │ │ - -- .000003476s elapsed
│ │ │ │ + -- .000005598s elapsed
│ │ │ │ + -- .000007122s elapsed
│ │ │ │ + -- .000006739s elapsed
│ │ │ │ + -- .00000619s elapsed
│ │ │ │ + -- .000008272s elapsed
│ │ │ │  
│ │ │ │       +----------------------------------------------------+
│ │ │ │       |   1 5   5 5                                        |
│ │ │ │       | - - - - - -                                        |
│ │ │ │       |   2 2   2 2                                        |
│ │ │ │  o6 = |x   x x   x                                         |
│ │ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_solve__Frobenius__Ideal.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : solveFrobeniusIdeal I
│ │ │ - -- .000004529s elapsed
│ │ │ + -- .000006559s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -113,15 +113,15 @@
│ │ │              
│ │ │                
    i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : solveFrobeniusIdeal(I, W)
│ │ │ - -- .000004469s elapsed
│ │ │ + -- .000006912s elapsed
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  Here is [_S_S_T, Example 2.3.16]:
│ │ │ │  i1 : R = QQ[t_1..t_5];
│ │ │ │  i2 : I = ideal(t_1+t_2+t_3+t_4+t_5, t_1+t_2-t_4, t_2+t_3-t_4, t_1*t_3,
│ │ │ │  t_2*t_4);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : solveFrobeniusIdeal I
│ │ │ │ - -- .000004529s elapsed
│ │ │ │ + -- .000006559s elapsed
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │  
│ │ │ │  
│ │ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │         1             1             1             3                 2
│ │ │ │       - -logX logX  - -logX logX  - -logX logX  - -logX logX  + logX }
│ │ │ │         2    4    0   4    4    1   2    4    2   4    4    3       4
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ │ - -- .000004469s elapsed
│ │ │ │ + -- .000006912s elapsed
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │  
│ │ │ │  
│ │ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_bracket.out
│ │ │ @@ -85,19 +85,16 @@
│ │ │  
│ │ │  o13 = 600
│ │ │  
│ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │  
│ │ │  i15 : H'
│ │ │  
│ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │        -----------------------------------------------------------------------
│ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3 
│ │ │        -----------------------------------------------------------------------
│ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -151,22 +148,25 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │           17      5   10    4 9    5 10      17      19      3   8    2 6  
│ │ │        -----------------------------------------------------------------------
│ │ │        T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │         3 8    4 9      14      17      3   6    3 6      11      12      5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ -       7      5 7    4 10      12      20
│ │ │ +      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ +       7      5 7    4 10      12      20      3   7    4 6    3 7      11  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ +         13      2   9      2 9      16
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : H#(H'_0)
│ │ │  
│ │ │ -o16 = 1
│ │ │ +o16 = -1
│ │ │  
│ │ │  o16 : S[T ..T  ]
│ │ │           1   99
│ │ │  
│ │ │  i17 : bracketMatrix(A,1,2)
│ │ │  
│ │ │  o17 = | 0    -T_8 -T_6 -T_7 -T_10 |
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket.html
│ │ │ @@ -215,19 +215,16 @@
│ │ │                
    i14 : H' = select(keys H, k->H#k != 0);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : H'
│ │ │  
│ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13  
│ │ │        -----------------------------------------------------------------------
│ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3 
│ │ │        -----------------------------------------------------------------------
│ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17  
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -281,25 +278,28 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │           17      5   10    4 9    5 10      17      19      3   8    2 6  
│ │ │        -----------------------------------------------------------------------
│ │ │        T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │         3 8    4 9      14      17      3   6    3 6      11      12      5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ -       7      5 7    4 10      12      20
│ │ │ +      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ +       7      5 7    4 10      12      20      3   7    4 6    3 7      11  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ +         13      2   9      2 9      16
│ │ │  
│ │ │  o15 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : H#(H'_0)
│ │ │  
│ │ │ -o16 = 1
│ │ │ +o16 = -1
│ │ │  
│ │ │  o16 : S[T ..T  ]
│ │ │           1   99
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -118,19 +118,16 @@
│ │ │ │  37   38   39   40   41   42   43   44
│ │ │ │  i13 : #keys H
│ │ │ │  
│ │ │ │  o13 = 600
│ │ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │ │  i15 : H'
│ │ │ │  
│ │ │ │ -o15 = {({T , T }, T T  + T T  - z*T   + y*T  ), ({T , T }, - T T  + y*T  ),
│ │ │ │ -          3   7    4 6    3 7      11      13      2   9      2 9      16
│ │ │ │ -      -----------------------------------------------------------------------
│ │ │ │ -      ({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ │ -         1   9      5 6    1 9      14      17      4   7      4 7      13
│ │ │ │ +o15 = {({T , T }, - T T  - T T  - z*T   + x*T  ), ({T , T }, - T T  + z*T  ),
│ │ │ │ +          1   9      5 6    1 9      14      17      4   7      4 7      13
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        ({T , T  }, T T  - T T   + x*T  ), ({T , T }, - T T  + z*T  ), ({T ,
│ │ │ │           1   10    4 6    1 10      20      5   9      5 9      16      3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        T }, T T  - z*T   + x*T  ), ({T , T }, - T T  - T T  - z*T   + x*T  ),
│ │ │ │         7    3 7      11      12      5   6      5 6    1 9      14      17
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -184,21 +181,24 @@
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        z*T  ), ({T , T  }, T T  - T T   - z*T   + z*T  ), ({T , T }, T T  +
│ │ │ │           17      5   10    4 9    5 10      17      19      3   8    2 6
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        T T  + T T  + y*T   - z*T  ), ({T , T }, T T  + y*T   - z*T  ), ({T ,
│ │ │ │         3 8    4 9      14      17      3   6    3 6      11      12      5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T }, - T T  - T T   + z*T   + z*T  )}
│ │ │ │ -       7      5 7    4 10      12      20
│ │ │ │ +      T }, - T T  - T T   + z*T   + z*T  ), ({T , T }, T T  + T T  - z*T   +
│ │ │ │ +       7      5 7    4 10      12      20      3   7    4 6    3 7      11
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +      y*T  ), ({T , T }, - T T  + y*T  )}
│ │ │ │ +         13      2   9      2 9      16
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : H#(H'_0)
│ │ │ │  
│ │ │ │ -o16 = 1
│ │ │ │ +o16 = -1
│ │ │ │  
│ │ │ │  o16 : S[T ..T  ]
│ │ │ │           1   99
│ │ │ │  From this we see that [T_5, T_6] sends T_37 to -1 in kk.
│ │ │ │  Another, often simpler view of the pairing is given by _b_r_a_c_k_e_t_M_a_t_r_i_x, where the
│ │ │ │  rows and columns correspond to the generators of Pi^d and Pi^e, and the entries
│ │ │ │  are the bracket products, interpreted as elements of Pi^{d+e}. Note the anti-
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_cone_lp__Arrangement_cm__Ring__Element_rp.out
│ │ │ @@ -44,15 +44,15 @@
│ │ │  
│ │ │  o13 = {x, y, x - y, 0, - x + y, x}
│ │ │  
│ │ │  o13 : Hyperplane Arrangement 
│ │ │  
│ │ │  i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {x - y, y, x}
│ │ │ +o14 = {y, x, x - y}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
│ │ │  
│ │ │  i15 : assert isCentral cA''
│ │ │  
│ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_type__B_lp__Z__Z_cm__Ring_rp.out
│ │ │ @@ -33,16 +33,16 @@
│ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │         1   1    2   1    2   2
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       2   1   1    2
│ │ │ +o6 = {x  + x , x , x }
│ │ │ +       1    2   2   1
│ │ │  
│ │ │  o6 : Hyperplane Arrangement 
│ │ │  
│ │ │  i7 : ring A3
│ │ │  
│ │ │       ZZ
│ │ │  o7 = --[x ..x ]
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_cone_lp__Arrangement_cm__Ring__Element_rp.html
│ │ │ @@ -167,15 +167,15 @@
│ │ │  o13 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {x - y, y, x}
│ │ │ +o14 = {y, x, x - y}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : assert isCentral cA''
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  i13 : cone(A, x)
│ │ │ │  
│ │ │ │  o13 = {x, y, x - y, 0, - x + y, x}
│ │ │ │  
│ │ │ │  o13 : Hyperplane Arrangement
│ │ │ │  i14 : cA'' = trim cone(A, x)
│ │ │ │  
│ │ │ │ -o14 = {x - y, y, x}
│ │ │ │ +o14 = {y, x, x - y}
│ │ │ │  
│ │ │ │  o14 : Hyperplane Arrangement
│ │ │ │  i15 : assert isCentral cA''
│ │ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ │ │  When the second input is a _S_y_m_b_o_l, this method creates a new ring from the
│ │ │ │  underlying ring of $A$ by adjoining the symbol as a variable and constructs the
│ │ │ │  cone in this new ring.
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_type__B_lp__Z__Z_cm__Ring_rp.html
│ │ │ @@ -125,16 +125,16 @@
│ │ │  o5 : Hyperplane Arrangement
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       2   1   1    2
│ │ │ +o6 = {x  + x , x , x }
│ │ │ +       1    2   2   1
│ │ │  
│ │ │  o6 : Hyperplane Arrangement
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : ring A3
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,16 +51,16 @@
│ │ │ │  
│ │ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │ │         1   1    2   1    2   2
│ │ │ │  
│ │ │ │  o5 : Hyperplane Arrangement
│ │ │ │  i6 : trim A3
│ │ │ │  
│ │ │ │ -o6 = {x , x , x  + x }
│ │ │ │ -       2   1   1    2
│ │ │ │ +o6 = {x  + x , x , x }
│ │ │ │ +       1    2   2   1
│ │ │ │  
│ │ │ │  o6 : Hyperplane Arrangement
│ │ │ │  i7 : ring A3
│ │ │ │  
│ │ │ │       ZZ
│ │ │ │  o7 = --[x ..x ]
│ │ │ │        2  1   2
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  i3 : R = S/f
│ │ │  
│ │ │  o3 = R
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : time R' = integralClosure R
│ │ │ - -- used 0.698745s (cpu); 0.38885s (thread); 0s (gc)
│ │ │ + -- used 0.920114s (cpu); 0.449067s (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 0.754967s (cpu); 0.404479s (thread); 0s (gc)
│ │ │ + -- used 0.99982s (cpu); 0.452185s (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 0.827843s (cpu); 0.398359s (thread); 0s (gc)
│ │ │ + -- used 1.8998s (cpu); 0.565232s (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 0.924032s (cpu); 0.455257s (thread); 0s (gc)
│ │ │ + -- used 1.25072s (cpu); 0.534691s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
│ │ │  
│ │ │  i22 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -266,15 +266,15 @@
│ │ │  i25 : R = S/f
│ │ │  
│ │ │  o25 = R
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │  
│ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 1.60396s (cpu); 0.735674s (thread); 0s (gc)
│ │ │ + -- used 2.3137s (cpu); 0.913945s (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 0.443271s (cpu); 0.317668s (thread); 0s (gc)
│ │ │ + -- used 0.615835s (cpu); 0.415006s (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 0.0446842s (cpu); 0.0446813s (thread); 0s (gc)
│ │ │ + -- used 0.057937s (cpu); 0.0579329s (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 0.0458552s (cpu); 0.0458557s (thread); 0s (gc)
│ │ │ + -- used 0.0586484s (cpu); 0.0586479s (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 0.0601559s (cpu); 0.0601556s (thread); 0s (gc)
│ │ │ + -- used 0.0803793s (cpu); 0.0803744s (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 0.0460727s (cpu); 0.0460695s (thread); 0s (gc)
│ │ │ + -- used 0.0551216s (cpu); 0.0551165s (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 0.0590566s (cpu); 0.0590562s (thread); 0s (gc)
│ │ │ + -- used 0.0717521s (cpu); 0.0717533s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
│ │ │  
│ │ │  i57 : icFractions R
│ │ │  
│ │ │ @@ -632,15 +632,15 @@
│ │ │  i66 : R = S/I
│ │ │  
│ │ │  o66 = R
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │  
│ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.155005s (cpu); 0.0885792s (thread); 0s (gc)
│ │ │ + -- used 0.235564s (cpu); 0.117736s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
│ │ │  
│ │ │  i68 : icFractions R
│ │ │  
│ │ │ @@ -721,15 +721,15 @@
│ │ │  i77 : R = S/I
│ │ │  
│ │ │  o77 = R
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │  
│ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.40797s (cpu); 0.335975s (thread); 0s (gc)
│ │ │ + -- used 0.551397s (cpu); 0.429892s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
│ │ │  
│ │ │  i79 : icFractions R
│ │ │  
│ │ │ @@ -749,15 +749,15 @@
│ │ │  i81 : R = S/sub(I,S)
│ │ │  
│ │ │  o81 = R
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │  
│ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.477713s (cpu); 0.348731s (thread); 0s (gc)
│ │ │ + -- used 0.685043s (cpu); 0.458663s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -777,20 +777,20 @@
│ │ │  i85 : R = S/sub(I,S)
│ │ │  
│ │ │  o85 = R
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000503134 sec #minors 4]
│ │ │ + [jacobian time .00106561 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .202766 sec  #fractions 6]
│ │ │ - [step 1:   time .208362 sec  #fractions 6]
│ │ │ - -- used 0.414861s (cpu); 0.29138s (thread); 0s (gc)
│ │ │ + [step 0:   time .294765 sec  #fractions 6]
│ │ │ + [step 1:   time .32024 sec  #fractions 6]
│ │ │ + -- used 0.622392s (cpu); 0.371539s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
│ │ │  
│ │ │  i87 : icFractions R
│ │ │  
│ │ │ @@ -810,20 +810,20 @@
│ │ │  i89 : R = S/sub(I,S)
│ │ │  
│ │ │  o89 = R
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │  
│ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000556724 sec #minors 4]
│ │ │ + [jacobian time .000664759 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0898557 sec  #fractions 6]
│ │ │ - [step 1:   time .358152 sec  #fractions 6]
│ │ │ - -- used 0.451949s (cpu); 0.337301s (thread); 0s (gc)
│ │ │ + [step 0:   time .110822 sec  #fractions 6]
│ │ │ + [step 1:   time .644635 sec  #fractions 6]
│ │ │ + -- used 0.760044s (cpu); 0.508975s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -843,20 +843,20 @@
│ │ │  i93 : R = S/sub(I,S)
│ │ │  
│ │ │  o93 = R
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000601899 sec #minors 1]
│ │ │ + [jacobian time .000915101 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .107839 sec  #fractions 6]
│ │ │ - [step 1:   time .407801 sec  #fractions 6]
│ │ │ - -- used 0.519219s (cpu); 0.403685s (thread); 0s (gc)
│ │ │ + [step 0:   time .148018 sec  #fractions 6]
│ │ │ + [step 1:   time .621665 sec  #fractions 6]
│ │ │ + -- used 0.774786s (cpu); 0.537187s (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 .000574116 sec #minors 3]
│ │ │ + [jacobian time .00063632 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00254135 seconds
│ │ │ -      idlizer1:  .0079901 seconds
│ │ │ -      idlizer2:  .00902264 seconds
│ │ │ -      minpres:   .00831893 seconds
│ │ │ -  time .0394227 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00305325 seconds
│ │ │ +      idlizer1:  .0104949 seconds
│ │ │ +      idlizer2:  .0118057 seconds
│ │ │ +      minpres:   .0115236 seconds
│ │ │ +  time .051421 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00259288 seconds
│ │ │ -      idlizer1:  .0123173 seconds
│ │ │ -      idlizer2:  .0126212 seconds
│ │ │ -      minpres:   .0119117 seconds
│ │ │ -  time .0507836 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00301374 seconds
│ │ │ +      idlizer1:  .0146384 seconds
│ │ │ +      idlizer2:  .0127716 seconds
│ │ │ +      minpres:   .0139366 seconds
│ │ │ +  time .0579154 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00227546 seconds
│ │ │ -      idlizer1:  .0113888 seconds
│ │ │ -      idlizer2:  .0095679 seconds
│ │ │ -      minpres:   .00912872 seconds
│ │ │ -  time .0432745 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00286536 seconds
│ │ │ +      idlizer1:  .0148254 seconds
│ │ │ +      idlizer2:  .012257 seconds
│ │ │ +      minpres:   .0113539 seconds
│ │ │ +  time .05453 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00229888 seconds
│ │ │ -      idlizer1:  .106805 seconds
│ │ │ -      idlizer2:  .0129682 seconds
│ │ │ -      minpres:   .0158987 seconds
│ │ │ -  time .150218 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00290437 seconds
│ │ │ +      idlizer1:  .173365 seconds
│ │ │ +      idlizer2:  .0157225 seconds
│ │ │ +      minpres:   .0184812 seconds
│ │ │ +  time .225126 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00219614 seconds
│ │ │ -      idlizer1:  .00843593 seconds
│ │ │ -      idlizer2:  .0157831 seconds
│ │ │ -      minpres:   .0123428 seconds
│ │ │ -  time .0509782 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00296973 seconds
│ │ │ +      idlizer1:  .0112753 seconds
│ │ │ +      idlizer2:  .0193338 seconds
│ │ │ +      minpres:   .014307 seconds
│ │ │ +  time .0626702 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00252353 seconds
│ │ │ -      idlizer1:  .00895437 seconds
│ │ │ -  time .0185368 sec  #fractions 5]
│ │ │ - -- used 0.35735s (cpu); 0.313767s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00317438 seconds
│ │ │ +      idlizer1:  .0104077 seconds
│ │ │ +  time .0219221 sec  #fractions 5]
│ │ │ + -- used 0.478459s (cpu); 0.37763s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │  i3 : trim ideal R'
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out
│ │ │ @@ -13,26 +13,26 @@
│ │ │  
│ │ │                  2      2    2        2   2 2     2
│ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : time integralClosure J
│ │ │ - -- used 0.903752s (cpu); 0.748642s (thread); 0s (gc)
│ │ │ + -- used 1.62892s (cpu); 0.911024s (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 0.634764s (cpu); 0.502541s (thread); 0s (gc)
│ │ │ + -- used 1.20986s (cpu); 0.654028s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time R' = integralClosure R
│ │ │ - -- used 0.698745s (cpu); 0.38885s (thread); 0s (gc)
│ │ │ + -- used 0.920114s (cpu); 0.449067s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -186,15 +186,15 @@ │ │ │ │ │ │ o9 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.754967s (cpu); 0.404479s (thread); 0s (gc)
│ │ │ + -- used 0.99982s (cpu); 0.452185s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -273,15 +273,15 @@ │ │ │ │ │ │ o15 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.827843s (cpu); 0.398359s (thread); 0s (gc)
│ │ │ + -- used 1.8998s (cpu); 0.565232s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -348,15 +348,15 @@ │ │ │ │ │ │ o20 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.924032s (cpu); 0.455257s (thread); 0s (gc)
│ │ │ + -- used 1.25072s (cpu); 0.534691s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -423,15 +423,15 @@ │ │ │ │ │ │ o25 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 1.60396s (cpu); 0.735674s (thread); 0s (gc)
│ │ │ + -- used 2.3137s (cpu); 0.913945s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -498,15 +498,15 @@ │ │ │ │ │ │ o30 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.443271s (cpu); 0.317668s (thread); 0s (gc)
│ │ │ + -- used 0.615835s (cpu); 0.415006s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -573,15 +573,15 @@ │ │ │ │ │ │ o35 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i36 : time R' = integralClosure R
│ │ │ - -- used 0.0446842s (cpu); 0.0446813s (thread); 0s (gc)
│ │ │ + -- used 0.057937s (cpu); 0.0579329s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -643,15 +643,15 @@ │ │ │ │ │ │ o40 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0458552s (cpu); 0.0458557s (thread); 0s (gc)
│ │ │ + -- used 0.0586484s (cpu); 0.0586479s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -695,15 +695,15 @@ │ │ │ │ │ │ o45 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0601559s (cpu); 0.0601556s (thread); 0s (gc)
│ │ │ + -- used 0.0803793s (cpu); 0.0803744s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -746,15 +746,15 @@ │ │ │ │ │ │ o50 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0460727s (cpu); 0.0460695s (thread); 0s (gc)
│ │ │ + -- used 0.0551216s (cpu); 0.0551165s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -798,15 +798,15 @@ │ │ │ │ │ │ o55 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.0590566s (cpu); 0.0590562s (thread); 0s (gc)
│ │ │ + -- used 0.0717521s (cpu); 0.0717533s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -932,15 +932,15 @@ │ │ │ │ │ │ o66 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.155005s (cpu); 0.0885792s (thread); 0s (gc)
│ │ │ + -- used 0.235564s (cpu); 0.117736s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1056,15 +1056,15 @@ │ │ │ │ │ │ o77 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.40797s (cpu); 0.335975s (thread); 0s (gc)
│ │ │ + -- used 0.551397s (cpu); 0.429892s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1098,15 +1098,15 @@ │ │ │ │ │ │ o81 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.477713s (cpu); 0.348731s (thread); 0s (gc)
│ │ │ + -- used 0.685043s (cpu); 0.458663s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1140,20 +1140,20 @@ │ │ │ │ │ │ o85 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000503134 sec #minors 4]
│ │ │ + [jacobian time .00106561 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .202766 sec  #fractions 6]
│ │ │ - [step 1:   time .208362 sec  #fractions 6]
│ │ │ - -- used 0.414861s (cpu); 0.29138s (thread); 0s (gc)
│ │ │ + [step 0:   time .294765 sec  #fractions 6]
│ │ │ + [step 1:   time .32024 sec  #fractions 6]
│ │ │ + -- used 0.622392s (cpu); 0.371539s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1187,20 +1187,20 @@ │ │ │ │ │ │ o89 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000556724 sec #minors 4]
│ │ │ + [jacobian time .000664759 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .0898557 sec  #fractions 6]
│ │ │ - [step 1:   time .358152 sec  #fractions 6]
│ │ │ - -- used 0.451949s (cpu); 0.337301s (thread); 0s (gc)
│ │ │ + [step 0:   time .110822 sec  #fractions 6]
│ │ │ + [step 1:   time .644635 sec  #fractions 6]
│ │ │ + -- used 0.760044s (cpu); 0.508975s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1237,20 +1237,20 @@ │ │ │ │ │ │ o93 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000601899 sec #minors 1]
│ │ │ + [jacobian time .000915101 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .107839 sec  #fractions 6]
│ │ │ - [step 1:   time .407801 sec  #fractions 6]
│ │ │ - -- used 0.519219s (cpu); 0.403685s (thread); 0s (gc)
│ │ │ + [step 0:   time .148018 sec  #fractions 6]
│ │ │ + [step 1:   time .621665 sec  #fractions 6]
│ │ │ + -- used 0.774786s (cpu); 0.537187s (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 0.698745s (cpu); 0.38885s (thread); 0s (gc) │ │ │ │ + -- used 0.920114s (cpu); 0.449067s (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 0.754967s (cpu); 0.404479s (thread); 0s (gc) │ │ │ │ + -- used 0.99982s (cpu); 0.452185s (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 0.827843s (cpu); 0.398359s (thread); 0s (gc) │ │ │ │ + -- used 1.8998s (cpu); 0.565232s (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 0.924032s (cpu); 0.455257s (thread); 0s (gc) │ │ │ │ + -- used 1.25072s (cpu); 0.534691s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 = R' │ │ │ │ │ │ │ │ o21 : QuotientRing │ │ │ │ i22 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -363,15 +363,15 @@ │ │ │ │ o24 : Ideal of S │ │ │ │ i25 : R = S/f │ │ │ │ │ │ │ │ o25 = R │ │ │ │ │ │ │ │ o25 : QuotientRing │ │ │ │ i26 : time R' = integralClosure (R, Strategy => RadicalCodim1) │ │ │ │ - -- used 1.60396s (cpu); 0.735674s (thread); 0s (gc) │ │ │ │ + -- used 2.3137s (cpu); 0.913945s (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 0.443271s (cpu); 0.317668s (thread); 0s (gc) │ │ │ │ + -- used 0.615835s (cpu); 0.415006s (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 0.0446842s (cpu); 0.0446813s (thread); 0s (gc) │ │ │ │ + -- used 0.057937s (cpu); 0.0579329s (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 0.0458552s (cpu); 0.0458557s (thread); 0s (gc) │ │ │ │ + -- used 0.0586484s (cpu); 0.0586479s (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 0.0601559s (cpu); 0.0601556s (thread); 0s (gc) │ │ │ │ + -- used 0.0803793s (cpu); 0.0803744s (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 0.0460727s (cpu); 0.0460695s (thread); 0s (gc) │ │ │ │ + -- used 0.0551216s (cpu); 0.0551165s (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 0.0590566s (cpu); 0.0590562s (thread); 0s (gc) │ │ │ │ + -- used 0.0717521s (cpu); 0.0717533s (thread); 0s (gc) │ │ │ │ │ │ │ │ o56 = R' │ │ │ │ │ │ │ │ o56 : QuotientRing │ │ │ │ i57 : icFractions R │ │ │ │ │ │ │ │ b*d │ │ │ │ @@ -754,15 +754,15 @@ │ │ │ │ o65 : BettiTally │ │ │ │ i66 : R = S/I │ │ │ │ │ │ │ │ o66 = R │ │ │ │ │ │ │ │ o66 : QuotientRing │ │ │ │ i67 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.155005s (cpu); 0.0885792s (thread); 0s (gc) │ │ │ │ + -- used 0.235564s (cpu); 0.117736s (thread); 0s (gc) │ │ │ │ │ │ │ │ o67 = R' │ │ │ │ │ │ │ │ o67 : QuotientRing │ │ │ │ i68 : icFractions R │ │ │ │ │ │ │ │ 2 2 │ │ │ │ @@ -838,15 +838,15 @@ │ │ │ │ o76 : BettiTally │ │ │ │ i77 : R = S/I │ │ │ │ │ │ │ │ o77 = R │ │ │ │ │ │ │ │ o77 : QuotientRing │ │ │ │ i78 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.40797s (cpu); 0.335975s (thread); 0s (gc) │ │ │ │ + -- used 0.551397s (cpu); 0.429892s (thread); 0s (gc) │ │ │ │ │ │ │ │ o78 = R' │ │ │ │ │ │ │ │ o78 : QuotientRing │ │ │ │ i79 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -862,15 +862,15 @@ │ │ │ │ o80 : PolynomialRing │ │ │ │ i81 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o81 = R │ │ │ │ │ │ │ │ o81 : QuotientRing │ │ │ │ i82 : time R' = integralClosure(R, Strategy => AllCodimensions) │ │ │ │ - -- used 0.477713s (cpu); 0.348731s (thread); 0s (gc) │ │ │ │ + -- used 0.685043s (cpu); 0.458663s (thread); 0s (gc) │ │ │ │ │ │ │ │ o82 = R' │ │ │ │ │ │ │ │ o82 : QuotientRing │ │ │ │ i83 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -886,20 +886,20 @@ │ │ │ │ o84 : PolynomialRing │ │ │ │ i85 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o85 = R │ │ │ │ │ │ │ │ o85 : QuotientRing │ │ │ │ i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1) │ │ │ │ - [jacobian time .000503134 sec #minors 4] │ │ │ │ + [jacobian time .00106561 sec #minors 4] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .202766 sec #fractions 6] │ │ │ │ - [step 1: time .208362 sec #fractions 6] │ │ │ │ - -- used 0.414861s (cpu); 0.29138s (thread); 0s (gc) │ │ │ │ + [step 0: time .294765 sec #fractions 6] │ │ │ │ + [step 1: time .32024 sec #fractions 6] │ │ │ │ + -- used 0.622392s (cpu); 0.371539s (thread); 0s (gc) │ │ │ │ │ │ │ │ o86 = R' │ │ │ │ │ │ │ │ o86 : QuotientRing │ │ │ │ i87 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -915,20 +915,20 @@ │ │ │ │ o88 : PolynomialRing │ │ │ │ i89 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o89 = R │ │ │ │ │ │ │ │ o89 : QuotientRing │ │ │ │ i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1) │ │ │ │ - [jacobian time .000556724 sec #minors 4] │ │ │ │ + [jacobian time .000664759 sec #minors 4] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .0898557 sec #fractions 6] │ │ │ │ - [step 1: time .358152 sec #fractions 6] │ │ │ │ - -- used 0.451949s (cpu); 0.337301s (thread); 0s (gc) │ │ │ │ + [step 0: time .110822 sec #fractions 6] │ │ │ │ + [step 1: time .644635 sec #fractions 6] │ │ │ │ + -- used 0.760044s (cpu); 0.508975s (thread); 0s (gc) │ │ │ │ │ │ │ │ o90 = R' │ │ │ │ │ │ │ │ o90 : QuotientRing │ │ │ │ i91 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -947,20 +947,20 @@ │ │ │ │ i93 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o93 = R │ │ │ │ │ │ │ │ o93 : QuotientRing │ │ │ │ i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, │ │ │ │ StartWithOneMinor}, Verbosity => 1) │ │ │ │ - [jacobian time .000601899 sec #minors 1] │ │ │ │ + [jacobian time .000915101 sec #minors 1] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .107839 sec #fractions 6] │ │ │ │ - [step 1: time .407801 sec #fractions 6] │ │ │ │ - -- used 0.519219s (cpu); 0.403685s (thread); 0s (gc) │ │ │ │ + [step 0: time .148018 sec #fractions 6] │ │ │ │ + [step 1: time .621665 sec #fractions 6] │ │ │ │ + -- used 0.774786s (cpu); 0.537187s (thread); 0s (gc) │ │ │ │ │ │ │ │ o94 = R' │ │ │ │ │ │ │ │ o94 : QuotientRing │ │ │ │ i95 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 2 3 2 │ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ @@ -71,52 +71,52 @@ │ │ │ │ │ │ 
    i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .000574116 sec #minors 3]
│ │ │ + [jacobian time .00063632 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00254135 seconds
│ │ │ -      idlizer1:  .0079901 seconds
│ │ │ -      idlizer2:  .00902264 seconds
│ │ │ -      minpres:   .00831893 seconds
│ │ │ -  time .0394227 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00305325 seconds
│ │ │ +      idlizer1:  .0104949 seconds
│ │ │ +      idlizer2:  .0118057 seconds
│ │ │ +      minpres:   .0115236 seconds
│ │ │ +  time .051421 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00259288 seconds
│ │ │ -      idlizer1:  .0123173 seconds
│ │ │ -      idlizer2:  .0126212 seconds
│ │ │ -      minpres:   .0119117 seconds
│ │ │ -  time .0507836 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00301374 seconds
│ │ │ +      idlizer1:  .0146384 seconds
│ │ │ +      idlizer2:  .0127716 seconds
│ │ │ +      minpres:   .0139366 seconds
│ │ │ +  time .0579154 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00227546 seconds
│ │ │ -      idlizer1:  .0113888 seconds
│ │ │ -      idlizer2:  .0095679 seconds
│ │ │ -      minpres:   .00912872 seconds
│ │ │ -  time .0432745 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00286536 seconds
│ │ │ +      idlizer1:  .0148254 seconds
│ │ │ +      idlizer2:  .012257 seconds
│ │ │ +      minpres:   .0113539 seconds
│ │ │ +  time .05453 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00229888 seconds
│ │ │ -      idlizer1:  .106805 seconds
│ │ │ -      idlizer2:  .0129682 seconds
│ │ │ -      minpres:   .0158987 seconds
│ │ │ -  time .150218 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00290437 seconds
│ │ │ +      idlizer1:  .173365 seconds
│ │ │ +      idlizer2:  .0157225 seconds
│ │ │ +      minpres:   .0184812 seconds
│ │ │ +  time .225126 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .00219614 seconds
│ │ │ -      idlizer1:  .00843593 seconds
│ │ │ -      idlizer2:  .0157831 seconds
│ │ │ -      minpres:   .0123428 seconds
│ │ │ -  time .0509782 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00296973 seconds
│ │ │ +      idlizer1:  .0112753 seconds
│ │ │ +      idlizer2:  .0193338 seconds
│ │ │ +      minpres:   .014307 seconds
│ │ │ +  time .0626702 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00252353 seconds
│ │ │ -      idlizer1:  .00895437 seconds
│ │ │ -  time .0185368 sec  #fractions 5]
│ │ │ - -- used 0.35735s (cpu); 0.313767s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00317438 seconds
│ │ │ +      idlizer1:  .0104077 seconds
│ │ │ +  time .0219221 sec  #fractions 5]
│ │ │ + -- used 0.478459s (cpu); 0.37763s (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 .000574116 sec #minors 3] │ │ │ │ + [jacobian time .00063632 sec #minors 3] │ │ │ │ integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ [step 0: │ │ │ │ - radical (use minprimes) .00254135 seconds │ │ │ │ - idlizer1: .0079901 seconds │ │ │ │ - idlizer2: .00902264 seconds │ │ │ │ - minpres: .00831893 seconds │ │ │ │ - time .0394227 sec #fractions 4] │ │ │ │ + radical (use minprimes) .00305325 seconds │ │ │ │ + idlizer1: .0104949 seconds │ │ │ │ + idlizer2: .0118057 seconds │ │ │ │ + minpres: .0115236 seconds │ │ │ │ + time .051421 sec #fractions 4] │ │ │ │ [step 1: │ │ │ │ - radical (use minprimes) .00259288 seconds │ │ │ │ - idlizer1: .0123173 seconds │ │ │ │ - idlizer2: .0126212 seconds │ │ │ │ - minpres: .0119117 seconds │ │ │ │ - time .0507836 sec #fractions 4] │ │ │ │ + radical (use minprimes) .00301374 seconds │ │ │ │ + idlizer1: .0146384 seconds │ │ │ │ + idlizer2: .0127716 seconds │ │ │ │ + minpres: .0139366 seconds │ │ │ │ + time .0579154 sec #fractions 4] │ │ │ │ [step 2: │ │ │ │ - radical (use minprimes) .00227546 seconds │ │ │ │ - idlizer1: .0113888 seconds │ │ │ │ - idlizer2: .0095679 seconds │ │ │ │ - minpres: .00912872 seconds │ │ │ │ - time .0432745 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00286536 seconds │ │ │ │ + idlizer1: .0148254 seconds │ │ │ │ + idlizer2: .012257 seconds │ │ │ │ + minpres: .0113539 seconds │ │ │ │ + time .05453 sec #fractions 5] │ │ │ │ [step 3: │ │ │ │ - radical (use minprimes) .00229888 seconds │ │ │ │ - idlizer1: .106805 seconds │ │ │ │ - idlizer2: .0129682 seconds │ │ │ │ - minpres: .0158987 seconds │ │ │ │ - time .150218 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00290437 seconds │ │ │ │ + idlizer1: .173365 seconds │ │ │ │ + idlizer2: .0157225 seconds │ │ │ │ + minpres: .0184812 seconds │ │ │ │ + time .225126 sec #fractions 5] │ │ │ │ [step 4: │ │ │ │ - radical (use minprimes) .00219614 seconds │ │ │ │ - idlizer1: .00843593 seconds │ │ │ │ - idlizer2: .0157831 seconds │ │ │ │ - minpres: .0123428 seconds │ │ │ │ - time .0509782 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00296973 seconds │ │ │ │ + idlizer1: .0112753 seconds │ │ │ │ + idlizer2: .0193338 seconds │ │ │ │ + minpres: .014307 seconds │ │ │ │ + time .0626702 sec #fractions 5] │ │ │ │ [step 5: │ │ │ │ - radical (use minprimes) .00252353 seconds │ │ │ │ - idlizer1: .00895437 seconds │ │ │ │ - time .0185368 sec #fractions 5] │ │ │ │ - -- used 0.35735s (cpu); 0.313767s (thread); 0s (gc) │ │ │ │ + radical (use minprimes) .00317438 seconds │ │ │ │ + idlizer1: .0104077 seconds │ │ │ │ + time .0219221 sec #fractions 5] │ │ │ │ + -- used 0.478459s (cpu); 0.37763s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = R' │ │ │ │ │ │ │ │ o2 : QuotientRing │ │ │ │ i3 : trim ideal R' │ │ │ │ │ │ │ │ 3 2 2 2 4 4 │ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html │ │ │ @@ -109,29 +109,29 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time integralClosure J
│ │ │ - -- used 0.903752s (cpu); 0.748642s (thread); 0s (gc)
│ │ │ + -- used 1.62892s (cpu); 0.911024s (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 0.634764s (cpu); 0.502541s (thread); 0s (gc)
│ │ │ + -- used 1.20986s (cpu); 0.654028s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,25 +46,25 @@
│ │ │ │  i3 : J = ideal jacobian ideal F
│ │ │ │  
│ │ │ │                  2      2    2        2   2 2     2
│ │ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : time integralClosure J
│ │ │ │ - -- used 0.903752s (cpu); 0.748642s (thread); 0s (gc)
│ │ │ │ + -- used 1.62892s (cpu); 0.911024s (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 0.634764s (cpu); 0.502541s (thread); 0s (gc)
│ │ │ │ + -- used 1.20986s (cpu); 0.654028s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o3 : DiagonalAction
│ │ │  
│ │ │  i4 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00262479s elapsed
│ │ │ + -- .00320262s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T 
│ │ │ @@ -51,10 +51,10 @@
│ │ │           0   1
│ │ │  
│ │ │  i6 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000463999s elapsed
│ │ │ + -- .000549151s elapsed
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hsop_spalgorithms.out
│ │ │ @@ -23,23 +23,23 @@
│ │ │  o3 = QQ[x..z] <- {| 0 -1 0  |, | 0 -1 0 |}
│ │ │                    | 1 0  0  |  | 1 0  0 |
│ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.867404s (cpu); 0.587595s (thread); 0s (gc)
│ │ │ + -- used 1.07522s (cpu); 0.668292s (thread); 0s (gc)
│ │ │  
│ │ │ -       2   2    2   2 2
│ │ │ -o4 = {z , x  + y , x y }
│ │ │ +       2   2    2   3       3
│ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.673557s (cpu); 0.37708s (thread); 0s (gc)
│ │ │ + -- used 0.976228s (cpu); 0.497661s (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 0.0312315s (cpu); 0.0312351s (thread); 0s (gc)
│ │ │ + -- used 0.0298017s (cpu); 0.0298065s (thread); 0s (gc)
│ │ │  
│ │ │ -          3       3
│ │ │ -o6 = {1, x y - x*y }
│ │ │ +          4    4
│ │ │ +o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 1.9744s (cpu); 1.31709s (thread); 0s (gc)
│ │ │ + -- used 2.76799s (cpu); 1.6821s (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
│ │ │ - -- .720592s elapsed
│ │ │ + -- .671574s 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.
│ │ │  
│ │ │ - -- .577507s elapsed
│ │ │ + -- .558109s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.out
│ │ │ @@ -14,28 +14,28 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .610383s elapsed
│ │ │ + -- .60342s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ - -- .0779819s elapsed
│ │ │ + -- .0840421s 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
│ │ │  
│ │ │ -                  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 }
│ │ │ +                                   3    3    3
│ │ │ +o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : K=GF(101)
│ │ │  
│ │ │  o5 = K
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_primary__Invariants_lp..._cm__Degree__Vector_eq_gt..._rp.out
│ │ │ @@ -16,13 +16,16 @@
│ │ │                    | 0 0 1 |  | 1 0 0 |
│ │ │                    | 1 0 0 |  | 0 0 1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : primaryInvariants(S3,DegreeVector=>{3,3,4})
│ │ │  
│ │ │ -       2       2    2     2       2      2          4    4    4
│ │ │ -o4 = {x y + x*y  + x z + y z + x*z  + y*z , x*y*z, x  + y  + z }
│ │ │ +       3    3    3   2       2    2     2       2      2   3       3    3   
│ │ │ +o4 = {x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z , x y + x*y  + x z +
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +      3       3      3
│ │ │ +     y z + x*z  + y*z }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o4 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00262479s elapsed
│ │ │ + -- .00320262s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (z z  + z   + z  )T + (z z  + z  + z  + z   + z  z   + z  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (z z  + z z  + z z  + z z   + 1 + z   + z  z  + z  z   + z  z   + z  )T 
│ │ │ @@ -124,15 +124,15 @@
│ │ │  
│ │ │  o6 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000463999s elapsed
    │ │ │ + -- .000549151s 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) │ │ │ │ - -- .00262479s elapsed │ │ │ │ + -- .00320262s elapsed │ │ │ │ │ │ │ │ -1 -1 2 2 -2 -1 -1 -2 2 │ │ │ │ o5 = 1 + (z z + z + z )T + (z z + z + z + z + z z + z )T + │ │ │ │ 0 1 1 0 0 1 0 1 1 0 1 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 3 2 2 -1 -3 -1 -1 -2 -2 -1 -3 3 │ │ │ │ (z z + z z + z z + z z + 1 + z + z z + z z + z z + z )T │ │ │ │ @@ -54,13 +54,13 @@ │ │ │ │ │ │ │ │ o5 : ZZ[z ..z ][T] │ │ │ │ 0 1 │ │ │ │ i6 : T.cache.?equivariantHilbert │ │ │ │ │ │ │ │ o6 = true │ │ │ │ i7 : elapsedTime equivariantHilbertSeries(T, Order => 5); │ │ │ │ - -- .000463999s elapsed │ │ │ │ + -- .000549151s elapsed │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _e_q_u_i_v_a_r_i_a_n_t_H_i_l_b_e_r_t is a _s_y_m_b_o_l. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.25.11+ds/M2/Macaulay2/packages/InvariantRing/AbelianGroupsDoc.m2:185:0. │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html │ │ │ @@ -92,26 +92,26 @@ │ │ │ │ │ │ │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.867404s (cpu); 0.587595s (thread); 0s (gc)
│ │ │ + -- used 1.07522s (cpu); 0.668292s (thread); 0s (gc)
│ │ │  
│ │ │ -       2   2    2   2 2
│ │ │ -o4 = {z , x  + y , x y }
│ │ │ +       2   2    2   3       3
│ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.673557s (cpu); 0.37708s (thread); 0s (gc)
│ │ │ + -- used 0.976228s (cpu); 0.497661s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,26 +168,26 @@
│ │ │
    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.0312315s (cpu); 0.0312351s (thread); 0s (gc)
│ │ │ + -- used 0.0298017s (cpu); 0.0298065s (thread); 0s (gc)
│ │ │  
│ │ │ -          3       3
│ │ │ -o6 = {1, x y - x*y }
│ │ │ +          4    4
│ │ │ +o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 1.9744s (cpu); 1.31709s (thread); 0s (gc)
│ │ │ + -- used 2.76799s (cpu); 1.6821s (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 0.867404s (cpu); 0.587595s (thread); 0s (gc)
│ │ │ │ + -- used 1.07522s (cpu); 0.668292s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -       2   2    2   2 2
│ │ │ │ -o4 = {z , x  + y , x y }
│ │ │ │ +       2   2    2   3       3
│ │ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 0.673557s (cpu); 0.37708s (thread); 0s (gc)
│ │ │ │ + -- used 0.976228s (cpu); 0.497661s (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 0.0312315s (cpu); 0.0312351s (thread); 0s (gc)
│ │ │ │ + -- used 0.0298017s (cpu); 0.0298065s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -          3       3
│ │ │ │ -o6 = {1, x y - x*y }
│ │ │ │ +          4    4
│ │ │ │ +o6 = {1, x  + y }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ │ - -- used 1.9744s (cpu); 1.31709s (thread); 0s (gc)
│ │ │ │ + -- used 2.76799s (cpu); 1.6821s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4
│ │ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : elapsedTime invariants S4
│ │ │ - -- .720592s elapsed
│ │ │ + -- .671574s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -117,15 +117,15 @@
│ │ │
    		│ │ │ 
    i5 : elapsedTime invariants(S4,DegreeBound=>4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .577507s elapsed
│ │ │ + -- .558109s 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
│ │ │ │ - -- .720592s elapsed
│ │ │ │ + -- .671574s 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.
│ │ │ │  
│ │ │ │ - -- .577507s elapsed
│ │ │ │ + -- .558109s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Use__Linear__Algebra_eq_gt..._rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : elapsedTime invariants S4
│ │ │ - -- .610383s elapsed
│ │ │ + -- .60342s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ - -- .0779819s elapsed
│ │ │ + -- .0840421s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,27 +35,27 @@
│ │ │ │  o3 = R <- {| 0 1 0 0 |, | 0 0 0 1 |}
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .610383s elapsed
│ │ │ │ + -- .60342s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │        4
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4,UseLinearAlgebra=>true)
│ │ │ │ - -- .0779819s elapsed
│ │ │ │ + -- .0840421s 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
│ │ │ @@ -101,16 +101,16 @@
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : primaryInvariants S3
│ │ │  
│ │ │ -                  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 }
│ │ │ +                                   3    3    3
│ │ │ +o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │

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

    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,16 +38,16 @@ │ │ │ │ o3 = QQ[x..z] <- {| 0 1 0 |, | 0 1 0 |} │ │ │ │ | 0 0 1 | | 1 0 0 | │ │ │ │ | 1 0 0 | | 0 0 1 | │ │ │ │ │ │ │ │ o3 : FiniteGroupAction │ │ │ │ i4 : primaryInvariants S3 │ │ │ │ │ │ │ │ - 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 } │ │ │ │ + 3 3 3 │ │ │ │ +o4 = {x + y + z, x*y + x*z + y*z, x + y + z } │ │ │ │ │ │ │ │ o4 : List │ │ │ │ Below, the invariant ring QQ[x,y,z]S3 is calculated with K being the field with │ │ │ │ 101 elements. │ │ │ │ i5 : K=GF(101) │ │ │ │ │ │ │ │ o5 = K │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants_lp..._cm__Degree__Vector_eq_gt..._rp.html │ │ │ @@ -97,16 +97,19 @@ │ │ │ o3 : FiniteGroupAction │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : primaryInvariants(S3,DegreeVector=>{3,3,4})
│ │ │  
│ │ │ -       2       2    2     2       2      2          4    4    4
│ │ │ -o4 = {x y + x*y  + x z + y z + x*z  + y*z , x*y*z, x  + y  + z }
│ │ │ +       3    3    3   2       2    2     2       2      2   3       3    3   
│ │ │ +o4 = {x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z , x y + x*y  + x z +
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +      3       3      3
│ │ │ +     y z + x*z  + y*z }
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -34,16 +34,19 @@ │ │ │ │ o3 = QQ[x..z] <- {| 0 1 0 |, | 0 1 0 |} │ │ │ │ | 0 0 1 | | 1 0 0 | │ │ │ │ | 1 0 0 | | 0 0 1 | │ │ │ │ │ │ │ │ o3 : FiniteGroupAction │ │ │ │ i4 : primaryInvariants(S3,DegreeVector=>{3,3,4}) │ │ │ │ │ │ │ │ - 2 2 2 2 2 2 4 4 4 │ │ │ │ -o4 = {x y + x*y + x z + y z + x*z + y*z , x*y*z, x + y + z } │ │ │ │ + 3 3 3 2 2 2 2 2 2 3 3 3 │ │ │ │ +o4 = {x + y + z , x y + x*y + x z + y z + x*z + y*z , x y + x*y + x z + │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + 3 3 3 │ │ │ │ + y z + x*z + y*z } │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Currently users can only use _p_r_i_m_a_r_y_I_n_v_a_r_i_a_n_t_s to calculate a hsop for the │ │ │ │ invariant ring over a finite field by using the Dade algorithm. Users should │ │ │ │ enter the finite field as a _G_a_l_o_i_s_F_i_e_l_d or a quotient field of the form _Z_Z/ │ │ │ │ p and are advised to ensure that the ground field has cardinality greater than │ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/example-output/_is__Isomorphic.out │ │ │ @@ -156,20 +156,20 @@ │ │ │ {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 x_1 x_0 | {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 x_1 x_0^2 | │ │ │ {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 -x_2 x_1 -x_3 x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_3 x_2 x_1^2 | │ │ │ │ │ │ 40 │ │ │ o22 : S-module, subquotient of S │ │ │ │ │ │ i23 : elapsedTime isIsomorphic(T1, T2) │ │ │ - -- 1.39908s elapsed │ │ │ + -- 1.64791s elapsed │ │ │ │ │ │ o23 = true │ │ │ │ │ │ i24 : elapsedTime isomorphism(T1, T2) │ │ │ - -- .000024476s elapsed │ │ │ + -- .000031222s elapsed │ │ │ │ │ │ o24 = {-1} | 1 -3976 -13490 13495 -2886 2577 14757 -881 7677 │ │ │ {-1} | -2527 -13566 2778 -6934 -14806 4619 -13099 6022 -10907 │ │ │ {-1} | -15420 5642 1489 1354 4591 11881 -5253 7296 -1098 │ │ │ {-1} | 7909 -12428 -2260 -8465 12113 -6893 8411 4186 -9393 │ │ │ {-1} | -9615 2934 10440 5015 8145 -5585 1360 3295 12851 │ │ │ {-1} | -4881 -7984 12700 -10391 -10009 -14538 13207 262 -6500 │ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/html/_is__Isomorphic.html │ │ │ @@ -328,23 +328,23 @@ │ │ │ 40 │ │ │ o22 : S-module, subquotient of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.39908s elapsed
│ │ │ + -- 1.64791s elapsed
│ │ │  
│ │ │  o23 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .000024476s elapsed
│ │ │ + -- .000031222s elapsed
│ │ │  
│ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
│ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
│ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
│ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
│ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ │ ├── html2text {}
│ │ │ │ @@ -684,19 +684,19 @@
│ │ │ │  0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0
│ │ │ │  0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0
│ │ │ │  0   0   0     x_3 x_2 x_1^2 |
│ │ │ │  
│ │ │ │                                  40
│ │ │ │  o22 : S-module, subquotient of S
│ │ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ │ - -- 1.39908s elapsed
│ │ │ │ + -- 1.64791s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ │ - -- .000024476s elapsed
│ │ │ │ + -- .000031222s elapsed
│ │ │ │  
│ │ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677
│ │ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098
│ │ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393
│ │ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851
│ │ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ ├── ./usr/share/doc/Macaulay2/JSON/example-output/_from__J__S__O__N.out
│ │ │ @@ -39,19 +39,19 @@
│ │ │  
│ │ │  o8 = {1, 2, 3}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-50457-0/0.json
│ │ │ +o9 = /tmp/M2-79391-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-50457-0/0.json
│ │ │ +o10 = /tmp/M2-79391-0/0.json
│ │ │  
│ │ │  o10 : File
│ │ │  
│ │ │  i11 : fromJSON openIn jsonFile
│ │ │  
│ │ │  o11 = {1, 2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/JSON/html/_from__J__S__O__N.html
│ │ │ @@ -167,22 +167,22 @@
│ │ │            
    The input may also be a file containing JSON data.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-50457-0/0.json
    │ │ │ +o9 = /tmp/M2-79391-0/0.json │ │ │
    │ │ │ 
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-50457-0/0.json
│ │ │ +o10 = /tmp/M2-79391-0/0.json
│ │ │  
│ │ │  o10 : File
    │ │ │
    │ │ │ 
    i11 : fromJSON openIn jsonFile
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,18 +53,18 @@
│ │ │ │  
│ │ │ │  o8 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  The input may also be a file containing JSON data.
│ │ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-50457-0/0.json
│ │ │ │ +o9 = /tmp/M2-79391-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-50457-0/0.json
│ │ │ │ +o10 = /tmp/M2-79391-0/0.json
│ │ │ │  
│ │ │ │  o10 : File
│ │ │ │  i11 : fromJSON openIn jsonFile
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o11 : List
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out
│ │ │ @@ -17,24 +17,24 @@
│ │ │  o3 = ideal (y0*z0*x2 + x0*z0*y2 + x0*y0*z2 + z0*x1*y1 + y0*x1*z1 + x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1 + x0*z0*y1 + x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o3 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .00241278s elapsed
│ │ │ + -- .00267188s 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
│ │ │ - -- .359999s elapsed
│ │ │ + -- .266443s 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)
│ │ │ - -- .00975895s elapsed
│ │ │ + -- .0116689s 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)
│ │ │ - -- .00266585s elapsed
│ │ │ + -- .0030166s 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)
│ │ │ - -- .00236014s elapsed
│ │ │ + -- .00269773s 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)
│ │ │ - -- .0120516s elapsed
│ │ │ + -- .0156404s 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)
│ │ │ - -- .000626965s elapsed
│ │ │ + -- .000907625s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html
│ │ │ @@ -87,27 +87,27 @@
│ │ │          
    
│ │ │            
    However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is generated by the individual terms in the generators of the ideal of jets. This observation provides an alternative algorithm for computing radicals of jets of monomial ideals, which can be faster than the default radical computation in Macaulay2.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,23 +27,23 @@
│ │ │ │  However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial
│ │ │ │  ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is
│ │ │ │  generated by the individual terms in the generators of the ideal of jets. This
│ │ │ │  observation provides an alternative algorithm for computing radicals of jets of
│ │ │ │  monomial ideals, which can be faster than the default radical computation in
│ │ │ │  Macaulay2.
│ │ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ │ - -- .00241278s elapsed
│ │ │ │ + -- .00267188s 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
│ │ │ │ - -- .359999s elapsed
│ │ │ │ + -- .266443s elapsed
│ │ │ │  
│ │ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │ │  
│ │ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  For a monomial hypersurface, [GS06, Theorem 3.2] describes the minimal primes
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o7 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -146,26 +146,26 @@
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -290,15 +290,15 @@
│ │ │  
│ │ │  o23 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .00241278s elapsed
│ │ │ + -- .00267188s 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
│ │ │ - -- .359999s elapsed
│ │ │ + -- .266443s 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)
│ │ │ - -- .00975895s elapsed
│ │ │ + -- .0116689s 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)
│ │ │ - -- .00266585s elapsed
│ │ │ + -- .0030166s 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)
│ │ │ - -- .00236014s elapsed
│ │ │ + -- .00269773s 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)
│ │ │ - -- .0120516s elapsed
│ │ │ + -- .0156404s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -324,15 +324,15 @@
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime jets(2,f)
│ │ │ - -- .000626965s elapsed
│ │ │ + -- .000907625s 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)
│ │ │ │ - -- .00975895s elapsed
│ │ │ │ + -- .0116689s 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)
│ │ │ │ - -- .00266585s elapsed
│ │ │ │ + -- .0030166s 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)
│ │ │ │ - -- .00236014s elapsed
│ │ │ │ + -- .00269773s 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)
│ │ │ │ - -- .0120516s elapsed
│ │ │ │ + -- .0156404s 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)
│ │ │ │ - -- .000626965s elapsed
│ │ │ │ + -- .000907625s elapsed
│ │ │ │  
│ │ │ │                                     QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  2                    2
│ │ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2,
│ │ │ │  2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                              2                 2
│ │ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │        32003  0   5   0   5         32003  0   5   0   5          32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5          32003  0   5   0   5
│ │ │                                                                                                                                                                                                                                                                                           
│ │ │       0                            1                             2                              3                              4                              5                              6                              7                              8                             9
│ │ │  
│ │ │  o3 : Complex
│ │ │  
│ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0236672s elapsed
│ │ │ + -- .0289114s 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)
│ │ │ - -- .00234144s elapsed
│ │ │ - -- .0146779s elapsed
│ │ │ - -- .0232469s elapsed
│ │ │ - -- .0850424s elapsed
│ │ │ - -- .00382558s elapsed
│ │ │ + -- .00292396s elapsed
│ │ │ + -- .00773808s elapsed
│ │ │ + -- .0273359s elapsed
│ │ │ + -- .0548439s elapsed
│ │ │ + -- .00422154s 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)
│ │ │ - -- .00264214s elapsed
│ │ │ - -- .00671366s elapsed
│ │ │ - -- .0243345s elapsed
│ │ │ - -- .0286816s elapsed
│ │ │ - -- .00369202s elapsed
│ │ │ - -- .420974s elapsed
│ │ │ + -- .0030297s elapsed
│ │ │ + -- .00688633s elapsed
│ │ │ + -- .027006s elapsed
│ │ │ + -- .0109151s elapsed
│ │ │ + -- .00391359s elapsed
│ │ │ + -- .392449s 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
│ │ │ - -- .180171s elapsed
│ │ │ + -- .214537s 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);
│ │ │ - -- .00685927s elapsed
│ │ │ - -- .0180639s elapsed
│ │ │ - -- .0911599s elapsed
│ │ │ - -- 1.23617s elapsed
│ │ │ - -- .586255s elapsed
│ │ │ - -- .0383612s elapsed
│ │ │ - -- .0067379s elapsed
│ │ │ - -- 6.09192s elapsed
│ │ │ + -- .00532269s elapsed
│ │ │ + -- .0200809s elapsed
│ │ │ + -- .106871s elapsed
│ │ │ + -- 1.08678s elapsed
│ │ │ + -- .469657s elapsed
│ │ │ + -- .0453141s elapsed
│ │ │ + -- .0081838s elapsed
│ │ │ + -- 6.36253s 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)
│ │ │ - -- .00264923s elapsed
│ │ │ - -- .00889819s elapsed
│ │ │ - -- .0257235s elapsed
│ │ │ - -- .00999915s elapsed
│ │ │ - -- .00370523s elapsed
│ │ │ + -- .00311883s elapsed
│ │ │ + -- .00746631s elapsed
│ │ │ + -- .0264546s elapsed
│ │ │ + -- .0110235s elapsed
│ │ │ + -- .00428519s 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
│ │ │ - -- .185433s elapsed
│ │ │ + -- .232498s 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);
│ │ │ - -- .00480544s elapsed
│ │ │ - -- .0187963s elapsed
│ │ │ - -- .0913313s elapsed
│ │ │ - -- 1.28201s elapsed
│ │ │ - -- .692733s elapsed
│ │ │ - -- .0429653s elapsed
│ │ │ - -- .00652921s elapsed
│ │ │ - -- 6.72795s elapsed
│ │ │ + -- .00526949s elapsed
│ │ │ + -- .020366s elapsed
│ │ │ + -- .119037s elapsed
│ │ │ + -- 1.1561s elapsed
│ │ │ + -- .470965s elapsed
│ │ │ + -- .043711s elapsed
│ │ │ + -- .00902826s elapsed
│ │ │ + -- 6.34639s 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)
│ │ │ - -- .00624298s elapsed
│ │ │ - -- .0136333s elapsed
│ │ │ + -- .00835134s elapsed
│ │ │ + -- .013873s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000327194s elapsed
│ │ │ + -- .00028654s elapsed
│ │ │  1
│ │ │ - -- .000136907s elapsed
│ │ │ + -- .000185432s elapsed
│ │ │  1
│ │ │ - -- .000132328s elapsed
│ │ │ + -- .000188627s elapsed
│ │ │  1
│ │ │ - -- .000144721s elapsed
│ │ │ + -- .000184763s elapsed
│ │ │  1
│ │ │ - -- .000134663s elapsed
│ │ │ + -- .000180944s elapsed
│ │ │  2
│ │ │ - -- .000146294s elapsed
│ │ │ + -- .000189348s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000158026s elapsed
│ │ │ + -- .000196318s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000171752s elapsed
│ │ │ + -- .00021251s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000148419s elapsed
│ │ │ + -- .000190274s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000135754s elapsed
│ │ │ + -- .000225895s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000126858s elapsed
│ │ │ + -- .000157541s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000129172s elapsed
│ │ │ + -- .000194815s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000130054s elapsed
│ │ │ + -- .000177877s elapsed
│ │ │  2
│ │ │ - -- .000121047s elapsed
│ │ │ + -- .000165428s elapsed
│ │ │  2
│ │ │ - -- .000127559s elapsed
│ │ │ + -- .000162315s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000127209s elapsed
│ │ │ + -- .000233775s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000159349s elapsed
│ │ │ + -- .000170383s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000134152s elapsed
│ │ │ + -- .000170659s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000144321s elapsed
│ │ │ + -- .00018404s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000142818s elapsed
│ │ │ + -- .000156819s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000127849s elapsed
│ │ │ + -- .000153345s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120276s elapsed
│ │ │ + -- .000150494s elapsed
│ │ │  2
│ │ │ - -- .000147958s elapsed
│ │ │ + -- .000174933s elapsed
│ │ │  1
│ │ │ - -- .000126007s elapsed
│ │ │ + -- .000149865s elapsed
│ │ │  1
│ │ │ - -- .000127409s elapsed
│ │ │ + -- .000172348s elapsed
│ │ │  1
│ │ │ - -- .000124584s elapsed
│ │ │ + -- .000172294s 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
│ │ │ - -- .238508s elapsed
│ │ │ - -- .302491s elapsed
│ │ │ - -- .206777s elapsed
│ │ │ - -- .290534s elapsed
│ │ │ - -- .340601s elapsed
│ │ │ - -- .391058s elapsed
│ │ │ + -- .182577s elapsed
│ │ │ + -- .20292s elapsed
│ │ │ + -- .186752s elapsed
│ │ │ + -- .390872s elapsed
│ │ │ + -- .204045s elapsed
│ │ │ + -- .251665s 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)
│ │ │ - -- .00264488s elapsed
│ │ │ - -- .00654608s elapsed
│ │ │ - -- .0240881s elapsed
│ │ │ - -- .00901472s elapsed
│ │ │ - -- .0141597s elapsed
│ │ │ + -- .0292424s elapsed
│ │ │ + -- .0112616s elapsed
│ │ │ + -- .0275194s elapsed
│ │ │ + -- .0104555s elapsed
│ │ │ + -- .00443724s 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)
│ │ │ - -- .257455s elapsed
│ │ │ + -- .273186s 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)
│ │ │ - -- .00265049s elapsed
│ │ │ - -- .00673866s elapsed
│ │ │ - -- .0333516s elapsed
│ │ │ - -- .0100248s elapsed
│ │ │ - -- .00383599s elapsed
│ │ │ + -- .00284613s elapsed
│ │ │ + -- .00773682s elapsed
│ │ │ + -- .0313114s elapsed
│ │ │ + -- .0107948s elapsed
│ │ │ + -- .00455676s 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)
│ │ │ - -- .0162559s elapsed
│ │ │ + -- .0221523s 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
│ │ │ - -- .000046116s elapsed
│ │ │ + -- .000066675s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000082144s elapsed
│ │ │ + -- .0001203s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000071754s elapsed
│ │ │ + -- .000084858s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000081182s elapsed
│ │ │ + -- .00010081s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000082916s elapsed
│ │ │ + -- .00010873s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000103855s elapsed
│ │ │ + -- .000113594s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000023434s elapsed
│ │ │ + -- .000031226s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000071143s elapsed
│ │ │ + -- .000076502s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .00007983s elapsed
│ │ │ + -- .000100636s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000093766s elapsed
│ │ │ + -- .00011083s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000088076s elapsed
│ │ │ + -- .000105735s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000095389s elapsed
│ │ │ + -- .000087389s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .00007482s elapsed
│ │ │ + -- .000104167s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000063289s elapsed
│ │ │ + -- .000084419s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000070302s elapsed
│ │ │ + -- .000080289s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000023885s elapsed
│ │ │ + -- .000035956s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000066264s elapsed
│ │ │ + -- .000089896s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000024747s elapsed
│ │ │ + -- .000030692s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o3 : Complex
    
│ │ │              
    
│ │ │                
    i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0236672s elapsed
│ │ │ + -- .0289114s elapsed
│ │ │  
│ │ │  o5 = 350
    
│ │ │              
    
│ │ │                
    i6 : betti F_a, betti F
│ │ │ @@ -141,19 +141,19 @@
│ │ │  
│ │ │  o9 = 14
    
│ │ │              
    
│ │ │                
    i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00234144s elapsed
│ │ │ - -- .0146779s elapsed
│ │ │ - -- .0232469s elapsed
│ │ │ - -- .0850424s elapsed
│ │ │ - -- .00382558s elapsed
│ │ │ + -- .00292396s elapsed
│ │ │ + -- .00773808s elapsed
│ │ │ + -- .0273359s elapsed
│ │ │ + -- .0548439s elapsed
│ │ │ + -- .00422154s 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
│ │ │ │ - -- .0236672s elapsed
│ │ │ │ + -- .0289114s 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)
│ │ │ │ - -- .00234144s elapsed
│ │ │ │ - -- .0146779s elapsed
│ │ │ │ - -- .0232469s elapsed
│ │ │ │ - -- .0850424s elapsed
│ │ │ │ - -- .00382558s elapsed
│ │ │ │ + -- .00292396s elapsed
│ │ │ │ + -- .00773808s elapsed
│ │ │ │ + -- .0273359s elapsed
│ │ │ │ + -- .0548439s elapsed
│ │ │ │ + -- .00422154s elapsed
│ │ │ │  
│ │ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html
│ │ │ @@ -83,20 +83,20 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00264214s elapsed
│ │ │ - -- .00671366s elapsed
│ │ │ - -- .0243345s elapsed
│ │ │ - -- .0286816s elapsed
│ │ │ - -- .00369202s elapsed
│ │ │ - -- .420974s elapsed
│ │ │ + -- .0030297s elapsed
│ │ │ + -- .00688633s elapsed
│ │ │ + -- .027006s elapsed
│ │ │ + -- .0109151s elapsed
│ │ │ + -- .00391359s elapsed
│ │ │ + -- .392449s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -112,15 +112,15 @@
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .180171s elapsed
│ │ │ + -- .214537s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -140,22 +140,22 @@
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00685927s elapsed
│ │ │ - -- .0180639s elapsed
│ │ │ - -- .0911599s elapsed
│ │ │ - -- 1.23617s elapsed
│ │ │ - -- .586255s elapsed
│ │ │ - -- .0383612s elapsed
│ │ │ - -- .0067379s elapsed
│ │ │ - -- 6.09192s elapsed
    
│ │ │ + -- .00532269s elapsed
│ │ │ + -- .0200809s elapsed
│ │ │ + -- .106871s elapsed
│ │ │ + -- 1.08678s elapsed
│ │ │ + -- .469657s elapsed
│ │ │ + -- .0453141s elapsed
│ │ │ + -- .0081838s elapsed
│ │ │ + -- 6.36253s 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)
│ │ │ │ - -- .00264214s elapsed
│ │ │ │ - -- .00671366s elapsed
│ │ │ │ - -- .0243345s elapsed
│ │ │ │ - -- .0286816s elapsed
│ │ │ │ - -- .00369202s elapsed
│ │ │ │ - -- .420974s elapsed
│ │ │ │ + -- .0030297s elapsed
│ │ │ │ + -- .00688633s elapsed
│ │ │ │ + -- .027006s elapsed
│ │ │ │ + -- .0109151s elapsed
│ │ │ │ + -- .00391359s elapsed
│ │ │ │ + -- .392449s 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
│ │ │ │ - -- .180171s elapsed
│ │ │ │ + -- .214537s 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);
│ │ │ │ - -- .00685927s elapsed
│ │ │ │ - -- .0180639s elapsed
│ │ │ │ - -- .0911599s elapsed
│ │ │ │ - -- 1.23617s elapsed
│ │ │ │ - -- .586255s elapsed
│ │ │ │ - -- .0383612s elapsed
│ │ │ │ - -- .0067379s elapsed
│ │ │ │ - -- 6.09192s elapsed
│ │ │ │ + -- .00532269s elapsed
│ │ │ │ + -- .0200809s elapsed
│ │ │ │ + -- .106871s elapsed
│ │ │ │ + -- 1.08678s elapsed
│ │ │ │ + -- .469657s elapsed
│ │ │ │ + -- .0453141s elapsed
│ │ │ │ + -- .0081838s elapsed
│ │ │ │ + -- 6.36253s elapsed
│ │ │ │  i7 : carpetBettiTable(h,7)
│ │ │ │  
│ │ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ │ │           2: .  .   .   .    .    . 1155 1408 891 320 55  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html
│ │ │ @@ -80,19 +80,19 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00264923s elapsed
│ │ │ - -- .00889819s elapsed
│ │ │ - -- .0257235s elapsed
│ │ │ - -- .00999915s elapsed
│ │ │ - -- .00370523s elapsed
│ │ │ + -- .00311883s elapsed
│ │ │ + -- .00746631s elapsed
│ │ │ + -- .0264546s elapsed
│ │ │ + -- .0110235s elapsed
│ │ │ + -- .00428519s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -134,15 +134,15 @@
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .185433s elapsed
│ │ │ + -- .232498s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -162,22 +162,22 @@
│ │ │  
│ │ │  o6 : BettiTally
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00480544s elapsed
│ │ │ - -- .0187963s elapsed
│ │ │ - -- .0913313s elapsed
│ │ │ - -- 1.28201s elapsed
│ │ │ - -- .692733s elapsed
│ │ │ - -- .0429653s elapsed
│ │ │ - -- .00652921s elapsed
│ │ │ - -- 6.72795s elapsed
    
│ │ │ + -- .00526949s elapsed
│ │ │ + -- .020366s elapsed
│ │ │ + -- .119037s elapsed
│ │ │ + -- 1.1561s elapsed
│ │ │ + -- .470965s elapsed
│ │ │ + -- .043711s elapsed
│ │ │ + -- .00902826s elapsed
│ │ │ + -- 6.34639s 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)
│ │ │ │ - -- .00264923s elapsed
│ │ │ │ - -- .00889819s elapsed
│ │ │ │ - -- .0257235s elapsed
│ │ │ │ - -- .00999915s elapsed
│ │ │ │ - -- .00370523s elapsed
│ │ │ │ + -- .00311883s elapsed
│ │ │ │ + -- .00746631s elapsed
│ │ │ │ + -- .0264546s elapsed
│ │ │ │ + -- .0110235s elapsed
│ │ │ │ + -- .00428519s 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
│ │ │ │ - -- .185433s elapsed
│ │ │ │ + -- .232498s 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);
│ │ │ │ - -- .00480544s elapsed
│ │ │ │ - -- .0187963s elapsed
│ │ │ │ - -- .0913313s elapsed
│ │ │ │ - -- 1.28201s elapsed
│ │ │ │ - -- .692733s elapsed
│ │ │ │ - -- .0429653s elapsed
│ │ │ │ - -- .00652921s elapsed
│ │ │ │ - -- 6.72795s elapsed
│ │ │ │ + -- .00526949s elapsed
│ │ │ │ + -- .020366s elapsed
│ │ │ │ + -- .119037s elapsed
│ │ │ │ + -- 1.1561s elapsed
│ │ │ │ + -- .470965s elapsed
│ │ │ │ + -- .043711s elapsed
│ │ │ │ + -- .00902826s elapsed
│ │ │ │ + -- 6.34639s elapsed
│ │ │ │  i8 : keys h
│ │ │ │  
│ │ │ │  o8 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : carpetBettiTable(h,7)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html
│ │ │ @@ -80,82 +80,82 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : d=carpetDet(a,b)
│ │ │ - -- .00624298s elapsed
│ │ │ - -- .0136333s elapsed
│ │ │ + -- .00835134s elapsed
│ │ │ + -- .013873s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000327194s elapsed
│ │ │ + -- .00028654s elapsed
│ │ │  1
│ │ │ - -- .000136907s elapsed
│ │ │ + -- .000185432s elapsed
│ │ │  1
│ │ │ - -- .000132328s elapsed
│ │ │ + -- .000188627s elapsed
│ │ │  1
│ │ │ - -- .000144721s elapsed
│ │ │ + -- .000184763s elapsed
│ │ │  1
│ │ │ - -- .000134663s elapsed
│ │ │ + -- .000180944s elapsed
│ │ │  2
│ │ │ - -- .000146294s elapsed
│ │ │ + -- .000189348s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000158026s elapsed
│ │ │ + -- .000196318s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000171752s elapsed
│ │ │ + -- .00021251s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000148419s elapsed
│ │ │ + -- .000190274s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000135754s elapsed
│ │ │ + -- .000225895s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000126858s elapsed
│ │ │ + -- .000157541s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000129172s elapsed
│ │ │ + -- .000194815s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000130054s elapsed
│ │ │ + -- .000177877s elapsed
│ │ │  2
│ │ │ - -- .000121047s elapsed
│ │ │ + -- .000165428s elapsed
│ │ │  2
│ │ │ - -- .000127559s elapsed
│ │ │ + -- .000162315s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000127209s elapsed
│ │ │ + -- .000233775s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000159349s elapsed
│ │ │ + -- .000170383s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000134152s elapsed
│ │ │ + -- .000170659s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000144321s elapsed
│ │ │ + -- .00018404s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000142818s elapsed
│ │ │ + -- .000156819s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000127849s elapsed
│ │ │ + -- .000153345s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120276s elapsed
│ │ │ + -- .000150494s elapsed
│ │ │  2
│ │ │ - -- .000147958s elapsed
│ │ │ + -- .000174933s elapsed
│ │ │  1
│ │ │ - -- .000126007s elapsed
│ │ │ + -- .000149865s elapsed
│ │ │  1
│ │ │ - -- .000127409s elapsed
│ │ │ + -- .000172348s elapsed
│ │ │  1
│ │ │ - -- .000124584s elapsed
│ │ │ + -- .000172294s 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)
│ │ │ │ - -- .00624298s elapsed
│ │ │ │ - -- .0136333s elapsed
│ │ │ │ + -- .00835134s elapsed
│ │ │ │ + -- .013873s elapsed
│ │ │ │  (number Of blocks, 26)
│ │ │ │ - -- .000327194s elapsed
│ │ │ │ + -- .00028654s elapsed
│ │ │ │  1
│ │ │ │ - -- .000136907s elapsed
│ │ │ │ + -- .000185432s elapsed
│ │ │ │  1
│ │ │ │ - -- .000132328s elapsed
│ │ │ │ + -- .000188627s elapsed
│ │ │ │  1
│ │ │ │ - -- .000144721s elapsed
│ │ │ │ + -- .000184763s elapsed
│ │ │ │  1
│ │ │ │ - -- .000134663s elapsed
│ │ │ │ + -- .000180944s elapsed
│ │ │ │  2
│ │ │ │ - -- .000146294s elapsed
│ │ │ │ + -- .000189348s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000158026s elapsed
│ │ │ │ + -- .000196318s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000171752s elapsed
│ │ │ │ + -- .00021251s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000148419s elapsed
│ │ │ │ + -- .000190274s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000135754s elapsed
│ │ │ │ + -- .000225895s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000126858s elapsed
│ │ │ │ + -- .000157541s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000129172s elapsed
│ │ │ │ + -- .000194815s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000130054s elapsed
│ │ │ │ + -- .000177877s elapsed
│ │ │ │  2
│ │ │ │ - -- .000121047s elapsed
│ │ │ │ + -- .000165428s elapsed
│ │ │ │  2
│ │ │ │ - -- .000127559s elapsed
│ │ │ │ + -- .000162315s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000127209s elapsed
│ │ │ │ + -- .000233775s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000159349s elapsed
│ │ │ │ + -- .000170383s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000134152s elapsed
│ │ │ │ + -- .000170659s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000144321s elapsed
│ │ │ │ + -- .00018404s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000142818s elapsed
│ │ │ │ + -- .000156819s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000127849s elapsed
│ │ │ │ + -- .000153345s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000120276s elapsed
│ │ │ │ + -- .000150494s elapsed
│ │ │ │  2
│ │ │ │ - -- .000147958s elapsed
│ │ │ │ + -- .000174933s elapsed
│ │ │ │  1
│ │ │ │ - -- .000126007s elapsed
│ │ │ │ + -- .000149865s elapsed
│ │ │ │  1
│ │ │ │ - -- .000127409s elapsed
│ │ │ │ + -- .000172348s elapsed
│ │ │ │  1
│ │ │ │ - -- .000124584s elapsed
│ │ │ │ + -- .000172294s elapsed
│ │ │ │  1
│ │ │ │  
│ │ │ │  o2 = 3131031158784
│ │ │ │  i3 : factor d
│ │ │ │  
│ │ │ │        32 6
│ │ │ │  o3 = 2  3
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html
│ │ │ @@ -85,20 +85,20 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
    
│ │ │                
    i2 : computeBound 3
│ │ │ - -- .238508s elapsed
│ │ │ - -- .302491s elapsed
│ │ │ - -- .206777s elapsed
│ │ │ - -- .290534s elapsed
│ │ │ - -- .340601s elapsed
│ │ │ - -- .391058s elapsed
│ │ │ + -- .182577s elapsed
│ │ │ + -- .20292s elapsed
│ │ │ + -- .186752s elapsed
│ │ │ + -- .390872s elapsed
│ │ │ + -- .204045s elapsed
│ │ │ + -- .251665s 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
│ │ │ │ - -- .238508s elapsed
│ │ │ │ - -- .302491s elapsed
│ │ │ │ - -- .206777s elapsed
│ │ │ │ - -- .290534s elapsed
│ │ │ │ - -- .340601s elapsed
│ │ │ │ - -- .391058s elapsed
│ │ │ │ + -- .182577s elapsed
│ │ │ │ + -- .20292s elapsed
│ │ │ │ + -- .186752s elapsed
│ │ │ │ + -- .390872s elapsed
│ │ │ │ + -- .204045s elapsed
│ │ │ │ + -- .251665s elapsed
│ │ │ │  
│ │ │ │  o2 = 6
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_l_a_t_i_v_e_E_q_u_a_t_i_o_n_s -- compute the relative quadrics
│ │ │ │  ********** WWaayyss ttoo uussee ccoommppuutteeBBoouunndd:: **********
│ │ │ │      * computeBound(ZZ)
│ │ │ │      * computeBound(ZZ,ZZ,ZZ)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html
│ │ │ @@ -90,19 +90,19 @@
│ │ │  
│ │ │  o2 : Sequence
    │ │ │
    │ │ │ 
    i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00264488s elapsed
│ │ │ - -- .00654608s elapsed
│ │ │ - -- .0240881s elapsed
│ │ │ - -- .00901472s elapsed
│ │ │ - -- .0141597s elapsed
│ │ │ + -- .0292424s elapsed
│ │ │ + -- .0112616s elapsed
│ │ │ + -- .0275194s elapsed
│ │ │ + -- .0104555s elapsed
│ │ │ + -- .00443724s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .257455s elapsed
│ │ │ + -- .273186s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -178,19 +178,19 @@
│ │ │  
│ │ │  o7 : Sequence
    │ │ │
    │ │ │ 
    i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00265049s elapsed
│ │ │ - -- .00673866s elapsed
│ │ │ - -- .0333516s elapsed
│ │ │ - -- .0100248s elapsed
│ │ │ - -- .00383599s elapsed
│ │ │ + -- .00284613s elapsed
│ │ │ + -- .00773682s elapsed
│ │ │ + -- .0313114s elapsed
│ │ │ + -- .0107948s elapsed
│ │ │ + -- .00455676s 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)
│ │ │ │ - -- .00264488s elapsed
│ │ │ │ - -- .00654608s elapsed
│ │ │ │ - -- .0240881s elapsed
│ │ │ │ - -- .00901472s elapsed
│ │ │ │ - -- .0141597s elapsed
│ │ │ │ + -- .0292424s elapsed
│ │ │ │ + -- .0112616s elapsed
│ │ │ │ + -- .0275194s elapsed
│ │ │ │ + -- .0104555s elapsed
│ │ │ │ + -- .00443724s 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)
│ │ │ │ - -- .257455s elapsed
│ │ │ │ + -- .273186s 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)
│ │ │ │ - -- .00265049s elapsed
│ │ │ │ - -- .00673866s elapsed
│ │ │ │ - -- .0333516s elapsed
│ │ │ │ - -- .0100248s elapsed
│ │ │ │ - -- .00383599s elapsed
│ │ │ │ + -- .00284613s elapsed
│ │ │ │ + -- .00773682s elapsed
│ │ │ │ + -- .0313114s elapsed
│ │ │ │ + -- .0107948s elapsed
│ │ │ │ + -- .00455676s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html
│ │ │ @@ -78,172 +78,172 @@
│ │ │  
│ │ │  o1 = 4
    │ │ │
    │ │ │ 
    i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0162559s elapsed
│ │ │ + -- .0221523s 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
│ │ │ - -- .000046116s elapsed
│ │ │ + -- .000066675s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000082144s elapsed
│ │ │ + -- .0001203s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000071754s elapsed
│ │ │ + -- .000084858s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000081182s elapsed
│ │ │ + -- .00010081s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000082916s elapsed
│ │ │ + -- .00010873s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000103855s elapsed
│ │ │ + -- .000113594s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000023434s elapsed
│ │ │ + -- .000031226s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000071143s elapsed
│ │ │ + -- .000076502s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .00007983s elapsed
│ │ │ + -- .000100636s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000093766s elapsed
│ │ │ + -- .00011083s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000088076s elapsed
│ │ │ + -- .000105735s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000095389s elapsed
│ │ │ + -- .000087389s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .00007482s elapsed
│ │ │ + -- .000104167s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000063289s elapsed
│ │ │ + -- .000084419s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000070302s elapsed
│ │ │ + -- .000080289s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000023885s elapsed
│ │ │ + -- .000035956s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000066264s elapsed
│ │ │ + -- .000089896s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000024747s elapsed
│ │ │ + -- .000030692s 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)
│ │ │ │ - -- .0162559s elapsed
│ │ │ │ + -- .0221523s 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
│ │ │ │ - -- .000046116s elapsed
│ │ │ │ + -- .000066675s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . 2
│ │ │ │ - -- .000082144s elapsed
│ │ │ │ + -- .0001203s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . .
│ │ │ │      9: . 2
│ │ │ │ - -- .000071754s elapsed
│ │ │ │ + -- .000084858s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │      7: 2 .
│ │ │ │      8: 1 .
│ │ │ │      9: . 1
│ │ │ │     10: . 2
│ │ │ │ - -- .000081182s elapsed
│ │ │ │ + -- .00010081s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (-3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      7: 1 .
│ │ │ │      8: 1 .
│ │ │ │      9: 2 2
│ │ │ │     10: . 1
│ │ │ │     11: . 1
│ │ │ │ - -- .000082916s elapsed
│ │ │ │ + -- .00010873s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      8: 1 .
│ │ │ │      9: 2 1
│ │ │ │     10: 1 2
│ │ │ │     11: . 1
│ │ │ │ - -- .000103855s elapsed
│ │ │ │ + -- .000113594s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      9: 1 1
│ │ │ │ - -- .000023434s elapsed
│ │ │ │ + -- .000031226s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      9: 1 1
│ │ │ │     10: 1 1
│ │ │ │ - -- .000071143s elapsed
│ │ │ │ + -- .000076502s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .00007983s elapsed
│ │ │ │ + -- .000100636s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 2 1
│ │ │ │     11: 1 2
│ │ │ │     12: . 1
│ │ │ │ - -- .000093766s elapsed
│ │ │ │ + -- .00011083s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 1 .
│ │ │ │     11: 2 2
│ │ │ │     12: . 1
│ │ │ │     13: . 1
│ │ │ │ - -- .000088076s elapsed
│ │ │ │ + -- .000105735s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000095389s elapsed
│ │ │ │ + -- .000087389s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │     10: 2 .
│ │ │ │     11: 1 .
│ │ │ │     12: . 1
│ │ │ │     13: . 2
│ │ │ │ - -- .00007482s elapsed
│ │ │ │ + -- .000104167s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     10: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000063289s elapsed
│ │ │ │ + -- .000084419s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     11: 2 .
│ │ │ │     12: . .
│ │ │ │     13: . 2
│ │ │ │ - -- .000070302s elapsed
│ │ │ │ + -- .000080289s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000023885s elapsed
│ │ │ │ + -- .000035956s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     12: 2 .
│ │ │ │     13: . 2
│ │ │ │ - -- .000066264s elapsed
│ │ │ │ + -- .000089896s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     13: 1 1
│ │ │ │ - -- .000024747s elapsed
│ │ │ │ + -- .000030692s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │  
│ │ │ │         6      32    32
│ │ │ │  o2 = (3 , (e )  (e )  )
│ │ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -7,55 +7,55 @@
│ │ │  
│ │ │  i2 : m = syz m1;
│ │ │  
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i3 : time LLL m;
│ │ │ - -- used 0.00871639s (cpu); 0.00871305s (thread); 0s (gc)
│ │ │ + -- used 0.0129193s (cpu); 0.0129198s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0284007s (cpu); 0.028402s (thread); 0s (gc)
│ │ │ + -- used 0.0341349s (cpu); 0.0341439s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.110205s (cpu); 0.110209s (thread); 0s (gc)
│ │ │ + -- used 0.139245s (cpu); 0.13906s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0107045s (cpu); 0.0107299s (thread); 0s (gc)
│ │ │ + -- used 0.0127287s (cpu); 0.0127373s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0479164s (cpu); 0.0478833s (thread); 0s (gc)
│ │ │ + -- used 0.0643493s (cpu); 0.0643594s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0563528s (cpu); 0.0563532s (thread); 0s (gc)
│ │ │ + -- used 0.0616462s (cpu); 0.0616564s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.321235s (cpu); 0.321236s (thread); 0s (gc)
│ │ │ + -- used 0.338777s (cpu); 0.338786s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.112857s (cpu); 0.11286s (thread); 0s (gc)
│ │ │ + -- used 0.152398s (cpu); 0.152407s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -139,78 +139,78 @@
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i3 : time LLL m;
│ │ │ - -- used 0.00871639s (cpu); 0.00871305s (thread); 0s (gc)
│ │ │ + -- used 0.0129193s (cpu); 0.0129198s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0284007s (cpu); 0.028402s (thread); 0s (gc)
│ │ │ + -- used 0.0341349s (cpu); 0.0341439s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.110205s (cpu); 0.110209s (thread); 0s (gc)
│ │ │ + -- used 0.139245s (cpu); 0.13906s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0107045s (cpu); 0.0107299s (thread); 0s (gc)
│ │ │ + -- used 0.0127287s (cpu); 0.0127373s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0479164s (cpu); 0.0478833s (thread); 0s (gc)
│ │ │ + -- used 0.0643493s (cpu); 0.0643594s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0563528s (cpu); 0.0563532s (thread); 0s (gc)
│ │ │ + -- used 0.0616462s (cpu); 0.0616564s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.321235s (cpu); 0.321236s (thread); 0s (gc)
│ │ │ + -- used 0.338777s (cpu); 0.338786s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.112857s (cpu); 0.11286s (thread); 0s (gc)
│ │ │ + -- used 0.152398s (cpu); 0.152407s (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 0.00871639s (cpu); 0.00871305s (thread); 0s (gc) │ │ │ │ + -- used 0.0129193s (cpu); 0.0129198s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o3 : Matrix ZZ <-- ZZ │ │ │ │ i4 : time LLL(m, Strategy=>CohenEngine); │ │ │ │ - -- used 0.0284007s (cpu); 0.028402s (thread); 0s (gc) │ │ │ │ + -- used 0.0341349s (cpu); 0.0341439s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ i5 : time LLL(m, Strategy=>CohenTopLevel); │ │ │ │ - -- used 0.110205s (cpu); 0.110209s (thread); 0s (gc) │ │ │ │ + -- used 0.139245s (cpu); 0.13906s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o5 : Matrix ZZ <-- ZZ │ │ │ │ i6 : time LLL(m, Strategy=>{Givens,RealFP}); │ │ │ │ - -- used 0.0107045s (cpu); 0.0107299s (thread); 0s (gc) │ │ │ │ + -- used 0.0127287s (cpu); 0.0127373s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o6 : Matrix ZZ <-- ZZ │ │ │ │ i7 : time LLL(m, Strategy=>{Givens,RealQP}); │ │ │ │ - -- used 0.0479164s (cpu); 0.0478833s (thread); 0s (gc) │ │ │ │ + -- used 0.0643493s (cpu); 0.0643594s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o7 : Matrix ZZ <-- ZZ │ │ │ │ i8 : time LLL(m, Strategy=>{Givens,RealXD}); │ │ │ │ - -- used 0.0563528s (cpu); 0.0563532s (thread); 0s (gc) │ │ │ │ + -- used 0.0616462s (cpu); 0.0616564s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o8 : Matrix ZZ <-- ZZ │ │ │ │ i9 : time LLL(m, Strategy=>{Givens,RealRR}); │ │ │ │ - -- used 0.321235s (cpu); 0.321236s (thread); 0s (gc) │ │ │ │ + -- used 0.338777s (cpu); 0.338786s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o9 : Matrix ZZ <-- ZZ │ │ │ │ i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP}); │ │ │ │ - -- used 0.112857s (cpu); 0.11286s (thread); 0s (gc) │ │ │ │ + -- used 0.152398s (cpu); 0.152407s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o10 : Matrix ZZ <-- ZZ │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ For most of the options, the columns do not need to be linearly independent. │ │ │ │ The strategies CohenEngine and CohenTopLevel currently require the columns to │ │ │ │ be linearly independent. │ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ @@ -16,14 +16,14 @@ │ │ │ │ │ │ 3 8 │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i5 : P = convexHull(M); │ │ │ │ │ │ i6 : time areIsomorphic(P,P); │ │ │ - -- used 0.57961s (cpu); 0.456304s (thread); 0s (gc) │ │ │ + -- used 1.04118s (cpu); 0.578589s (thread); 0s (gc) │ │ │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ - -- used 0.446402s (cpu); 0.350861s (thread); 0s (gc) │ │ │ + -- used 0.903389s (cpu); 0.406243s (thread); 0s (gc) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ @@ -120,21 +120,21 @@ │ │ │ │ │ │ 
    i5 : P = convexHull(M);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.57961s (cpu); 0.456304s (thread); 0s (gc)
    │ │ │ + -- used 1.04118s (cpu); 0.578589s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.446402s (cpu); 0.350861s (thread); 0s (gc)
    │ │ │ + -- used 0.903389s (cpu); 0.406243s (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 0.57961s (cpu); 0.456304s (thread); 0s (gc) │ │ │ │ + -- used 1.04118s (cpu); 0.578589s (thread); 0s (gc) │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ │ - -- used 0.446402s (cpu); 0.350861s (thread); 0s (gc) │ │ │ │ + -- used 0.903389s (cpu); 0.406243s (thread); 0s (gc) │ │ │ │ ********** WWaayyss ttoo uussee aarreeIIssoommoorrpphhiicc:: ********** │ │ │ │ * areIsomorphic(Matrix,Matrix) │ │ │ │ * areIsomorphic(Polyhedron,Polyhedron) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _a_r_e_I_s_o_m_o_r_p_h_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ @@ -29,21 +29,21 @@ │ │ │ i5 : findRegion({{0,0},{4,4}},M,f) │ │ │ │ │ │ o5 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : elapsedTime findRegion({{0,0},{4,4}},M,f) │ │ │ - -- .106878s elapsed │ │ │ + -- .0992923s 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}}) │ │ │ - -- .0125467s elapsed │ │ │ + -- .0155386s elapsed │ │ │ │ │ │ o7 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ @@ -30,21 +30,21 @@ │ │ │ i5 : apply(L, d -> isLinearComplex res prune truncate(d,M)) │ │ │ │ │ │ o5 = {true, true} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M) │ │ │ - -- 4.37365s elapsed │ │ │ + -- 3.55977s elapsed │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ - -- .0297842s elapsed │ │ │ + -- .0282688s elapsed │ │ │ │ │ │ o7 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html │ │ │ @@ -125,25 +125,25 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ - -- .106878s elapsed
│ │ │ + -- .0992923s 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}})
│ │ │ - -- .0125467s elapsed
│ │ │ + -- .0155386s 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) │ │ │ │ - -- .106878s elapsed │ │ │ │ + -- .0992923s 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}}) │ │ │ │ - -- .0125467s elapsed │ │ │ │ + -- .0155386s elapsed │ │ │ │ │ │ │ │ o7 = {{1, 2}, {3, 1}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** CCoonnttrriibbuuttoorrss ********** │ │ │ │ Mahrud Sayrafi contributed to the code for this function. │ │ │ │ ********** CCaavveeaatt ********** │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ @@ -123,25 +123,25 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ - -- 4.37365s elapsed
│ │ │ + -- 3.55977s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .0297842s elapsed
│ │ │ + -- .0282688s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,21 +48,21 @@ │ │ │ │ o5 = {true, true} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ The output is a list of the minimal multidegrees $d$ such that the sum of the │ │ │ │ positive coordinates of $b-d$ is at most $i$ for all degrees $b$ appearing in │ │ │ │ the i-th step of the resolution of $M$. │ │ │ │ i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M) │ │ │ │ - -- 4.37365s elapsed │ │ │ │ + -- 3.55977s elapsed │ │ │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ │ - -- .0297842s elapsed │ │ │ │ + -- .0282688s elapsed │ │ │ │ │ │ │ │ o7 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ In general linearTruncationsBound will not find the minimal degrees where $M$ │ │ │ │ has a linear resolution but will be faster than repeatedly truncating $M$. │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ │ │ │ o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 | │ │ │ │ │ │ 1 │ │ │ o4 : RP-module, quotient of RP │ │ │ │ │ │ i5 : elapsedTime hilbertSamuelFunction(M, 0, 6) │ │ │ - -- .188967s elapsed │ │ │ + -- .211418s 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 │ │ │ - -- .0743369s elapsed │ │ │ + -- .0209733s elapsed │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ o11 : List │ │ │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ - -- .248024s elapsed │ │ │ + -- .314252s elapsed │ │ │ │ │ │ o12 = {6, 12, 18, 24, 30, 36} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html │ │ │ @@ -111,15 +111,15 @@ │ │ │ 1 │ │ │ o4 : RP-module, quotient of RP │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .188967s elapsed
│ │ │ + -- .211418s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -163,25 +163,25 @@ │ │ │ │ │ │ o10 : Ideal of RP │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0743369s elapsed
│ │ │ + -- .0209733s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .248024s elapsed
│ │ │ + -- .314252s 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) │ │ │ │ - -- .188967s elapsed │ │ │ │ + -- .211418s 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 │ │ │ │ - -- .0743369s elapsed │ │ │ │ + -- .0209733s elapsed │ │ │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ │ │ o11 : List │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ │ - -- .248024s elapsed │ │ │ │ + -- .314252s elapsed │ │ │ │ │ │ │ │ o12 = {6, 12, 18, 24, 30, 36} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Hilbert-Samuel function with respect to a parameter ideal other than the │ │ │ │ maximal ideal can be slower. │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Command.out │ │ │ @@ -5,12 +5,12 @@ │ │ │ i2 : f │ │ │ │ │ │ o2 = 1073741824 │ │ │ │ │ │ i3 : (c = Command "date";) │ │ │ │ │ │ i4 : c │ │ │ -Mon Jan 19 17:30:24 UTC 2026 │ │ │ +Sun Jan 25 00:43:41 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/M2-11638-0/0.dbm │ │ │ +o1 = /tmp/M2-13228-0/0.dbm │ │ │ │ │ │ i2 : x = openDatabaseOut filename │ │ │ │ │ │ -o2 = /tmp/M2-11638-0/0.dbm │ │ │ +o2 = /tmp/M2-13228-0/0.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" => 42970092554 } │ │ │ +o1 = HashTable{"bytesAlloc" => 43074037370 } │ │ │ "GC_free_space_divisor" => 3 │ │ │ "GC_LARGE_ALLOC_WARN_INTERVAL" => 1 │ │ │ "gcCpuTimeSecs" => 0 │ │ │ - "heapSize" => 206467072 │ │ │ - "numGCs" => 795 │ │ │ - "numGCThreads" => 6 │ │ │ + "heapSize" => 225644544 │ │ │ + "numGCs" => 783 │ │ │ + "numGCThreads" => 16 │ │ │ │ │ │ o1 : HashTable │ │ │ │ │ │ i2 : s#"heapSize" │ │ │ │ │ │ -o2 = 206467072 │ │ │ +o2 = 225644544 │ │ │ │ │ │ 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 0.00907699s (cpu); 0.00906961s (thread); 0s (gc) │ │ │ + -- used 0.00592172s (cpu); 0.00591156s (thread); 0s (gc) │ │ │ │ │ │ o8 : Ideal of R │ │ │ │ │ │ i9 : time K = truncate(8, I, MinimalGenerators => true); │ │ │ - -- used 0.0809478s (cpu); 0.0809554s (thread); 0s (gc) │ │ │ + -- used 0.0509569s (cpu); 0.0509699s (thread); 0s (gc) │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ │ │ i10 : numgens J │ │ │ │ │ │ o10 = 1067 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out │ │ │ @@ -3,13 +3,13 @@ │ │ │ i1 : M = random(RR^200, RR^200); │ │ │ │ │ │ 200 200 │ │ │ o1 : Matrix RR <-- RR │ │ │ 53 53 │ │ │ │ │ │ i2 : time SVD(M); │ │ │ - -- used 0.0247234s (cpu); 0.0247216s (thread); 0s (gc) │ │ │ + -- used 0.0402253s (cpu); 0.0402245s (thread); 0s (gc) │ │ │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ - -- used 0.0245265s (cpu); 0.0245324s (thread); 0s (gc) │ │ │ + -- used 0.039368s (cpu); 0.0393845s (thread); 0s (gc) │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out │ │ │ @@ -351,15 +351,15 @@ │ │ │ | b e h k n q | │ │ │ | c f i l o r | │ │ │ │ │ │ 3 │ │ │ o58 : R-module, quotient of R │ │ │ │ │ │ i59 : time C = resolution M │ │ │ - -- used 0.00196552s (cpu); 0.00195841s (thread); 0s (gc) │ │ │ + -- used 0.00218428s (cpu); 0.00217334s (thread); 0s (gc) │ │ │ │ │ │ 3 6 15 18 6 │ │ │ o59 = R <-- R <-- R <-- R <-- R <-- 0 │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ o59 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ @@ -14,10 +14,10 @@ │ │ │ │ │ │ i4 : peek read f │ │ │ │ │ │ o4 = "hi there" │ │ │ │ │ │ i5 : atEndOfFile f │ │ │ │ │ │ -o5 = false │ │ │ +o5 = true │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out │ │ │ @@ -1,9 +1,9 @@ │ │ │ -- -*- M2-comint -*- hash: 1330379359420 │ │ │ │ │ │ i1 : benchmark "sqrt 2p100000" │ │ │ │ │ │ -o1 = .0002898228374663042 │ │ │ +o1 = .0003637207220832036 │ │ │ │ │ │ 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' │ │ │ - -- .00625565s elapsed │ │ │ + -- .00454321s elapsed │ │ │ │ │ │ o4 = 3 │ │ │ │ │ │ i5 : elapsedTime pdim' M │ │ │ - -- .000001804s elapsed │ │ │ + -- .000002686s elapsed │ │ │ │ │ │ o5 = 3 │ │ │ │ │ │ i6 : peek M.cache │ │ │ │ │ │ o6 = CacheTable{cache => MutableHashTable{} } │ │ │ isHomogeneous => true │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ @@ -18,57 +18,57 @@ │ │ │ │ │ │ o4 = <> │ │ │ │ │ │ o4 : Task │ │ │ │ │ │ i5 : n │ │ │ │ │ │ -o5 = 710974 │ │ │ +o5 = 1095396 │ │ │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ o6 = 0 │ │ │ │ │ │ i7 : t │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ o7 : Task │ │ │ │ │ │ i8 : n │ │ │ │ │ │ -o8 = 1452916 │ │ │ +o8 = 2221986 │ │ │ │ │ │ i9 : isReady t │ │ │ │ │ │ o9 = false │ │ │ │ │ │ i10 : cancelTask t │ │ │ │ │ │ i11 : sleep 2 │ │ │ -stdio:2:39:(3):[1]: error: interrupted │ │ │ +stdio:2:25:(3):[1]: error: interrupted │ │ │ │ │ │ o11 = 0 │ │ │ │ │ │ i12 : t │ │ │ │ │ │ o12 = <> │ │ │ │ │ │ o12 : Task │ │ │ │ │ │ i13 : n │ │ │ │ │ │ -o13 = 1453128 │ │ │ +o13 = 2222213 │ │ │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ o14 = 0 │ │ │ │ │ │ i15 : n │ │ │ │ │ │ -o15 = 1453128 │ │ │ +o15 = 2222213 │ │ │ │ │ │ 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/M2-10460-0/0 │ │ │ +o1 = /tmp/M2-10830-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-10460-0/0 │ │ │ +o2 = /tmp/M2-10830-0/0 │ │ │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ -o3 = /tmp/M2-10460-0/0/ │ │ │ +o3 = /tmp/M2-10830-0/0/ │ │ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ -o4 = /tmp/M2-10460-0/0/ │ │ │ +o4 = /tmp/M2-10830-0/0/ │ │ │ │ │ │ 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") -- .153064s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .128193s elapsed │ │ │ │ │ │ i3 : check FirstPackage │ │ │ - -- capturing check(0, "FirstPackage") -- .150274s elapsed │ │ │ - -- capturing check(1, "FirstPackage") -- .147265s elapsed │ │ │ + -- capturing check(0, "FirstPackage") -- .127121s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .123325s elapsed │ │ │ │ │ │ i4 : check_1 "FirstPackage" │ │ │ - -- capturing check(1, "FirstPackage") -- .140205s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .134544s elapsed │ │ │ │ │ │ i5 : check "FirstPackage" │ │ │ - -- capturing check(0, "FirstPackage") -- .151078s elapsed │ │ │ - -- capturing check(1, "FirstPackage") -- .150374s elapsed │ │ │ + -- capturing check(0, "FirstPackage") -- .139643s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .143099s 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") -- .147771s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .142709s 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") -- .137448s elapsed │ │ │ - -- capturing check(1, "FirstPackage") -- .148236s elapsed │ │ │ + -- capturing check(0, "FirstPackage") -- .13571s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .132292s 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") -- .138202s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .138938s elapsed │ │ │ │ │ │ i12 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ @@ -1,25 +1,25 @@ │ │ │ -- -*- M2-comint -*- hash: 10365735446967377456 │ │ │ │ │ │ i1 : run "uname -a" │ │ │ -Linux sbuild 6.12.63+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1 (2025-12-30) x86_64 GNU/Linux │ │ │ +Linux sbuild 6.12.63+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1 (2025-12-30) 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 6.12.63+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ +o3 = "Linux sbuild 6.12.63+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ 6.12.63-1 (2025-12-30) 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 0x7f5cd0c81540 │ │ │ + -- registering gb 5 at 0x7fda3bfe9540 │ │ │ │ │ │ -- [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 0.171852s (cpu); 0.169834s (thread); 0s (gc) │ │ │ + -- used 0.224047s (cpu); 0.223729s (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 0.0037732s (cpu); 0.00284882s (thread); 0s (gc) │ │ │ + -- used 0.00288335s (cpu); 0.00322848s (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/M2-11182-0/0/ │ │ │ +o1 = /tmp/M2-12292-0/0/ │ │ │ │ │ │ i2 : dst = temporaryFileName() | "/" │ │ │ │ │ │ -o2 = /tmp/M2-11182-0/1/ │ │ │ +o2 = /tmp/M2-12292-0/1/ │ │ │ │ │ │ i3 : makeDirectory (src|"a/") │ │ │ │ │ │ -o3 = /tmp/M2-11182-0/0/a/ │ │ │ +o3 = /tmp/M2-12292-0/0/a/ │ │ │ │ │ │ i4 : makeDirectory (src|"b/") │ │ │ │ │ │ -o4 = /tmp/M2-11182-0/0/b/ │ │ │ +o4 = /tmp/M2-12292-0/0/b/ │ │ │ │ │ │ i5 : makeDirectory (src|"b/c/") │ │ │ │ │ │ -o5 = /tmp/M2-11182-0/0/b/c/ │ │ │ +o5 = /tmp/M2-12292-0/0/b/c/ │ │ │ │ │ │ i6 : src|"a/f" << "hi there" << close │ │ │ │ │ │ -o6 = /tmp/M2-11182-0/0/a/f │ │ │ +o6 = /tmp/M2-12292-0/0/a/f │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : src|"a/g" << "hi there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11182-0/0/a/g │ │ │ +o7 = /tmp/M2-12292-0/0/a/g │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : src|"b/c/g" << "ho there" << close │ │ │ │ │ │ -o8 = /tmp/M2-11182-0/0/b/c/g │ │ │ +o8 = /tmp/M2-12292-0/0/b/c/g │ │ │ │ │ │ o8 : File │ │ │ │ │ │ i9 : stack findFiles src │ │ │ │ │ │ -o9 = /tmp/M2-11182-0/0/ │ │ │ - /tmp/M2-11182-0/0/b/ │ │ │ - /tmp/M2-11182-0/0/b/c/ │ │ │ - /tmp/M2-11182-0/0/b/c/g │ │ │ - /tmp/M2-11182-0/0/a/ │ │ │ - /tmp/M2-11182-0/0/a/g │ │ │ - /tmp/M2-11182-0/0/a/f │ │ │ +o9 = /tmp/M2-12292-0/0/ │ │ │ + /tmp/M2-12292-0/0/a/ │ │ │ + /tmp/M2-12292-0/0/a/g │ │ │ + /tmp/M2-12292-0/0/a/f │ │ │ + /tmp/M2-12292-0/0/b/ │ │ │ + /tmp/M2-12292-0/0/b/c/ │ │ │ + /tmp/M2-12292-0/0/b/c/g │ │ │ │ │ │ i10 : copyDirectory(src,dst,Verbose=>true) │ │ │ - -- copying: /tmp/M2-11182-0/0/b/c/g -> /tmp/M2-11182-0/1/b/c/g │ │ │ - -- copying: /tmp/M2-11182-0/0/a/g -> /tmp/M2-11182-0/1/a/g │ │ │ - -- copying: /tmp/M2-11182-0/0/a/f -> /tmp/M2-11182-0/1/a/f │ │ │ + -- copying: /tmp/M2-12292-0/0/a/g -> /tmp/M2-12292-0/1/a/g │ │ │ + -- copying: /tmp/M2-12292-0/0/a/f -> /tmp/M2-12292-0/1/a/f │ │ │ + -- copying: /tmp/M2-12292-0/0/b/c/g -> /tmp/M2-12292-0/1/b/c/g │ │ │ │ │ │ i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true) │ │ │ - -- skipping: /tmp/M2-11182-0/0/b/c/g not newer than /tmp/M2-11182-0/1/b/c/g │ │ │ - -- skipping: /tmp/M2-11182-0/0/a/g not newer than /tmp/M2-11182-0/1/a/g │ │ │ - -- skipping: /tmp/M2-11182-0/0/a/f not newer than /tmp/M2-11182-0/1/a/f │ │ │ + -- skipping: /tmp/M2-12292-0/0/a/g not newer than /tmp/M2-12292-0/1/a/g │ │ │ + -- skipping: /tmp/M2-12292-0/0/a/f not newer than /tmp/M2-12292-0/1/a/f │ │ │ + -- skipping: /tmp/M2-12292-0/0/b/c/g not newer than /tmp/M2-12292-0/1/b/c/g │ │ │ │ │ │ i12 : stack findFiles dst │ │ │ │ │ │ -o12 = /tmp/M2-11182-0/1/ │ │ │ - /tmp/M2-11182-0/1/a/ │ │ │ - /tmp/M2-11182-0/1/a/f │ │ │ - /tmp/M2-11182-0/1/a/g │ │ │ - /tmp/M2-11182-0/1/b/ │ │ │ - /tmp/M2-11182-0/1/b/c/ │ │ │ - /tmp/M2-11182-0/1/b/c/g │ │ │ +o12 = /tmp/M2-12292-0/1/ │ │ │ + /tmp/M2-12292-0/1/a/ │ │ │ + /tmp/M2-12292-0/1/a/g │ │ │ + /tmp/M2-12292-0/1/a/f │ │ │ + /tmp/M2-12292-0/1/b/ │ │ │ + /tmp/M2-12292-0/1/b/c/ │ │ │ + /tmp/M2-12292-0/1/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/M2-10967-0/0 │ │ │ +o1 = /tmp/M2-11857-0/0 │ │ │ │ │ │ i2 : dst = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-10967-0/1 │ │ │ +o2 = /tmp/M2-11857-0/1 │ │ │ │ │ │ i3 : src << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10967-0/0 │ │ │ +o3 = /tmp/M2-11857-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : copyFile(src,dst,Verbose=>true) │ │ │ - -- copying: /tmp/M2-10967-0/0 -> /tmp/M2-10967-0/1 │ │ │ + -- copying: /tmp/M2-11857-0/0 -> /tmp/M2-11857-0/1 │ │ │ │ │ │ i5 : get dst │ │ │ │ │ │ o5 = hi there │ │ │ │ │ │ i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true) │ │ │ - -- skipping: /tmp/M2-10967-0/0 not newer than /tmp/M2-10967-0/1 │ │ │ + -- skipping: /tmp/M2-11857-0/0 not newer than /tmp/M2-11857-0/1 │ │ │ │ │ │ i7 : src << "ho there" << close │ │ │ │ │ │ -o7 = /tmp/M2-10967-0/0 │ │ │ +o7 = /tmp/M2-11857-0/0 │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true) │ │ │ - -- skipping: /tmp/M2-10967-0/0 not newer than /tmp/M2-10967-0/1 │ │ │ + -- skipping: /tmp/M2-11857-0/0 not newer than /tmp/M2-11857-0/1 │ │ │ │ │ │ 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 = 355.748080184 │ │ │ +o1 = 380.525134373 │ │ │ │ │ │ o1 : RR (of precision 53) │ │ │ │ │ │ i2 : for i from 0 to 1000000 do 223131321321*324234324324; │ │ │ │ │ │ i3 : t2 = cpuTime() │ │ │ │ │ │ -o3 = 357.599866817 │ │ │ +o3 = 381.599535454 │ │ │ │ │ │ o3 : RR (of precision 53) │ │ │ │ │ │ i4 : t2-t1 │ │ │ │ │ │ -o4 = 1.851786633000017 │ │ │ +o4 = 1.074401081000019 │ │ │ │ │ │ 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 = 1768843899 │ │ │ +o1 = 1769301879 │ │ │ │ │ │ i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 ) │ │ │ │ │ │ -o2 = 56.05243177477978 │ │ │ +o2 = 56.06694458324112 │ │ │ │ │ │ o2 : RR (of precision 53) │ │ │ │ │ │ i3 : 12 * (oo - floor oo) │ │ │ │ │ │ -o3 = .6291812973573201 │ │ │ +o3 = .8033349988934901 │ │ │ │ │ │ o3 : RR (of precision 53) │ │ │ │ │ │ i4 : run "date" │ │ │ -Mon Jan 19 17:31:39 UTC 2026 │ │ │ +Sun Jan 25 00:44:39 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 │ │ │ - -- 1.00017s elapsed │ │ │ + -- 1.00133s elapsed │ │ │ │ │ │ o1 = 0 │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ @@ -1,14 +1,14 @@ │ │ │ -- -*- M2-comint -*- hash: 1731106803207298715 │ │ │ │ │ │ i1 : elapsedTiming sleep 1 │ │ │ │ │ │ o1 = 0 │ │ │ - -- 1.00018 seconds │ │ │ + -- 1.00015 seconds │ │ │ │ │ │ o1 : Time │ │ │ │ │ │ i2 : peek oo │ │ │ │ │ │ -o2 = Time{1.00018, 0} │ │ │ +o2 = Time{1.00015, 0} │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ @@ -6,15 +6,15 @@ │ │ │ │ │ │ 3 3 2 3 │ │ │ o2 = ideal (- s - s*t + x - 1, - t - 3t - t + y, - s*t + z) │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ i3 : time leadTerm gens gb I │ │ │ - -- used 0.461542s (cpu); 0.27877s (thread); 0s (gc) │ │ │ + -- used 0.142667s (cpu); 0.142666s (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 0.489181s (cpu); 0.266854s (thread); 0s (gc) │ │ │ + -- used 0.461277s (cpu); 0.209215s (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 0.29671s (cpu); 0.11819s (thread); 0s (gc) │ │ │ + -- used 0.0323307s (cpu); 0.0323384s (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 0.418187s (cpu); 0.235941s (thread); 0s (gc) │ │ │ + -- used 0.1103s (cpu); 0.110306s (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 0.00229141s (cpu); 0.00229176s (thread); 0s (gc) │ │ │ + -- used 0.00190719s (cpu); 0.00191013s (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 0.0542115s (cpu); 0.0542197s (thread); 0s (gc) │ │ │ + -- used 0.037301s (cpu); 0.0373166s (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/M2-10188-0/91-rundir/ │ │ │ + source directory => /tmp/M2-10308-0/91-rundir/ │ │ │ source file => stdio │ │ │ test inputs => MutableList{} │ │ │ │ │ │ i7 : dictionaryPath │ │ │ │ │ │ o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary, │ │ │ ------------------------------------------------------------------------ │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Exists.out │ │ │ @@ -1,20 +1,20 @@ │ │ │ -- -*- M2-comint -*- hash: 7475038936570224899 │ │ │ │ │ │ i1 : fn = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10555-0/0 │ │ │ +o1 = /tmp/M2-11025-0/0 │ │ │ │ │ │ i2 : fileExists fn │ │ │ │ │ │ o2 = false │ │ │ │ │ │ i3 : fn << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10555-0/0 │ │ │ +o3 = /tmp/M2-11025-0/0 │ │ │ │ │ │ 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/M2-12147-0/0 │ │ │ +o1 = /tmp/M2-14267-0/0 │ │ │ │ │ │ o1 : File │ │ │ │ │ │ i2 : fileLength f │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ i3 : close f │ │ │ │ │ │ -o3 = /tmp/M2-12147-0/0 │ │ │ +o3 = /tmp/M2-14267-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : filename = toString f │ │ │ │ │ │ -o4 = /tmp/M2-12147-0/0 │ │ │ +o4 = /tmp/M2-14267-0/0 │ │ │ │ │ │ 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/M2-11372-0/0 │ │ │ +o1 = /tmp/M2-12682-0/0 │ │ │ │ │ │ i2 : f = fn << "hi there" │ │ │ │ │ │ -o2 = /tmp/M2-11372-0/0 │ │ │ +o2 = /tmp/M2-12682-0/0 │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : fileMode f │ │ │ │ │ │ o3 = 420 │ │ │ │ │ │ i4 : close f │ │ │ │ │ │ -o4 = /tmp/M2-11372-0/0 │ │ │ +o4 = /tmp/M2-12682-0/0 │ │ │ │ │ │ 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/M2-10986-0/0 │ │ │ +o1 = /tmp/M2-11896-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-10986-0/0 │ │ │ +o2 = /tmp/M2-11896-0/0 │ │ │ │ │ │ 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/M2-10851-0/0 │ │ │ +o1 = /tmp/M2-11621-0/0 │ │ │ │ │ │ i2 : f = fn << "hi there" │ │ │ │ │ │ -o2 = /tmp/M2-10851-0/0 │ │ │ +o2 = /tmp/M2-11621-0/0 │ │ │ │ │ │ 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/M2-10851-0/0 │ │ │ +o6 = /tmp/M2-11621-0/0 │ │ │ │ │ │ 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/M2-11974-0/0 │ │ │ +o1 = /tmp/M2-13914-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-11974-0/0 │ │ │ +o2 = /tmp/M2-13914-0/0 │ │ │ │ │ │ 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 = 62 │ │ │ +o1 = 52 │ │ │ │ │ │ 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 0x7f43e8412e00 │ │ │ + -- registering gb 0 at 0x7fa4a8edee00 │ │ │ │ │ │ -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3 │ │ │ -- number of monomials = 9 │ │ │ -- #reduction steps = 6 │ │ │ -- #spairs done = 6 │ │ │ -- ncalls = 0 │ │ │ -- nloop = 0 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_get.out │ │ │ @@ -10,11 +10,11 @@ │ │ │ │ │ │ o2 = hi there │ │ │ │ │ │ i3 : removeFile "test-file" │ │ │ │ │ │ i4 : get "!date" │ │ │ │ │ │ -o4 = Mon Jan 19 17:30:51 UTC 2026 │ │ │ +o4 = Sun Jan 25 00:44:01 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/M2-10377-0/0 │ │ │ +o2 = /tmp/M2-10667-0/0 │ │ │ │ │ │ i3 : fn << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10377-0/0 │ │ │ +o3 = /tmp/M2-10667-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : isDirectory fn │ │ │ │ │ │ o4 = false │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ @@ -75,15 +75,15 @@ │ │ │ o17 = false │ │ │ │ │ │ i18 : isPrime(m*m*m1*m1*m2^6) │ │ │ │ │ │ o18 = false │ │ │ │ │ │ i19 : elapsedTime facs = factor(m*m1) │ │ │ - -- 4.51779s elapsed │ │ │ + -- 5.87053s 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 │ │ │ - -- .0563902s elapsed │ │ │ + -- .0586494s elapsed │ │ │ │ │ │ o23 = true │ │ │ │ │ │ i24 : elapsedTime isPseudoprime m3 │ │ │ - -- .000102792s elapsed │ │ │ + -- .000143174s 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/M2-12185-0/0 │ │ │ +o1 = /tmp/M2-14345-0/0 │ │ │ │ │ │ i2 : fn << "hi there" << close │ │ │ │ │ │ -o2 = /tmp/M2-12185-0/0 │ │ │ +o2 = /tmp/M2-14345-0/0 │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : isRegularFile fn │ │ │ │ │ │ o3 = true │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_make__Directory_lp__String_rp.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 5113372159204571746 │ │ │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10719-0/0 │ │ │ +o1 = /tmp/M2-11349-0/0 │ │ │ │ │ │ i2 : makeDirectory (dir|"/a/b/c") │ │ │ │ │ │ -o2 = /tmp/M2-10719-0/0/a/b/c │ │ │ +o2 = /tmp/M2-11349-0/0/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 1.29904s (cpu); 0.825275s (thread); 0s (gc) │ │ │ + -- used 0.885923s (cpu); 0.678937s (thread); 0s (gc) │ │ │ │ │ │ o2 = 514229 │ │ │ │ │ │ i3 : fib = memoize fib │ │ │ │ │ │ o3 = fib │ │ │ │ │ │ o3 : FunctionClosure │ │ │ │ │ │ i4 : time fib 28 │ │ │ - -- used 0.000178845s (cpu); 0.000178345s (thread); 0s (gc) │ │ │ + -- used 8.9328e-05s (cpu); 7.8284e-05s (thread); 0s (gc) │ │ │ │ │ │ o4 = 514229 │ │ │ │ │ │ i5 : time fib 28 │ │ │ - -- used 7.644e-06s (cpu); 7.043e-06s (thread); 0s (gc) │ │ │ + -- used 3.096e-06s (cpu); 2.905e-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 │ │ │ @@ -4,31 +4,31 @@ │ │ │ │ │ │ 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) } │ │ │ + {19 => (codim, BettiTally) } │ │ │ {20 => (truncate, BettiTally, ZZ, ZZ) } │ │ │ - {21 => (truncate, BettiTally, ZZ, InfiniteNumber) } │ │ │ - {22 => (codim, BettiTally) } │ │ │ + {21 => (truncate, BettiTally, InfiniteNumber, ZZ) } │ │ │ + {22 => (truncate, BettiTally, ZZ, InfiniteNumber) } │ │ │ {23 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)} │ │ │ {24 => (dual, BettiTally) } │ │ │ {25 => (^, Ring, BettiTally) } │ │ │ │ │ │ o1 : NumberedVerticalList │ │ │ │ │ │ i2 : methods resolution │ │ │ @@ -61,19 +61,19 @@ │ │ │ {2 => (++, Module, Module) } │ │ │ │ │ │ o4 : NumberedVerticalList │ │ │ │ │ │ i5 : methods( Matrix, Matrix ) │ │ │ │ │ │ o5 = {0 => (diff', Matrix, Matrix) } │ │ │ - {1 => (contract', Matrix, Matrix) } │ │ │ + {1 => (diff, Matrix, Matrix) } │ │ │ {2 => (contract, Matrix, Matrix) } │ │ │ - {3 => (-, Matrix, Matrix) } │ │ │ - {4 => (+, Matrix, Matrix) } │ │ │ - {5 => (diff, Matrix, Matrix) } │ │ │ + {3 => (+, Matrix, Matrix) } │ │ │ + {4 => (-, Matrix, Matrix) } │ │ │ + {5 => (contract', 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 => (remainder, Matrix, Matrix) } │ │ │ - {25 => (%, Matrix, Matrix) } │ │ │ + {24 => (%, Matrix, Matrix) } │ │ │ + {25 => (remainder, Matrix, Matrix) } │ │ │ {26 => (pushout, Matrix, Matrix) } │ │ │ {27 => (solve, Matrix, Matrix) } │ │ │ - {28 => (intersect, Matrix, Matrix, Matrix, Matrix) } │ │ │ - {29 => (tensor, Matrix, Matrix) } │ │ │ - {30 => (intersect, Matrix, Matrix) } │ │ │ - {31 => (pullback, Matrix, Matrix) } │ │ │ + {28 => (pullback, Matrix, Matrix) } │ │ │ + {29 => (intersect, Matrix, Matrix, Matrix, Matrix) } │ │ │ + {30 => (tensor, Matrix, Matrix) } │ │ │ + {31 => (intersect, 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 │ │ │ - -- 1.75782s elapsed │ │ │ + -- 2.40147s 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) │ │ │ - -- .720188s elapsed │ │ │ + -- 1.04931s 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) │ │ │ - -- .0298402s elapsed │ │ │ + -- .0405396s elapsed │ │ │ │ │ │ 0 1 2 3 4 │ │ │ o7 = total: 1 35 140 189 84 │ │ │ 0: 1 . . . . │ │ │ 1: . 35 140 189 84 │ │ │ │ │ │ o7 : BettiTally │ │ │ │ │ │ i8 : I = ideal I_*; │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5) │ │ │ - -- 1.16354s elapsed │ │ │ + -- 1.69433s 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/M2-10738-0/0/ │ │ │ +o1 = /tmp/M2-11388-0/0/ │ │ │ │ │ │ i2 : mkdir p │ │ │ │ │ │ i3 : isDirectory p │ │ │ │ │ │ o3 = true │ │ │ │ │ │ i4 : (fn = p | "foo") << "hi there" << close │ │ │ │ │ │ -o4 = /tmp/M2-10738-0/0/foo │ │ │ +o4 = /tmp/M2-11388-0/0/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/M2-10612-0/0 │ │ │ +o1 = /tmp/M2-11142-0/0 │ │ │ │ │ │ i2 : dst = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-10612-0/1 │ │ │ +o2 = /tmp/M2-11142-0/1 │ │ │ │ │ │ i3 : src << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-10612-0/0 │ │ │ +o3 = /tmp/M2-11142-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : moveFile(src,dst,Verbose=>true) │ │ │ ---moving: /tmp/M2-10612-0/0 -> /tmp/M2-10612-0/1 │ │ │ +--moving: /tmp/M2-11142-0/0 -> /tmp/M2-11142-0/1 │ │ │ │ │ │ i5 : get dst │ │ │ │ │ │ o5 = hi there │ │ │ │ │ │ i6 : bak = moveFile(dst,Verbose=>true) │ │ │ ---backup file created: /tmp/M2-10612-0/1.bak │ │ │ +--backup file created: /tmp/M2-11142-0/1.bak │ │ │ │ │ │ -o6 = /tmp/M2-10612-0/1.bak │ │ │ +o6 = /tmp/M2-11142-0/1.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 │ │ │ - -- .500151s elapsed │ │ │ + -- .500167s elapsed │ │ │ │ │ │ o1 = 0 │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ @@ -5,26 +5,26 @@ │ │ │ o1 = {1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : L = random toList (1..10000); │ │ │ │ │ │ i3 : elapsedTime apply(1..100, n -> sort L); │ │ │ - -- 1.07356s elapsed │ │ │ + -- .755934s elapsed │ │ │ │ │ │ i4 : elapsedTime parallelApply(1..100, n -> sort L); │ │ │ - -- .314344s elapsed │ │ │ + -- .205538s 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) │ │ │ @@ -101,15 +101,15 @@ │ │ │ │ │ │ o21 : Task │ │ │ │ │ │ i22 : addStartTask(F,G) │ │ │ │ │ │ i23 : schedule F │ │ │ │ │ │ -o23 = <> │ │ │ +o23 = <> │ │ │ │ │ │ o23 : Task │ │ │ │ │ │ i24 : taskResult F │ │ │ │ │ │ o24 = result of F │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ @@ -67,15 +67,15 @@ │ │ │ i3 : S = ring I │ │ │ │ │ │ o3 = S │ │ │ │ │ │ o3 : PolynomialRing │ │ │ │ │ │ i4 : elapsedTime minimalBetti I │ │ │ - -- 2.1171s elapsed │ │ │ + -- 2.46489s 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) │ │ │ - -- 1.87462s elapsed │ │ │ + -- 2.50184s 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) │ │ │ - -- 1.85815s elapsed │ │ │ + -- 2.53058s 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) │ │ │ - -- 2.14686s elapsed │ │ │ + -- 2.84364s elapsed │ │ │ │ │ │ 1 35 241 841 1781 2464 2294 1432 576 135 14 │ │ │ o13 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ o13 : Complex │ │ │ @@ -150,15 +150,15 @@ │ │ │ o14 = 1 │ │ │ │ │ │ i15 : I = ideal I_*; │ │ │ │ │ │ o15 : Ideal of S │ │ │ │ │ │ i16 : elapsedTime freeResolution(I, Strategy => Nonminimal) │ │ │ - -- 2.05195s elapsed │ │ │ + -- 2.79999s 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"); │ │ │ - -- 4.98038s elapsed │ │ │ + -- 4.52716s 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"); │ │ │ - -- 6.54385s elapsed │ │ │ + -- 9.18152s 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"); │ │ │ - -- 4.47844s elapsed │ │ │ + -- 4.00098s elapsed │ │ │ │ │ │ 1 108 │ │ │ o26 : Matrix S <-- S │ │ │ │ │ │ i27 : needsPackage "AssociativeAlgebras" │ │ │ │ │ │ o27 = AssociativeAlgebras │ │ │ @@ -233,15 +233,15 @@ │ │ │ o30 = ideal (5a + 2b*c + 3c*b, 3a*c + 5b + 2c*a, 2a*b + 3b*a + 5c ) │ │ │ │ │ │ ZZ │ │ │ o30 : Ideal of ---<|a, b, c|> │ │ │ 101 │ │ │ │ │ │ i31 : elapsedTime NCGB(I, 22); │ │ │ - -- 1.0147s elapsed │ │ │ + -- 1.10887s 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); │ │ │ - -- 1.17142s elapsed │ │ │ + -- 1.77207s 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 0.00152129s (cpu); 1.9947e-05s (thread); 0s (gc) │ │ │ + -- used 0.00273319s (cpu); 1.1568e-05s (thread); 0s (gc) │ │ │ │ │ │ 3 6 9 │ │ │ o28 = 1 - 3T + 3T - T │ │ │ │ │ │ o28 : ZZ[T] │ │ │ │ │ │ i29 : time gens gb I; │ │ │ │ │ │ - -- registering gb 19 at 0x7f2f23197540 │ │ │ + -- registering gb 19 at 0x7fdffa32a540 │ │ │ │ │ │ -- [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 0.0224323s (cpu); 0.0254628s (thread); 0s (gc) │ │ │ + -- -- used 0.0132534s (cpu); 0.0162859s (thread); 0s (gc) │ │ │ │ │ │ 1 11 │ │ │ o29 : Matrix R <-- R │ │ │ │ │ │ i30 : R = QQ[a..d]; │ │ │ │ │ │ i31 : I = ideal random(R^1, R^{3:-3}); │ │ │ │ │ │ - -- registering gb 20 at 0x7f2f23197380 │ │ │ + -- registering gb 20 at 0x7fdffa32a380 │ │ │ │ │ │ -- [gb]number of (nonminimal) gb elements = 0 │ │ │ -- number of monomials = 0 │ │ │ -- #reduction steps = 0 │ │ │ -- #spairs done = 0 │ │ │ -- ncalls = 0 │ │ │ -- nloop = 0 │ │ │ -- nsaved = 0 │ │ │ -- │ │ │ o31 : Ideal of R │ │ │ │ │ │ i32 : time p = poincare I │ │ │ │ │ │ - -- registering gb 21 at 0x7f2f23197000 │ │ │ + -- registering gb 21 at 0x7fdffa32a000 │ │ │ │ │ │ -- [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 0.00800071s (cpu); 0.00954054s (thread); 0s (gc) │ │ │ + -- -- used 0.00805159s (cpu); 0.0069751s (thread); 0s (gc) │ │ │ │ │ │ 3 6 9 │ │ │ o32 = 1 - 3T + 3T - T │ │ │ │ │ │ o32 : ZZ[T] │ │ │ │ │ │ i33 : S = QQ[a..d, MonomialOrder => Eliminate 2] │ │ │ @@ -254,27 +254,27 @@ │ │ │ │ │ │ i36 : gbTrace = 3 │ │ │ │ │ │ o36 = 3 │ │ │ │ │ │ i37 : time gens gb J; │ │ │ │ │ │ - -- registering gb 22 at 0x7f2f23367e00 │ │ │ + -- registering gb 22 at 0x7fdff9e35e00 │ │ │ │ │ │ -- [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 0.0879724s (cpu); 0.0884082s (thread); 0s (gc) │ │ │ + -- -- used 0.0520684s (cpu); 0.0555531s (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/M2-10929-0/0 │ │ │ +o3 = /tmp/M2-11779-0/0 │ │ │ │ │ │ i4 : fn << "hi there" << endl << close │ │ │ │ │ │ -o4 = /tmp/M2-10929-0/0 │ │ │ +o4 = /tmp/M2-11779-0/0 │ │ │ │ │ │ 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/M2-10929-0/0 │ │ │ +o9 = /tmp/M2-11779-0/0 │ │ │ │ │ │ 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/M2-10929-0/0 │ │ │ +o11 = /tmp/M2-11779-0/0 │ │ │ │ │ │ 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 = 10188 │ │ │ +o1 = 10308 │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out │ │ │ @@ -8,36 +8,36 @@ │ │ │ | 1 3 9 27 81 | │ │ │ │ │ │ 4 5 │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i2 : profileSummary │ │ │ │ │ │ -o2 = #run %time position │ │ │ - 1 94.7 ../../m2/matrix1.m2:279:4-282:58 │ │ │ - 1 92.28 ../../m2/matrix1.m2:281:22-281:43 │ │ │ - 1 44.62 ../../m2/matrix1.m2:193:25-193:52 │ │ │ - 1 30.92 ../../m2/matrix1.m2:114:5-156:72 │ │ │ - 1 29.74 ../../m2/matrix1.m2:140:10-155:16 │ │ │ - 1 22.38 ../../m2/matrix1.m2:181:4-181:42 │ │ │ - 1 21.12 ../../m2/matrix1.m2:45:10-49:22 │ │ │ - 1 21.01 ../../m2/set.m2:127:5-127:61 │ │ │ - 1 3.29 ../../m2/matrix1.m2:112:5-112:29 │ │ │ - 1 2.44 ../../m2/matrix1.m2:96:5-109:11 │ │ │ - 1 2.34 ../../m2/matrix1.m2:141:13-141:78 │ │ │ - 1 1.36 ../../m2/matrix1.m2:98:10-98:46 │ │ │ - 1 1.33 ../../m2/matrix1.m2:281:7-281:16 │ │ │ - 1 1.33 ../../m2/matrix1.m2:147:20-147:64 │ │ │ - 1 1.24 ../../m2/matrix1.m2:276:4-277:73 │ │ │ - 1 1.20 ../../m2/matrix1.m2:111:5-111:91 │ │ │ - 1 1.08 ../../m2/matrix1.m2:182:4-184:74 │ │ │ - 1 .74 ../../m2/modules.m2:279:4-279:52 │ │ │ - 20 .51 ../../m2/matrix1.m2:191:14-192:67 │ │ │ - 20 .48 ../../m2/matrix1.m2:47:43-47:71 │ │ │ - 1 .0037s elapsed total │ │ │ +o2 = #run %time position │ │ │ + 1 93.48 ../../m2/matrix1.m2:279:4-282:58 │ │ │ + 1 91.02 ../../m2/matrix1.m2:281:22-281:43 │ │ │ + 1 45.78 ../../m2/matrix1.m2:193:25-193:52 │ │ │ + 1 31.03 ../../m2/matrix1.m2:114:5-156:72 │ │ │ + 1 29.95 ../../m2/matrix1.m2:140:10-155:16 │ │ │ + 1 21.54 ../../m2/matrix1.m2:181:4-181:42 │ │ │ + 1 20.7 ../../m2/matrix1.m2:45:10-49:22 │ │ │ + 1 20.23 ../../m2/set.m2:127:5-127:61 │ │ │ + 1 3.31 ../../m2/matrix1.m2:112:5-112:29 │ │ │ + 1 2.72 ../../m2/matrix1.m2:96:5-109:11 │ │ │ + 1 2.44 ../../m2/matrix1.m2:141:13-141:78 │ │ │ + 1 1.6 ../../m2/matrix1.m2:111:5-111:91 │ │ │ + 1 1.59 ../../m2/matrix1.m2:147:20-147:64 │ │ │ + 1 1.52 ../../m2/matrix1.m2:281:7-281:16 │ │ │ + 1 1.39 ../../m2/matrix1.m2:98:10-98:46 │ │ │ + 1 1.34 ../../m2/matrix1.m2:276:4-277:73 │ │ │ + 20 1.21 ../../m2/matrix1.m2:191:14-192:67 │ │ │ + 1 1.04 ../../m2/matrix1.m2:182:4-184:74 │ │ │ + 1 .90 ../../m2/modules.m2:279:4-279:52 │ │ │ + 20 .8 ../../m2/matrix1.m2:47:43-47:71 │ │ │ + 1 .004s elapsed total │ │ │ │ │ │ i3 : coverageSummary │ │ │ │ │ │ o3 = covered lines: │ │ │ ../../m2/lists.m2:145:24-145:32 │ │ │ ../../m2/lists.m2:145:34-145:58 │ │ │ ../../m2/matrix.m2:12:5-12:35 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/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 0.168784s (cpu); 0.136714s (thread); 0s (gc) │ │ │ + -- used 0.275836s (cpu); 0.110111s (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 0.717081s (cpu); 0.414694s (thread); 0s (gc) │ │ │ + -- used 0.834012s (cpu); 0.328562s (thread); 0s (gc) │ │ │ │ │ │ o10 = ideal (x - 27x , x - 16x , x - 9x , x + 44x , x - 52x ) │ │ │ 4 5 3 5 2 5 1 5 0 5 │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ i11 : p=10007,kk=ZZ/p,R=kk[x_0..x_2] │ │ │ @@ -58,12 +58,12 @@ │ │ │ │ │ │ i14 : I=ideal random(n,R); │ │ │ │ │ │ o14 : Ideal of R │ │ │ │ │ │ i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c); │ │ │ #select(degs,d->d==1))),f->f>0)) │ │ │ - -- used 3.7226s (cpu); 2.08535s (thread); 0s (gc) │ │ │ + -- used 4.74283s (cpu); 2.23073s (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/M2-11562-0/0 │ │ │ +o1 = /tmp/M2-13072-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-11562-0/0 │ │ │ +o2 = /tmp/M2-13072-0/0 │ │ │ │ │ │ i3 : (fn = dir | "/" | "foo") << "hi there" << close │ │ │ │ │ │ -o3 = /tmp/M2-11562-0/0/foo │ │ │ +o3 = /tmp/M2-13072-0/0/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/M2-11104-0/0 │ │ │ +o1 = /tmp/M2-12134-0/0 │ │ │ │ │ │ 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/M2-11104-0/0 │ │ │ +o2 = /tmp/M2-12134-0/0 │ │ │ │ │ │ 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/M2-11104-0/0 │ │ │ +o7 = /tmp/M2-12134-0/0 │ │ │ │ │ │ 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/M2-11803-0/0 │ │ │ +o1 = /tmp/M2-13553-0/0 │ │ │ │ │ │ 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/M2-10188-0/86-rundir/ │ │ │ +o1 = /tmp/M2-10308-0/86-rundir/ │ │ │ │ │ │ i2 : p = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-11822-0/0 │ │ │ +o2 = /tmp/M2-13592-0/0 │ │ │ │ │ │ i3 : q = temporaryFileName() │ │ │ │ │ │ -o3 = /tmp/M2-11822-0/1 │ │ │ +o3 = /tmp/M2-13592-0/1 │ │ │ │ │ │ i4 : symlinkFile(p,q) │ │ │ │ │ │ i5 : p << close │ │ │ │ │ │ -o5 = /tmp/M2-11822-0/0 │ │ │ +o5 = /tmp/M2-13592-0/0 │ │ │ │ │ │ o5 : File │ │ │ │ │ │ i6 : readlink q │ │ │ │ │ │ -o6 = /tmp/M2-11822-0/0 │ │ │ +o6 = /tmp/M2-13592-0/0 │ │ │ │ │ │ i7 : realpath q │ │ │ │ │ │ -o7 = /tmp/M2-11822-0/0 │ │ │ +o7 = /tmp/M2-13592-0/0 │ │ │ │ │ │ i8 : removeFile p │ │ │ │ │ │ i9 : removeFile q │ │ │ │ │ │ i10 : realpath "" │ │ │ │ │ │ -o10 = /tmp/M2-10188-0/86-rundir/ │ │ │ +o10 = /tmp/M2-10308-0/86-rundir/ │ │ │ │ │ │ i11 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 1729384374372662693 │ │ │ │ │ │ i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --")) │ │ │ │ │ │ i2 : collectGarbage() │ │ │ ---finalization: (1)[5]: -- finalizing sequence #6 -- │ │ │ ---finalization: (2)[1]: -- finalizing sequence #2 -- │ │ │ ---finalization: (3)[8]: -- finalizing sequence #9 -- │ │ │ ---finalization: (4)[3]: -- finalizing sequence #4 -- │ │ │ ---finalization: (5)[2]: -- finalizing sequence #3 -- │ │ │ ---finalization: (6)[4]: -- finalizing sequence #5 -- │ │ │ ---finalization: (7)[6]: -- finalizing sequence #7 -- │ │ │ ---finalization: (8)[0]: -- finalizing sequence #1 -- │ │ │ ---finalization: (9)[7]: -- finalizing sequence #8 -- │ │ │ +--finalization: (1)[7]: -- finalizing sequence #8 -- │ │ │ +--finalization: (2)[2]: -- finalizing sequence #3 -- │ │ │ +--finalization: (3)[6]: -- finalizing sequence #7 -- │ │ │ +--finalization: (4)[0]: -- finalizing sequence #1 -- │ │ │ +--finalization: (5)[5]: -- finalizing sequence #6 -- │ │ │ +--finalization: (6)[3]: -- finalizing sequence #4 -- │ │ │ +--finalization: (7)[8]: -- finalizing sequence #9 -- │ │ │ +--finalization: (8)[1]: -- finalizing sequence #2 -- │ │ │ +--finalization: (9)[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/M2-10776-0/0 │ │ │ +o1 = /tmp/M2-11466-0/0 │ │ │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ -o2 = /tmp/M2-10776-0/0 │ │ │ +o2 = /tmp/M2-11466-0/0 │ │ │ │ │ │ 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/M2-10280-0/0 │ │ │ +o1 = /tmp/M2-10470-0/0 │ │ │ │ │ │ i2 : rootPath | fn │ │ │ │ │ │ -o2 = /tmp/M2-10280-0/0 │ │ │ +o2 = /tmp/M2-10470-0/0 │ │ │ │ │ │ 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/M2-11505-0/0 │ │ │ +o1 = /tmp/M2-12955-0/0 │ │ │ │ │ │ i2 : rootURI | fn │ │ │ │ │ │ -o2 = file:///tmp/M2-11505-0/0 │ │ │ +o2 = file:///tmp/M2-12955-0/0 │ │ │ │ │ │ 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/M2-11353-0/0 │ │ │ +o5 = /tmp/M2-12643-0/0 │ │ │ │ │ │ i6 : f << toString (p,m,M) << close │ │ │ │ │ │ -o6 = /tmp/M2-11353-0/0 │ │ │ +o6 = /tmp/M2-12643-0/0 │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : get f │ │ │ │ │ │ o7 = (x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix {{x^2, │ │ │ x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}}) │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_serial__Number.out │ │ │ @@ -1,15 +1,15 @@ │ │ │ -- -*- M2-comint -*- hash: 5271760183816554957 │ │ │ │ │ │ i1 : serialNumber asdf │ │ │ │ │ │ -o1 = 1426273 │ │ │ +o1 = 1526273 │ │ │ │ │ │ i2 : serialNumber foo │ │ │ │ │ │ -o2 = 1426275 │ │ │ +o2 = 1526275 │ │ │ │ │ │ i3 : serialNumber ZZ │ │ │ │ │ │ o3 = 1000050 │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out │ │ │ @@ -189,18 +189,18 @@ │ │ │ o25 = 40 │ │ │ │ │ │ i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A; │ │ │ │ │ │ i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B; │ │ │ │ │ │ i30 : time X = solve(A,B); │ │ │ - -- used 0.000215955s (cpu); 0.000209113s (thread); 0s (gc) │ │ │ + -- used 0.000256235s (cpu); 0.000227845s (thread); 0s (gc) │ │ │ │ │ │ i31 : time X = solve(A,B, MaximalRank=>true); │ │ │ - -- used 0.000155772s (cpu); 0.000155762s (thread); 0s (gc) │ │ │ + -- used 0.000121023s (cpu); 0.000120607s (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 0.233s (cpu); 0.233s (thread); 0s (gc) │ │ │ + -- used 0.143751s (cpu); 0.143759s (thread); 0s (gc) │ │ │ │ │ │ i39 : time X = solve(A,B, MaximalRank=>true); │ │ │ - -- used 0.492751s (cpu); 0.295313s (thread); 0s (gc) │ │ │ + -- used 0.144679s (cpu); 0.144699s (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/M2-11144-0/0/ │ │ │ +o1 = /tmp/M2-12214-0/0/ │ │ │ │ │ │ i2 : dst = temporaryFileName() | "/" │ │ │ │ │ │ -o2 = /tmp/M2-11144-0/1/ │ │ │ +o2 = /tmp/M2-12214-0/1/ │ │ │ │ │ │ i3 : makeDirectory (src|"a/") │ │ │ │ │ │ -o3 = /tmp/M2-11144-0/0/a/ │ │ │ +o3 = /tmp/M2-12214-0/0/a/ │ │ │ │ │ │ i4 : makeDirectory (src|"b/") │ │ │ │ │ │ -o4 = /tmp/M2-11144-0/0/b/ │ │ │ +o4 = /tmp/M2-12214-0/0/b/ │ │ │ │ │ │ i5 : makeDirectory (src|"b/c/") │ │ │ │ │ │ -o5 = /tmp/M2-11144-0/0/b/c/ │ │ │ +o5 = /tmp/M2-12214-0/0/b/c/ │ │ │ │ │ │ i6 : src|"a/f" << "hi there" << close │ │ │ │ │ │ -o6 = /tmp/M2-11144-0/0/a/f │ │ │ +o6 = /tmp/M2-12214-0/0/a/f │ │ │ │ │ │ o6 : File │ │ │ │ │ │ i7 : src|"a/g" << "hi there" << close │ │ │ │ │ │ -o7 = /tmp/M2-11144-0/0/a/g │ │ │ +o7 = /tmp/M2-12214-0/0/a/g │ │ │ │ │ │ o7 : File │ │ │ │ │ │ i8 : src|"b/c/g" << "ho there" << close │ │ │ │ │ │ -o8 = /tmp/M2-11144-0/0/b/c/g │ │ │ +o8 = /tmp/M2-12214-0/0/b/c/g │ │ │ │ │ │ o8 : File │ │ │ │ │ │ i9 : symlinkDirectory(src,dst,Verbose=>true) │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11144-0/1/b/c/g │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11144-0/1/a/g │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11144-0/1/a/f │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12214-0/1/a/g │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12214-0/1/a/f │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12214-0/1/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/M2-11144-0/1/b/c/g │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11144-0/1/a/g │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11144-0/1/a/f │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12214-0/1/a/g │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12214-0/1/a/f │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12214-0/1/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/M2-11201-0/0 │ │ │ +o1 = /tmp/M2-12331-0/0 │ │ │ │ │ │ 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/M2-12166-0/0.tex │ │ │ +o1 = /tmp/M2-14306-0/0.tex │ │ │ │ │ │ i2 : temporaryFileName () | ".html" │ │ │ │ │ │ -o2 = /tmp/M2-12166-0/1.html │ │ │ +o2 = /tmp/M2-14306-0/1.html │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out │ │ │ @@ -1,8 +1,8 @@ │ │ │ -- -*- M2-comint -*- hash: 1332435500723 │ │ │ │ │ │ i1 : time 3^30 │ │ │ - -- used 1.6912e-05s (cpu); 9.258e-06s (thread); 0s (gc) │ │ │ + -- used 2.6805e-05s (cpu); 6.149e-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 │ │ │ - -- .000014818 seconds │ │ │ + -- .000024213 seconds │ │ │ │ │ │ o1 : Time │ │ │ │ │ │ i2 : peek oo │ │ │ │ │ │ -o2 = Time{.000014818, 205891132094649} │ │ │ +o2 = Time{.000024213, 205891132094649} │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out │ │ │ @@ -34,15 +34,15 @@ │ │ │ "memtailor version" => 1.1 │ │ │ "mpfi version" => 1.5.4 │ │ │ "mpfr version" => 4.2.2 │ │ │ "mpsolve version" => 3.2.2 │ │ │ "mysql version" => not present │ │ │ "normaliz version" => 3.11.0 │ │ │ "ntl version" => 11.5.1 │ │ │ - "operating system release" => 6.12.63+deb13-amd64 │ │ │ + "operating system release" => 6.12.63+deb13-cloud-amd64 │ │ │ "operating system" => Linux │ │ │ "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings PruneComplex BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomPlaneCurves RandomSpaceCurves RandomGenus14Curves RandomCanonicalCurves RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing OldPolyhedra OldToricVectorBundles K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization NonminimalComplexes CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC MatrixFactorizations PathSignatures │ │ │ "pointer size" => 8 │ │ │ "python version" => 3.13.11 │ │ │ "readline version" => 8.3 │ │ │ "scscp version" => not present │ │ │ "tbb version" => 2022.3 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Command.html │ │ │ @@ -84,15 +84,15 @@ │ │ │ │ │ │ 
    i3 : (c = Command "date";)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : c
│ │ │ -Mon Jan 19 17:30:24 UTC 2026
│ │ │ +Sun Jan 25 00:43:41 UTC 2026
│ │ │  
│ │ │  o4 = 0
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ in a file), then it gets executed with empty argument list. │ │ │ │ i1 : (f = Command ( () -> 2^30 );) │ │ │ │ i2 : f │ │ │ │ │ │ │ │ o2 = 1073741824 │ │ │ │ i3 : (c = Command "date";) │ │ │ │ i4 : c │ │ │ │ -Mon Jan 19 17:30:24 UTC 2026 │ │ │ │ +Sun Jan 25 00:43:41 UTC 2026 │ │ │ │ │ │ │ │ o4 = 0 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_u_n -- run an external command │ │ │ │ * _A_f_t_e_r_E_v_a_l -- top level method applied after evaluation │ │ │ │ ********** MMeetthhooddss tthhaatt uussee aa ccoommmmaanndd:: ********** │ │ │ │ * code(Command) -- see _c_o_d_e -- display source code │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Database.html │ │ │ @@ -52,22 +52,22 @@ │ │ │
    │ │ │

    Description

    │ │ │ A database file is just like a hash table, except both the keys and values have to be strings. In this example we create a database file, store a few entries, remove an entry with remove, close the file, and then remove the file. │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-11638-0/0.dbm
    │ │ │ +o1 = /tmp/M2-13228-0/0.dbm │ │ │
    │ │ │ 
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-11638-0/0.dbm
│ │ │ +o2 = /tmp/M2-13228-0/0.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/M2-11638-0/0.dbm
│ │ │ │ +o1 = /tmp/M2-13228-0/0.dbm
│ │ │ │  i2 : x = openDatabaseOut filename
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11638-0/0.dbm
│ │ │ │ +o2 = /tmp/M2-13228-0/0.dbm
│ │ │ │  
│ │ │ │  o2 : Database
│ │ │ │  i3 : x#"first" = "hi there"
│ │ │ │  
│ │ │ │  o3 = hi there
│ │ │ │  i4 : x#"first"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html
│ │ │ @@ -53,33 +53,33 @@
│ │ │          
    Description
    
│ │ │          
    Macaulay2 uses the Hans Boehm garbage collector to reclaim unused memory.  The function GCstats provides information about its status, such as the total number of bytes allocated, the current heap size, the number of garbage collections done, the number of threads used in each collection, the total cpu time spent in garbage collection, etc.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 42970092554        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 43074037370        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 206467072
│ │ │ -               "numGCs" => 795
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 225644544
│ │ │ +               "numGCs" => 783
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
    
│ │ │              
    
│ │ │          
    The value returned is a hash table, from which individual bits of information can be easily extracted, as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 206467072
    
│ │ │ +o2 = 225644544
│ │ │              
    
│ │ │          
    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" => 42970092554        }
│ │ │ │ +o1 = HashTable{"bytesAlloc" => 43074037370        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 206467072
│ │ │ │ -               "numGCs" => 795
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 225644544
│ │ │ │ +               "numGCs" => 783
│ │ │ │ +               "numGCThreads" => 16
│ │ │ │  
│ │ │ │  o1 : HashTable
│ │ │ │  The value returned is a hash table, from which individual bits of information
│ │ │ │  can be easily extracted, as follows.
│ │ │ │  i2 : s#"heapSize"
│ │ │ │  
│ │ │ │ -o2 = 206467072
│ │ │ │ +o2 = 225644544
│ │ │ │  Any entries whose keys are all upper case give the values of environment
│ │ │ │  variables affecting the operation of the garbage collector that have been
│ │ │ │  specified by the user.
│ │ │ │  For further information about the individual items in the table, we refer the
│ │ │ │  user to the source code and documentation of the garbage collector.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Minimal__Generators.html
│ │ │ @@ -128,23 +128,23 @@
│ │ │  
│ │ │  o7 : Ideal of R
    │ │ │
    │ │ │ 
    i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00907699s (cpu); 0.00906961s (thread); 0s (gc)
│ │ │ + -- used 0.00592172s (cpu); 0.00591156s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
    │ │ │
    │ │ │ 
    i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0809478s (cpu); 0.0809554s (thread); 0s (gc)
│ │ │ + -- used 0.0509569s (cpu); 0.0509699s (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 0.00907699s (cpu); 0.00906961s (thread); 0s (gc)
│ │ │ │ + -- used 0.00592172s (cpu); 0.00591156s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ │ - -- used 0.0809478s (cpu); 0.0809554s (thread); 0s (gc)
│ │ │ │ + -- used 0.0509569s (cpu); 0.0509699s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : numgens J
│ │ │ │  
│ │ │ │  o10 = 1067
│ │ │ │  i11 : numgens K
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.html
│ │ │ @@ -68,21 +68,21 @@
│ │ │  o1 : Matrix RR      <-- RR
│ │ │                53          53
    │ │ │
    │ │ │ 
    i2 : time SVD(M);
│ │ │ - -- used 0.0247234s (cpu); 0.0247216s (thread); 0s (gc)
    │ │ │ + -- used 0.0402253s (cpu); 0.0402245s (thread); 0s (gc) │ │ │
    │ │ │ 
    i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0245265s (cpu); 0.0245324s (thread); 0s (gc)
    │ │ │ + -- used 0.039368s (cpu); 0.0393845s (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 0.0247234s (cpu); 0.0247216s (thread); 0s (gc) │ │ │ │ + -- used 0.0402253s (cpu); 0.0402245s (thread); 0s (gc) │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ │ - -- used 0.0245265s (cpu); 0.0245324s (thread); 0s (gc) │ │ │ │ + -- used 0.039368s (cpu); 0.0393845s (thread); 0s (gc) │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd DDiivviiddeeCCoonnqquueerr:: ********** │ │ │ │ * _S_V_D_(_._._._,_D_i_v_i_d_e_C_o_n_q_u_e_r_=_>_._._._) -- whether to use the LAPACK divide and │ │ │ │ conquer SVD algorithm │ │ │ │ ********** FFuurrtthheerr iinnffoorrmmaattiioonn ********** │ │ │ │ * Default value: _t_r_u_e │ │ │ │ * Function: _S_V_D -- singular value decomposition of a matrix │ │ │ │ * Option key: _D_i_v_i_d_e_C_o_n_q_u_e_r -- an optional argument │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_a_spfirst_sp__Macaulay2_spsession.html │ │ │ @@ -826,15 +826,15 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i59 : time C = resolution M
│ │ │ - -- used 0.00196552s (cpu); 0.00195841s (thread); 0s (gc)
│ │ │ + -- used 0.00218428s (cpu); 0.00217334s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R  <-- 0
│ │ │                                              
│ │ │        0      1      2       3       4      5
│ │ │  
│ │ │  o59 : ChainComplex
    │ │ │ ├── html2text {} │ │ │ │ @@ -390,15 +390,15 @@ │ │ │ │ | c f i l o r | │ │ │ │ │ │ │ │ 3 │ │ │ │ o58 : R-module, quotient of R │ │ │ │ We may use _r_e_s_o_l_u_t_i_o_n to produce a projective resolution of it, and _t_i_m_e to │ │ │ │ report the time required. │ │ │ │ i59 : time C = resolution M │ │ │ │ - -- used 0.00196552s (cpu); 0.00195841s (thread); 0s (gc) │ │ │ │ + -- used 0.00218428s (cpu); 0.00217334s (thread); 0s (gc) │ │ │ │ │ │ │ │ 3 6 15 18 6 │ │ │ │ o59 = R <-- R <-- R <-- R <-- R <-- 0 │ │ │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ │ │ │ │ │ o59 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ o4 = "hi there" │ │ │
    │ │ │ 
    i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
    │ │ │ +o5 = true │ │ │
    │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -23,13 +23,13 @@ │ │ │ │ │ │ │ │ o3 = false │ │ │ │ i4 : peek read f │ │ │ │ │ │ │ │ o4 = "hi there" │ │ │ │ i5 : atEndOfFile f │ │ │ │ │ │ │ │ -o5 = false │ │ │ │ +o5 = true │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _a_t_E_n_d_O_f_F_i_l_e_(_F_i_l_e_) -- test for end of file │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_files.m2:374:0. │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_benchmark.html │ │ │ @@ -68,15 +68,15 @@ │ │ │
    │ │ │

    Description

    │ │ │ Produces an accurate timing for the code contained in the string s. The value returned is the number of seconds. │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .0002898228374663042
│ │ │ +o1 = .0003637207220832036
│ │ │  
│ │ │  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 = .0002898228374663042 │ │ │ │ +o1 = .0003637207220832036 │ │ │ │ │ │ │ │ o1 : RR (of precision 53) │ │ │ │ The snippet of code provided will be run enough times to register meaningfully │ │ │ │ on the clock, and the garbage collector will be called beforehand. │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _b_e_n_c_h_m_a_r_k is a _f_u_n_c_t_i_o_n_ _c_l_o_s_u_r_e. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_caching_spcomputation_spresults.html │ │ │ @@ -69,23 +69,23 @@ │ │ │ 
    i3 : M = coker vars R;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .00625565s elapsed
│ │ │ + -- .00454321s elapsed
│ │ │  
│ │ │  o4 = 3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime pdim' M
│ │ │ - -- .000001804s elapsed
│ │ │ + -- .000002686s 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'
│ │ │ │ - -- .00625565s elapsed
│ │ │ │ + -- .00454321s elapsed
│ │ │ │  
│ │ │ │  o4 = 3
│ │ │ │  i5 : elapsedTime pdim' M
│ │ │ │ - -- .000001804s elapsed
│ │ │ │ + -- .000002686s elapsed
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : peek M.cache
│ │ │ │  
│ │ │ │  o6 = CacheTable{cache => MutableHashTable{}
│ │ │ │  }
│ │ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  o4 : Task
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : n
│ │ │  
│ │ │ -o5 = 710974
    │ │ │ +o5 = 1095396 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    │ │ │ @@ -127,15 +127,15 @@ │ │ │ o7 : Task │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : n
│ │ │  
│ │ │ -o8 = 1452916
    │ │ │ +o8 = 2221986 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    │ │ │ @@ -145,15 +145,15 @@ │ │ │ │ │ │ 
    i10 : cancelTask t
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : sleep 2
│ │ │ -stdio:2:39:(3):[1]: error: interrupted
│ │ │ +stdio:2:25:(3):[1]: error: interrupted
│ │ │  
│ │ │  o11 = 0
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : t
│ │ │ @@ -163,29 +163,29 @@
│ │ │  o12 : Task
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : n
│ │ │  
│ │ │ -o13 = 1453128
    │ │ │ +o13 = 2222213 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : n
│ │ │  
│ │ │ -o15 = 1453128
    │ │ │ +o15 = 2222213 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : isReady t
│ │ │  
│ │ │  o16 = false
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,48 +28,48 @@ │ │ │ │ i4 : t │ │ │ │ │ │ │ │ o4 = <> │ │ │ │ │ │ │ │ o4 : Task │ │ │ │ i5 : n │ │ │ │ │ │ │ │ -o5 = 710974 │ │ │ │ +o5 = 1095396 │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ │ │ o6 = 0 │ │ │ │ i7 : t │ │ │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ │ │ o7 : Task │ │ │ │ i8 : n │ │ │ │ │ │ │ │ -o8 = 1452916 │ │ │ │ +o8 = 2221986 │ │ │ │ i9 : isReady t │ │ │ │ │ │ │ │ o9 = false │ │ │ │ i10 : cancelTask t │ │ │ │ i11 : sleep 2 │ │ │ │ -stdio:2:39:(3):[1]: error: interrupted │ │ │ │ +stdio:2:25:(3):[1]: error: interrupted │ │ │ │ │ │ │ │ o11 = 0 │ │ │ │ i12 : t │ │ │ │ │ │ │ │ o12 = <> │ │ │ │ │ │ │ │ o12 : Task │ │ │ │ i13 : n │ │ │ │ │ │ │ │ -o13 = 1453128 │ │ │ │ +o13 = 2222213 │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ │ │ o14 = 0 │ │ │ │ i15 : n │ │ │ │ │ │ │ │ -o15 = 1453128 │ │ │ │ +o15 = 2222213 │ │ │ │ i16 : isReady t │ │ │ │ │ │ │ │ o16 = false │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _c_a_n_c_e_l_T_a_s_k_(_T_a_s_k_) -- stop a task │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_change__Directory.html │ │ │ @@ -71,36 +71,36 @@ │ │ │

    Change the current working directory to dir.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10460-0/0
    │ │ │ +o1 = /tmp/M2-10830-0/0 │ │ │
    │ │ │ 
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10460-0/0
    │ │ │ +o2 = /tmp/M2-10830-0/0 │ │ │
    │ │ │ 
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-10460-0/0/
    │ │ │ +o3 = /tmp/M2-10830-0/0/ │ │ │
    │ │ │ 
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-10460-0/0/
    │ │ │ +o4 = /tmp/M2-10830-0/0/ │ │ │
    │ │ │
    │ │ │

    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/M2-10460-0/0 │ │ │ │ +o1 = /tmp/M2-10830-0/0 │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ │ │ -o2 = /tmp/M2-10460-0/0 │ │ │ │ +o2 = /tmp/M2-10830-0/0 │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ │ │ -o3 = /tmp/M2-10460-0/0/ │ │ │ │ +o3 = /tmp/M2-10830-0/0/ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ │ │ -o4 = /tmp/M2-10460-0/0/ │ │ │ │ +o4 = /tmp/M2-10830-0/0/ │ │ │ │ If dir is omitted, then the current working directory is changed to the user's │ │ │ │ home directory. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_u_r_r_e_n_t_D_i_r_e_c_t_o_r_y -- current working directory │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_h_a_n_g_e_D_i_r_e_c_t_o_r_y is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_check.html │ │ │ @@ -95,40 +95,40 @@ │ │ │ o1 : Package │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .153064s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .128193s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .150274s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .147265s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .127121s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .123325s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .140205s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .134544s elapsed │ │ │
    │ │ │ 
    i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .151078s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .150374s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .139643s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .143099s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ @@ -140,15 +140,15 @@ │ │ │ │ │ │ o6 : TestInput │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .147771s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .142709s elapsed │ │ │
    │ │ │ 
    i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ @@ -156,16 +156,16 @@
│ │ │  
│ │ │  o8 : NumberedVerticalList
    │ │ │
    │ │ │ 
    i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .137448s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .148236s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .13571s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .132292s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ @@ -178,15 +178,15 @@ │ │ │ │ │ │ o10 : NumberedVerticalList │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .138202s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .138938s 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") -- .153064s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .128193s elapsed │ │ │ │ i3 : check FirstPackage │ │ │ │ - -- capturing check(0, "FirstPackage") -- .150274s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .147265s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .127121s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .123325s elapsed │ │ │ │ Alternatively, if the package is installed somewhere accessible, one can do the │ │ │ │ following. │ │ │ │ i4 : check_1 "FirstPackage" │ │ │ │ - -- capturing check(1, "FirstPackage") -- .140205s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .134544s elapsed │ │ │ │ i5 : check "FirstPackage" │ │ │ │ - -- capturing check(0, "FirstPackage") -- .151078s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .150374s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .139643s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .143099s 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") -- .147771s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .142709s 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") -- .137448s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .148236s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .13571s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .132292s 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") -- .138202s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .138938s elapsed │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Currently, if the package was only partially loaded because the documentation │ │ │ │ was obtainable from a database (see _b_e_g_i_n_D_o_c_u_m_e_n_t_a_t_i_o_n), then the package will │ │ │ │ be reloaded, this time completely, to ensure that all tests are considered; │ │ │ │ this may affect user objects of types declared by the package, as they may be │ │ │ │ not usable by the new instance of the package. In a future version, either the │ │ │ │ tests and the documentation will both be cached, or neither will. │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_communicating_spwith_spprograms.html │ │ │ @@ -50,15 +50,15 @@ │ │ │
    │ │ │

    communicating with programs

    │ │ │
    │ │ │ The most naive way to interact with another program is simply to run it, let it communicate directly with the user, and wait for it to finish. This is done with the run command. │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.63+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1 (2025-12-30) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.63+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1 (2025-12-30) 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. │ │ │ │ │ │ @@ -74,15 +74,15 @@ │ │ │ │ │ │
    │ │ │ More often, one wants to write Macaulay2 code to obtain and manipulate the output from the other program. If the program requires no input data, then we can use get with a file name whose first character is an exclamation point. In the following example, we also peek at the string to see whether it includes a newline character. │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : peek get "!uname -a"
│ │ │  
│ │ │ -o3 = "Linux sbuild 6.12.63+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +o3 = "Linux sbuild 6.12.63+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │       6.12.63-1 (2025-12-30) 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 {} │ │ │ │ @@ -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 3.01286s (cpu); 1.87175s (thread); 0s (gc) │ │ │ │ + -- used 3.98532s (cpu); 2.24454s (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 3.35764s (cpu); 2.18666s (thread); 0s (gc) │ │ │ │ + -- used 4.35559s (cpu); 2.40335s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 : MultirationalMap (automorphism of PP^8) │ │ │ │ i12 : g||W │ │ │ │ │ │ │ │ o12 = multi-rational map consisting of one single rational map │ │ │ │ source variety: 6-dimensional subvariety of PP^8 cut out by 2 │ │ │ │ hypersurfaces of degrees 1^1 2^1 │ │ │ │ @@ -144,15 +144,15 @@ │ │ │ │ i15 : Z = projectiveVariety pfaffians(4,A); │ │ │ │ │ │ │ │ o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9 │ │ │ │ i16 : ? Z │ │ │ │ │ │ │ │ o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2 │ │ │ │ i17 : time h = Z ===> GG_K(1,4) │ │ │ │ - -- used 7.61692s (cpu); 4.97834s (thread); 0s (gc) │ │ │ │ + -- used 7.17956s (cpu); 4.95993s (thread); 0s (gc) │ │ │ │ │ │ │ │ o17 = h │ │ │ │ │ │ │ │ o17 : MultirationalMap (isomorphism from PP^9 to PP^9) │ │ │ │ i18 : h || GG_K(1,4) │ │ │ │ │ │ │ │ o18 = multi-rational map consisting of one single rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html │ │ │ @@ -89,15 +89,15 @@ │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -25,24 +25,24 @@ │ │ │ │ │ │ │ │ o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x │ │ │ │ PP^7 to PP^7 x PP^7) │ │ │ │ i4 : Z = source Phi; │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7 │ │ │ │ i5 : time Phi Z; │ │ │ │ - -- used 0.0946816s (cpu); 0.0919773s (thread); 0s (gc) │ │ │ │ + -- used 0.134079s (cpu); 0.116376s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7 │ │ │ │ i6 : dim oo, degree oo, degrees oo │ │ │ │ │ │ │ │ o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)}) │ │ │ │ │ │ │ │ o6 : Sequence │ │ │ │ i7 : time Phi (point Z + point Z + point Z) │ │ │ │ - -- used 1.75371s (cpu); 1.28035s (thread); 0s (gc) │ │ │ │ + -- used 2.45168s (cpu); 1.52315s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of │ │ │ │ multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 │ │ │ │ │ │ │ │ o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7 │ │ │ │ i8 : dim oo, degree oo, degrees oo │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_cm__Option_rp.html │ │ │ @@ -93,31 +93,31 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ ├── 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 6.12.63+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1 (2025- │ │ │ │ -12-30) x86_64 GNU/Linux │ │ │ │ +Linux sbuild 6.12.63+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.63-1 │ │ │ │ +(2025-12-30) 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 │ │ │ │ @@ -26,15 +26,15 @@ │ │ │ │ More often, one wants to write Macaulay2 code to obtain and manipulate the │ │ │ │ output from the other program. If the program requires no input data, then we │ │ │ │ can use _g_e_t with a file name whose first character is an exclamation point. In │ │ │ │ the following example, we also peek at the string to see whether it includes a │ │ │ │ newline character. │ │ │ │ i3 : peek get "!uname -a" │ │ │ │ │ │ │ │ -o3 = "Linux sbuild 6.12.63+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ │ +o3 = "Linux sbuild 6.12.63+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ │ 6.12.63-1 (2025-12-30) 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 │ │ │ @@ -269,15 +269,15 @@ │ │ │ 1277 │ │ │
    │ │ │ 
    i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7f5cd0c81540
│ │ │ +   -- registering gb 5 at 0x7fda3bfe9540
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -373,15 +373,15 @@
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
    │ │ │
    │ │ │ 
    i33 : time betti gb f
│ │ │ - -- used 0.171852s (cpu); 0.169834s (thread); 0s (gc)
│ │ │ + -- used 0.224047s (cpu); 0.223729s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -417,15 +417,15 @@
│ │ │  
│ │ │  o35 : ZZ[T]
    │ │ │
    │ │ │ 
    i36 : time betti gb f
│ │ │ - -- used 0.0037732s (cpu); 0.00284882s (thread); 0s (gc)
│ │ │ + -- used 0.00288335s (cpu); 0.00322848s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -140,15 +140,15 @@
│ │ │ │  o23 = ideal (x*y - z , y  - w )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o23 : Ideal of ----[x..z, w]
│ │ │ │                 1277
│ │ │ │  i24 : gb I
│ │ │ │  
│ │ │ │ -   -- registering gb 5 at 0x7f5cd0c81540
│ │ │ │ +   -- registering gb 5 at 0x7fda3bfe9540
│ │ │ │  
│ │ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │ │     -- number of monomials                = 8
│ │ │ │     -- #reduction steps = 2
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │ @@ -213,15 +213,15 @@
│ │ │ │  
│ │ │ │  o31 : ZZ[T]
│ │ │ │  i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │ │  
│ │ │ │                1      4
│ │ │ │  o32 : Matrix R  <-- R
│ │ │ │  i33 : time betti gb f
│ │ │ │ - -- used 0.171852s (cpu); 0.169834s (thread); 0s (gc)
│ │ │ │ + -- used 0.224047s (cpu); 0.223729s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o33 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ │ │ @@ -245,15 +245,15 @@
│ │ │ │  i35 : poincare cokernel f = (1-T^3)*(1-T^3)*(1-T^5)*(1-T^6) -- cache poincare
│ │ │ │  
│ │ │ │              3    5     8     9    12     14    17
│ │ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : time betti gb f
│ │ │ │ - -- used 0.0037732s (cpu); 0.00284882s (thread); 0s (gc)
│ │ │ │ + -- used 0.00288335s (cpu); 0.00322848s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o36 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -80,112 +80,112 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11182-0/0/
    
│ │ │ +o1 = /tmp/M2-12292-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11182-0/1/
    
│ │ │ +o2 = /tmp/M2-12292-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11182-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12292-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11182-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12292-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11182-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12292-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11182-0/0/a/f
│ │ │ +o6 = /tmp/M2-12292-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11182-0/0/a/g
│ │ │ +o7 = /tmp/M2-12292-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11182-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12292-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-11182-0/0/
│ │ │ -     /tmp/M2-11182-0/0/b/
│ │ │ -     /tmp/M2-11182-0/0/b/c/
│ │ │ -     /tmp/M2-11182-0/0/b/c/g
│ │ │ -     /tmp/M2-11182-0/0/a/
│ │ │ -     /tmp/M2-11182-0/0/a/g
│ │ │ -     /tmp/M2-11182-0/0/a/f
    
│ │ │ +o9 = /tmp/M2-12292-0/0/
│ │ │ +     /tmp/M2-12292-0/0/a/
│ │ │ +     /tmp/M2-12292-0/0/a/g
│ │ │ +     /tmp/M2-12292-0/0/a/f
│ │ │ +     /tmp/M2-12292-0/0/b/
│ │ │ +     /tmp/M2-12292-0/0/b/c/
│ │ │ +     /tmp/M2-12292-0/0/b/c/g
│ │ │              
    
│ │ │                
    i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11182-0/0/b/c/g -> /tmp/M2-11182-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-11182-0/0/a/g -> /tmp/M2-11182-0/1/a/g
│ │ │ - -- copying: /tmp/M2-11182-0/0/a/f -> /tmp/M2-11182-0/1/a/f
    
│ │ │ + -- copying: /tmp/M2-12292-0/0/a/g -> /tmp/M2-12292-0/1/a/g
│ │ │ + -- copying: /tmp/M2-12292-0/0/a/f -> /tmp/M2-12292-0/1/a/f
│ │ │ + -- copying: /tmp/M2-12292-0/0/b/c/g -> /tmp/M2-12292-0/1/b/c/g
│ │ │              
    
│ │ │                
    i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11182-0/0/b/c/g not newer than /tmp/M2-11182-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-11182-0/0/a/g not newer than /tmp/M2-11182-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-11182-0/0/a/f not newer than /tmp/M2-11182-0/1/a/f
    
│ │ │ + -- skipping: /tmp/M2-12292-0/0/a/g not newer than /tmp/M2-12292-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-12292-0/0/a/f not newer than /tmp/M2-12292-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-12292-0/0/b/c/g not newer than /tmp/M2-12292-0/1/b/c/g
│ │ │              
    
│ │ │                
    i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-11182-0/1/
│ │ │ -      /tmp/M2-11182-0/1/a/
│ │ │ -      /tmp/M2-11182-0/1/a/f
│ │ │ -      /tmp/M2-11182-0/1/a/g
│ │ │ -      /tmp/M2-11182-0/1/b/
│ │ │ -      /tmp/M2-11182-0/1/b/c/
│ │ │ -      /tmp/M2-11182-0/1/b/c/g
    
│ │ │ +o12 = /tmp/M2-12292-0/1/
│ │ │ +      /tmp/M2-12292-0/1/a/
│ │ │ +      /tmp/M2-12292-0/1/a/g
│ │ │ +      /tmp/M2-12292-0/1/a/f
│ │ │ +      /tmp/M2-12292-0/1/b/
│ │ │ +      /tmp/M2-12292-0/1/b/c/
│ │ │ +      /tmp/M2-12292-0/1/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/M2-11182-0/0/
│ │ │ │ +o1 = /tmp/M2-12292-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11182-0/1/
│ │ │ │ +o2 = /tmp/M2-12292-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11182-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12292-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11182-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12292-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11182-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12292-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11182-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12292-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11182-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12292-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11182-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12292-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : stack findFiles src
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11182-0/0/
│ │ │ │ -     /tmp/M2-11182-0/0/b/
│ │ │ │ -     /tmp/M2-11182-0/0/b/c/
│ │ │ │ -     /tmp/M2-11182-0/0/b/c/g
│ │ │ │ -     /tmp/M2-11182-0/0/a/
│ │ │ │ -     /tmp/M2-11182-0/0/a/g
│ │ │ │ -     /tmp/M2-11182-0/0/a/f
│ │ │ │ +o9 = /tmp/M2-12292-0/0/
│ │ │ │ +     /tmp/M2-12292-0/0/a/
│ │ │ │ +     /tmp/M2-12292-0/0/a/g
│ │ │ │ +     /tmp/M2-12292-0/0/a/f
│ │ │ │ +     /tmp/M2-12292-0/0/b/
│ │ │ │ +     /tmp/M2-12292-0/0/b/c/
│ │ │ │ +     /tmp/M2-12292-0/0/b/c/g
│ │ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11182-0/0/b/c/g -> /tmp/M2-11182-0/1/b/c/g
│ │ │ │ - -- copying: /tmp/M2-11182-0/0/a/g -> /tmp/M2-11182-0/1/a/g
│ │ │ │ - -- copying: /tmp/M2-11182-0/0/a/f -> /tmp/M2-11182-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12292-0/0/a/g -> /tmp/M2-12292-0/1/a/g
│ │ │ │ + -- copying: /tmp/M2-12292-0/0/a/f -> /tmp/M2-12292-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-12292-0/0/b/c/g -> /tmp/M2-12292-0/1/b/c/g
│ │ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11182-0/0/b/c/g not newer than /tmp/M2-11182-0/1/b/c/g
│ │ │ │ - -- skipping: /tmp/M2-11182-0/0/a/g not newer than /tmp/M2-11182-0/1/a/g
│ │ │ │ - -- skipping: /tmp/M2-11182-0/0/a/f not newer than /tmp/M2-11182-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12292-0/0/a/g not newer than /tmp/M2-12292-0/1/a/g
│ │ │ │ + -- skipping: /tmp/M2-12292-0/0/a/f not newer than /tmp/M2-12292-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-12292-0/0/b/c/g not newer than /tmp/M2-12292-0/1/b/c/g
│ │ │ │  i12 : stack findFiles dst
│ │ │ │  
│ │ │ │ -o12 = /tmp/M2-11182-0/1/
│ │ │ │ -      /tmp/M2-11182-0/1/a/
│ │ │ │ -      /tmp/M2-11182-0/1/a/f
│ │ │ │ -      /tmp/M2-11182-0/1/a/g
│ │ │ │ -      /tmp/M2-11182-0/1/b/
│ │ │ │ -      /tmp/M2-11182-0/1/b/c/
│ │ │ │ -      /tmp/M2-11182-0/1/b/c/g
│ │ │ │ +o12 = /tmp/M2-12292-0/1/
│ │ │ │ +      /tmp/M2-12292-0/1/a/
│ │ │ │ +      /tmp/M2-12292-0/1/a/g
│ │ │ │ +      /tmp/M2-12292-0/1/a/f
│ │ │ │ +      /tmp/M2-12292-0/1/b/
│ │ │ │ +      /tmp/M2-12292-0/1/b/c/
│ │ │ │ +      /tmp/M2-12292-0/1/b/c/g
│ │ │ │  i13 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o13 = ho there
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o14 = rm
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__File_lp__String_cm__String_rp.html
│ │ │ @@ -78,65 +78,65 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,16 +31,16 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal(x, y, z)
│ │ │ │  
│ │ │ │  o2 = ideal (x, y, z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6)
│ │ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53622-0/0-in.ms -o /tmp/
│ │ │ │ -M2-53622-0/0-out.ms
│ │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84695-0/0-in.ms -o /tmp/
│ │ │ │ +M2-84695-0/0-out.ms
│ │ │ │  
│ │ │ │  --------------- INPUT DATA ---------------
│ │ │ │  #variables                       3
│ │ │ │  #equations                       3
│ │ │ │  #invalid equations               0
│ │ │ │  field characteristic             0
│ │ │ │  homogeneous input?               1
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  initial hash table size     131072 (2^17)
│ │ │ │  max pair selection             ALL
│ │ │ │  reduce gb                        1
│ │ │ │  #threads                         6
│ │ │ │  info level                       2
│ │ │ │  generate pbm files               0
│ │ │ │  ------------------------------------------
│ │ │ │ -Initial prime = 1147761463
│ │ │ │ +Initial prime = 1253775077
│ │ │ │  
│ │ │ │  Legend for f4 information
│ │ │ │  --------------------------------------------------------
│ │ │ │  deg       current degree of pairs selected in this round
│ │ │ │  sel       number of pairs selected in this round
│ │ │ │  pairs     total number of pairs in pair list
│ │ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ │ @@ -73,26 +73,26 @@
│ │ │ │  deg     sel   pairs        mat          density            new data
│ │ │ │  time(rd) in sec (real|cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  reduce final basis        3 x 3          33.33%        3 new       0 zero
│ │ │ │ -0.06 | 0.13
│ │ │ │ +0.00 | 0.00
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -overall(elapsed)        0.13 sec
│ │ │ │ -overall(cpu)            0.29 sec
│ │ │ │ +overall(elapsed)        0.00 sec
│ │ │ │ +overall(cpu)            0.00 sec
│ │ │ │  select                  0.00 sec   0.0%
│ │ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ │ -update                  0.07 sec  51.6%
│ │ │ │ -convert                 0.06 sec  48.3%
│ │ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ │ +symbolic prep.          0.00 sec   0.5%
│ │ │ │ +update                  0.00 sec  76.8%
│ │ │ │ +convert                 0.00 sec   3.3%
│ │ │ │ +linear algebra          0.00 sec   0.9%
│ │ │ │  reduce gb               0.00 sec   0.0%
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  size of basis                     3
│ │ │ │  #terms in basis                   3
│ │ │ │  #pairs reduced                    0
│ │ │ │ @@ -106,18 +106,18 @@
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  [3]
│ │ │ │  #polynomials to lift              3
│ │ │ │  -----------------------------------------
│ │ │ │ -New prime = 1086711877
│ │ │ │ +New prime = 1098899071
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -multi-mod overall(elapsed)      0.04 sec
│ │ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │ │  learning phase                  0.00 Gops/sec
│ │ │ │  application phase               0.00 Gops/sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  multi-modular steps
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │ @@ -136,15 +136,15 @@
│ │ │ │  CRT     (elapsed)               0.00 sec
│ │ │ │  ratrecon(elapsed)               0.00 sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │ -msolve overall time           0.25 sec (elapsed) /  0.63 sec (cpu)
│ │ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │  
│ │ │ │  o3 = | z y x |
│ │ │ │  
│ │ │ │               1      3
│ │ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out
│ │ │ @@ -23,19 +23,19 @@
│ │ │  o4 : RingMap S <-- R
│ │ │  
│ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00857534s elapsed
│ │ │ + -- .00258617s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00905284s elapsed
│ │ │ + -- .0117182s elapsed
│ │ │  
│ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │                        {0, 2, 0, 2, 0} => {}
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html
│ │ │ @@ -117,19 +117,19 @@
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,23 +20,23 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal"xy,yz,zx"
│ │ │ │  
│ │ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime jMult I
│ │ │ │ - -- .024652s elapsed
│ │ │ │ + -- .0287072s elapsed
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  i4 : elapsedTime monjMult I
│ │ │ │ - -- .132531s elapsed
│ │ │ │ + -- .0988799s elapsed
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ │ - -- .172768s elapsed
│ │ │ │ + -- .173869s elapsed
│ │ │ │  
│ │ │ │  o5 = HashTable{2 => 3}
│ │ │ │                 3 => 2
│ │ │ │  
│ │ │ │  o5 : HashTable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10967-0/0
    
│ │ │ +o1 = /tmp/M2-11857-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10967-0/1
    
│ │ │ +o2 = /tmp/M2-11857-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10967-0/0
│ │ │ +o3 = /tmp/M2-11857-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-10967-0/0 -> /tmp/M2-10967-0/1
    
│ │ │ + -- copying: /tmp/M2-11857-0/0 -> /tmp/M2-11857-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10967-0/0 not newer than /tmp/M2-10967-0/1
    
│ │ │ + -- skipping: /tmp/M2-11857-0/0 not newer than /tmp/M2-11857-0/1
│ │ │              
    
│ │ │                
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-10967-0/0
│ │ │ +o7 = /tmp/M2-11857-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-10967-0/0 not newer than /tmp/M2-10967-0/1
    
│ │ │ + -- skipping: /tmp/M2-11857-0/0 not newer than /tmp/M2-11857-0/1
│ │ │              
    
│ │ │                
    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/M2-10967-0/0
│ │ │ │ +o1 = /tmp/M2-11857-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10967-0/1
│ │ │ │ +o2 = /tmp/M2-11857-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10967-0/0
│ │ │ │ +o3 = /tmp/M2-11857-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-10967-0/0 -> /tmp/M2-10967-0/1
│ │ │ │ + -- copying: /tmp/M2-11857-0/0 -> /tmp/M2-11857-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-10967-0/0 not newer than /tmp/M2-10967-0/1
│ │ │ │ + -- skipping: /tmp/M2-11857-0/0 not newer than /tmp/M2-11857-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-10967-0/0
│ │ │ │ +o7 = /tmp/M2-11857-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-10967-0/0 not newer than /tmp/M2-10967-0/1
│ │ │ │ + -- skipping: /tmp/M2-11857-0/0 not newer than /tmp/M2-11857-0/1
│ │ │ │  i9 : get dst
│ │ │ │  
│ │ │ │  o9 = hi there
│ │ │ │  i10 : removeFile src
│ │ │ │  i11 : removeFile dst
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_D_i_r_e_c_t_o_r_y
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cpu__Time.html
│ │ │ @@ -64,38 +64,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 355.748080184
│ │ │ +o1 = 380.525134373
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │              
    
│ │ │                
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 357.599866817
│ │ │ +o3 = 381.599535454
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.851786633000017
│ │ │ +o4 = 1.074401081000019
│ │ │  
│ │ │  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 = 355.748080184
│ │ │ │ +o1 = 380.525134373
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 357.599866817
│ │ │ │ +o3 = 381.599535454
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 1.851786633000017
│ │ │ │ +o4 = 1.074401081000019
│ │ │ │  
│ │ │ │  o4 : RR (of precision 53)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_current__Time.html
│ │ │ @@ -64,48 +64,48 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1768843899
    
│ │ │ +o1 = 1769301879
│ │ │              
    
│ │ │          
    We can compute, roughly, how many years ago the epoch began as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 56.05243177477978
│ │ │ +o2 = 56.06694458324112
│ │ │  
│ │ │  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 = .6291812973573201
│ │ │ +o3 = .8033349988934901
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    Compare that to the current date, available from a standard Unix command.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : run "date"
│ │ │ -Mon Jan 19 17:31:39 UTC 2026
│ │ │ +Sun Jan 25 00:44:39 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 = 1768843899
│ │ │ │ +o1 = 1769301879
│ │ │ │  We can compute, roughly, how many years ago the epoch began as follows.
│ │ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │ │  
│ │ │ │ -o2 = 56.05243177477978
│ │ │ │ +o2 = 56.06694458324112
│ │ │ │  
│ │ │ │  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 = .6291812973573201
│ │ │ │ +o3 = .8033349988934901
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  Compare that to the current date, available from a standard Unix command.
│ │ │ │  i4 : run "date"
│ │ │ │ -Mon Jan 19 17:31:39 UTC 2026
│ │ │ │ +Sun Jan 25 00:44:39 UTC 2026
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_u_r_r_e_n_t_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:1849:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Time.html
│ │ │ @@ -59,15 +59,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the value of e.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime sleep 1
│ │ │ - -- 1.00017s elapsed
│ │ │ + -- 1.00133s 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
│ │ │ │ - -- 1.00017s elapsed
│ │ │ │ + -- 1.00133s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _G_C_s_t_a_t_s -- information about the status of the garbage collector
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Timing.html
│ │ │ @@ -54,24 +54,24 @@
│ │ │  elapsedTiming e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of time elapsed, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTiming sleep 1
│ │ │  
│ │ │  o1 = 0
│ │ │ -     -- 1.00018 seconds
│ │ │ +     -- 1.00015 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00018, 0}
    
│ │ │ +o2 = Time{1.00015, 0}
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,20 +10,20 @@
│ │ │ │  where t is the number of seconds of time elapsed, and v is the value of the
│ │ │ │  expression.
│ │ │ │  The default method for printing such timing results is to display the timing
│ │ │ │  separately in a comment below the computed value.
│ │ │ │  i1 : elapsedTiming sleep 1
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │ -     -- 1.00018 seconds
│ │ │ │ +     -- 1.00015 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{1.00018, 0}
│ │ │ │ +o2 = Time{1.00015, 0}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_i_m_e -- the class of all timing results
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elimination_spof_spvariables.html
│ │ │ @@ -65,15 +65,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : time leadTerm gens gb I
│ │ │ - -- used 0.461542s (cpu); 0.27877s (thread); 0s (gc)
│ │ │ + -- used 0.142667s (cpu); 0.142666s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.489181s (cpu); 0.266854s (thread); 0s (gc)
│ │ │ + -- used 0.461277s (cpu); 0.209215s (thread); 0s (gc)
│ │ │  
│ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │       ------------------------------------------------------------------------
│ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -245,15 +245,15 @@
│ │ │  
│ │ │  o11 : Ideal of R1
    
│ │ │              
    
│ │ │                
    i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.29671s (cpu); 0.11819s (thread); 0s (gc)
│ │ │ + -- used 0.0323307s (cpu); 0.0323384s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7  
│ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2  
│ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -337,15 +337,15 @@
│ │ │  
│ │ │  o16 : RingMap A <-- B
    
│ │ │              
    
│ │ │                
    i17 : time G = kernel F
│ │ │ - -- used 0.418187s (cpu); 0.235941s (thread); 0s (gc)
│ │ │ + -- used 0.1103s (cpu); 0.110306s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │        -----------------------------------------------------------------------
│ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -418,26 +418,26 @@
│ │ │  
│ │ │  o19 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00229141s (cpu); 0.00229176s (thread); 0s (gc)
│ │ │ + -- used 0.00190719s (cpu); 0.00191013s (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 0.0542115s (cpu); 0.0542197s (thread); 0s (gc)
│ │ │ + -- used 0.037301s (cpu); 0.0373166s (thread); 0s (gc)
│ │ │  
│ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2  
│ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │        -----------------------------------------------------------------------
│ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8   
│ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │        -----------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,15 +13,15 @@
│ │ │ │  i2 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o2 = ideal (- s  - s*t + x - 1, - t  - 3t  - t + y, - s*t  + z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time leadTerm gens gb I
│ │ │ │ - -- used 0.461542s (cpu); 0.27877s (thread); 0s (gc)
│ │ │ │ + -- used 0.142667s (cpu); 0.142666s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -89,15 +89,15 @@
│ │ │ │  i7 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time G = eliminate(I,{s,t})
│ │ │ │ - -- used 0.489181s (cpu); 0.266854s (thread); 0s (gc)
│ │ │ │ + -- used 0.461277s (cpu); 0.209215s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -156,15 +156,15 @@
│ │ │ │  Sometimes giving the variables different degrees will speed up the
│ │ │ │  computations. Here, we set the degrees of x, y, and z to be the total degrees.
│ │ │ │  i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
│ │ │ │  i11 : I1 = substitute(I,R1);
│ │ │ │  
│ │ │ │  o11 : Ideal of R1
│ │ │ │  i12 : time G = eliminate(I1,{s,t})
│ │ │ │ - -- used 0.29671s (cpu); 0.11819s (thread); 0s (gc)
│ │ │ │ + -- used 0.0323307s (cpu); 0.0323384s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7
│ │ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2
│ │ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -227,15 +227,15 @@
│ │ │ │  i16 : F = map(A,B,{s^3+s*t+1, t^3+3*t^2+t, s*t^3})
│ │ │ │  
│ │ │ │                     3             3     2         3
│ │ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │ │  
│ │ │ │  o16 : RingMap A <-- B
│ │ │ │  i17 : time G = kernel F
│ │ │ │ - -- used 0.418187s (cpu); 0.235941s (thread); 0s (gc)
│ │ │ │ + -- used 0.1103s (cpu); 0.110306s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -296,22 +296,22 @@
│ │ │ │  involve the variables s and t.
│ │ │ │  i19 : use ring I
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : PolynomialRing
│ │ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ │ - -- used 0.00229141s (cpu); 0.00229176s (thread); 0s (gc)
│ │ │ │ + -- used 0.00190719s (cpu); 0.00191013s (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 0.0542115s (cpu); 0.0542197s (thread); 0s (gc)
│ │ │ │ + -- used 0.037301s (cpu); 0.0373166s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2
│ │ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8
│ │ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_end__Package.html
│ │ │ @@ -154,15 +154,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-10188-0/91-rundir/
│ │ │ +             source directory => /tmp/M2-10308-0/91-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
    
│ │ │              
    
│ │ │                
    i7 : dictionaryPath
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,15 +77,15 @@
│ │ │ │                                      Version => 0.0
│ │ │ │               package prefix => /usr/
│ │ │ │               PackageIsLoaded => true
│ │ │ │               pkgname => Foo
│ │ │ │               private dictionary => Foo#"private dictionary"
│ │ │ │               processed documentation => MutableHashTable{}
│ │ │ │               raw documentation => MutableHashTable{}
│ │ │ │ -             source directory => /tmp/M2-10188-0/91-rundir/
│ │ │ │ +             source directory => /tmp/M2-10308-0/91-rundir/
│ │ │ │               source file => stdio
│ │ │ │               test inputs => MutableList{}
│ │ │ │  i7 : dictionaryPath
│ │ │ │  
│ │ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Truncations.Dictionary, Polyhedra.Dictionary, Saturation.Dictionary,
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Exists.html
│ │ │ @@ -68,29 +68,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -510,27 +510,27 @@
│ │ │  o36 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10555-0/0
    
│ │ │ +o1 = /tmp/M2-11025-0/0
│ │ │              
    
│ │ │                
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10555-0/0
│ │ │ +o3 = /tmp/M2-11025-0/0
│ │ │  
│ │ │  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/M2-10555-0/0
│ │ │ │ +o1 = /tmp/M2-11025-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10555-0/0
│ │ │ │ +o3 = /tmp/M2-11025-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : fileExists fn
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : removeFile fn
│ │ │ │  If fn refers to a symbolic link, then whether the file exists is determined by
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Length.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │          
    Description
    
│ │ │          
    The length of an open output file is determined from the internal count of the number of bytes written so far.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-12147-0/0
│ │ │ +o1 = /tmp/M2-14267-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │              
    
│ │ │                
    i2 : fileLength f
│ │ │ @@ -85,24 +85,24 @@
│ │ │  o2 = 8
    
│ │ │              
    
│ │ │                
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-12147-0/0
│ │ │ +o3 = /tmp/M2-14267-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-12147-0/0
    
│ │ │ +o4 = /tmp/M2-14267-0/0
│ │ │              
    
│ │ │                
    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/M2-12147-0/0
│ │ │ │ +o1 = /tmp/M2-14267-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12147-0/0
│ │ │ │ +o3 = /tmp/M2-14267-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12147-0/0
│ │ │ │ +o4 = /tmp/M2-14267-0/0
│ │ │ │  i5 : fileLength filename
│ │ │ │  
│ │ │ │  o5 = 8
│ │ │ │  i6 : get filename
│ │ │ │  
│ │ │ │  o6 = hi there
│ │ │ │  i7 : length oo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__File_rp.html
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -425,15 +425,15 @@
│ │ │  
│ │ │  o33 = 1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11372-0/0
    
│ │ │ +o1 = /tmp/M2-12682-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11372-0/0
│ │ │ +o2 = /tmp/M2-12682-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode f
│ │ │ @@ -92,15 +92,15 @@
│ │ │  o3 = 420
    
│ │ │              
    
│ │ │                
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-11372-0/0
│ │ │ +o4 = /tmp/M2-12682-0/0
│ │ │  
│ │ │  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/M2-11372-0/0
│ │ │ │ +o1 = /tmp/M2-12682-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11372-0/0
│ │ │ │ +o2 = /tmp/M2-12682-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11372-0/0
│ │ │ │ +o4 = /tmp/M2-12682-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _f_i_l_e_M_o_d_e_(_F_i_l_e_) -- get file mode
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__String_rp.html
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │ @@ -189,19 +189,19 @@
│ │ │            
│ │ │            
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10986-0/0
    
│ │ │ +o1 = /tmp/M2-11896-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-10986-0/0
│ │ │ +o2 = /tmp/M2-11896-0/0
│ │ │  
│ │ │  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/M2-10986-0/0
│ │ │ │ +o1 = /tmp/M2-11896-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10986-0/0
│ │ │ │ +o2 = /tmp/M2-11896-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__File_rp.html
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -245,23 +245,23 @@
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10851-0/0
    
│ │ │ +o1 = /tmp/M2-11621-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-10851-0/0
│ │ │ +o2 = /tmp/M2-11621-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = 7 + 7*8 + 7*64
│ │ │ @@ -108,15 +108,15 @@
│ │ │  o5 = 511
    
│ │ │              
    
│ │ │                
    i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-10851-0/0
│ │ │ +o6 = /tmp/M2-11621-0/0
│ │ │  
│ │ │  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/M2-10851-0/0
│ │ │ │ +o1 = /tmp/M2-11621-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10851-0/0
│ │ │ │ +o2 = /tmp/M2-11621-0/0
│ │ │ │  
│ │ │ │  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/M2-10851-0/0
│ │ │ │ +o6 = /tmp/M2-11621-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : fileMode fn
│ │ │ │  
│ │ │ │  o7 = 511
│ │ │ │  i8 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__String_rp.html
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11974-0/0
    
│ │ │ +o1 = /tmp/M2-13914-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11974-0/0
│ │ │ +o2 = /tmp/M2-13914-0/0
│ │ │  
│ │ │  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/M2-11974-0/0
│ │ │ │ +o1 = /tmp/M2-13914-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11974-0/0
│ │ │ │ +o2 = /tmp/M2-13914-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : fileMode(m|7,fn)
│ │ │ │  i5 : fileMode fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Time.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The value is the number of seconds since 00:00:00 1970-01-01 UTC, the beginning of the epoch, so the number of seconds ago a file or directory was modified may be found by using the following code.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 62
    
│ │ │ +o1 = 52
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 = 62
│ │ │ │ +o1 = 52
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │      * _f_i_l_e_ _m_a_n_i_p_u_l_a_t_i_o_n -- Unix file manipulation functions
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _f_i_l_e_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.html
│ │ │ @@ -120,15 +120,15 @@
│ │ │  o6 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f43e8412e00
│ │ │ +   -- registering gb 0 at 0x7fa4a8edee00
│ │ │  
│ │ │     -- [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 0x7f43e8412e00
│ │ │ │ +   -- registering gb 0 at 0x7fa4a8edee00
│ │ │ │  
│ │ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb
│ │ │ │  elements = 3
│ │ │ │     -- number of monomials                = 9
│ │ │ │     -- #reduction steps = 6
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_get.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │                
    i3 : removeFile "test-file"
    
│ │ │              
    
│ │ │                
    i4 : get "!date"
│ │ │  
│ │ │ -o4 = Mon Jan 19 17:30:51 UTC 2026
    
│ │ │ +o4 = Sun Jan 25 00:44:01 UTC 2026
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  o1 : File
│ │ │ │  i2 : get "test-file"
│ │ │ │  
│ │ │ │  o2 = hi there
│ │ │ │  i3 : removeFile "test-file"
│ │ │ │  i4 : get "!date"
│ │ │ │  
│ │ │ │ -o4 = Mon Jan 19 17:30:51 UTC 2026
│ │ │ │ +o4 = Sun Jan 25 00:44:01 UTC 2026
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d -- read from a file
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ │ │      * _c_l_o_s_e -- close a file
│ │ │ │      * _F_i_l_e_ _<_<_ _T_h_i_n_g -- print to a file
│ │ │ │  ********** WWaayyss ttoo uussee ggeett:: **********
│ │ │ │      * get(File)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instances.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │                 defaultPrecision => 53
│ │ │                 engineDebugLevel => 0
│ │ │                 errorDepth => 0
│ │ │                 gbTrace => 0
│ │ │                 interpreterDepth => 1
│ │ │                 lineNumber => 2
│ │ │                 loadDepth => 3
│ │ │ -               maxAllowableThreads => 7
│ │ │ +               maxAllowableThreads => 17
│ │ │                 maxExponent => 1073741823
│ │ │                 minExponent => -1073741824
│ │ │                 numTBBThreads => 0
│ │ │                 o1 => 2432902008176640000
│ │ │                 oo => 2432902008176640000
│ │ │                 printingAccuracy => -1
│ │ │                 printingLeadLimit => 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │                 defaultPrecision => 53
│ │ │ │                 engineDebugLevel => 0
│ │ │ │                 errorDepth => 0
│ │ │ │                 gbTrace => 0
│ │ │ │                 interpreterDepth => 1
│ │ │ │                 lineNumber => 2
│ │ │ │                 loadDepth => 3
│ │ │ │ -               maxAllowableThreads => 7
│ │ │ │ +               maxAllowableThreads => 17
│ │ │ │                 maxExponent => 1073741823
│ │ │ │                 minExponent => -1073741824
│ │ │ │                 numTBBThreads => 0
│ │ │ │                 o1 => 2432902008176640000
│ │ │ │                 oo => 2432902008176640000
│ │ │ │                 printingAccuracy => -1
│ │ │ │                 printingLeadLimit => 5
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Directory.html
│ │ │ @@ -75,22 +75,22 @@
│ │ │  o1 = true
    
│ │ │              
    
│ │ │                
    i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10377-0/0
    
│ │ │ +o2 = /tmp/M2-10667-0/0
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10377-0/0
│ │ │ +o3 = /tmp/M2-10667-0/0
│ │ │  
│ │ │  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/M2-10377-0/0
│ │ │ │ +o2 = /tmp/M2-10667-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10377-0/0
│ │ │ │ +o3 = /tmp/M2-10667-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : isDirectory fn
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Pseudoprime_lp__Z__Z_rp.html
│ │ │ @@ -211,15 +211,15 @@
│ │ │  
│ │ │  o18 = false
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 4.51779s elapsed
│ │ │ + -- 5.87053s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime m3
│ │ │ - -- .0563902s elapsed
│ │ │ + -- .0586494s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000102792s elapsed
│ │ │ + -- .000143174s elapsed
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │  i17 : isPrime (m*m1)
│ │ │ │  
│ │ │ │  o17 = false
│ │ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ │ - -- 4.51779s elapsed
│ │ │ │ + -- 5.87053s 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
│ │ │ │ - -- .0563902s elapsed
│ │ │ │ + -- .0586494s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000102792s elapsed
│ │ │ │ + -- .000143174s elapsed
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_P_r_i_m_e_(_Z_Z_) -- whether a integer or polynomial is prime
│ │ │ │      * _f_a_c_t_o_r_(_Z_Z_) -- factor a ring element
│ │ │ │      * _n_e_x_t_P_r_i_m_e_(_N_u_m_b_e_r_) -- compute the smallest prime greater than or equal to
│ │ │ │        a given number
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Regular__File.html
│ │ │ @@ -68,22 +68,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  In UNIX, a regular file is one that is not special in some way.  Special files include symbolic links and directories.  A regular file is a sequence of bytes stored permanently in a file system.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12185-0/0
    
│ │ │ +o1 = /tmp/M2-14345-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12185-0/0
│ │ │ +o2 = /tmp/M2-14345-0/0
│ │ │  
│ │ │  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/M2-12185-0/0
│ │ │ │ +o1 = /tmp/M2-14345-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12185-0/0
│ │ │ │ +o2 = /tmp/M2-14345-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : isRegularFile fn
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_make__Directory_lp__String_rp.html
│ │ │ @@ -76,22 +76,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the name of the newly made directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is made, with as many new path components as needed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10719-0/0
│ │ │ │ +o1 = /tmp/M2-11349-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10719-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-11349-0/0/a/b/c
│ │ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │ │  i5 : removeDirectory (dir|"/a")
│ │ │ │  A filename starting with ~/ will have the tilde replaced by the home directory.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_k_d_i_r
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Allowable__Threads.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10719-0/0
    
│ │ │ +o1 = /tmp/M2-11349-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-10719-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-11349-0/0/a/b/c
│ │ │              
    
│ │ │                
    i3 : removeDirectory (dir|"/a/b/c")
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
    
│ │ │ +o1 = 17
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,15 +9,15 @@
│ │ │ │      *   Usage:
│ │ │ │              maxAllowableThreads
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the maximum number to which _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s can be set
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : maxAllowableThreads
│ │ │ │  
│ │ │ │ -o1 = 7
│ │ │ │ +o1 = 17
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _m_a_x_A_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s is an _i_n_t_e_g_e_r.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_threads.m2:498:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_memoize.html
│ │ │ @@ -61,15 +61,15 @@
│ │ │  
│ │ │  o1 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i2 : time fib 28
│ │ │ - -- used 1.29904s (cpu); 0.825275s (thread); 0s (gc)
│ │ │ + -- used 0.885923s (cpu); 0.678937s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
    
│ │ │              
    
│ │ │                
    i3 : fib = memoize fib
│ │ │ @@ -78,23 +78,23 @@
│ │ │  
│ │ │  o3 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i4 : time fib 28
│ │ │ - -- used 0.000178845s (cpu); 0.000178345s (thread); 0s (gc)
│ │ │ + -- used 8.9328e-05s (cpu); 7.8284e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │              
    
│ │ │                
    i5 : time fib 28
│ │ │ - -- used 7.644e-06s (cpu); 7.043e-06s (thread); 0s (gc)
│ │ │ + -- used 3.096e-06s (cpu); 2.905e-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 1.29904s (cpu); 0.825275s (thread); 0s (gc)
│ │ │ │ + -- used 0.885923s (cpu); 0.678937s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 514229
│ │ │ │  i3 : fib = memoize fib
│ │ │ │  
│ │ │ │  o3 = fib
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : time fib 28
│ │ │ │ - -- used 0.000178845s (cpu); 0.000178345s (thread); 0s (gc)
│ │ │ │ + -- used 8.9328e-05s (cpu); 7.8284e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 7.644e-06s (cpu); 7.043e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.096e-06s (cpu); 2.905e-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
│ │ │ @@ -76,31 +76,31 @@
│ │ │  
│ │ │  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)            }
│ │ │ +     {19 => (codim, BettiTally)                                   }
│ │ │       {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {22 => (codim, BettiTally)                                   }
│ │ │ +     {21 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ +     {22 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │       {23 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │       {24 => (dual, BettiTally)                                    }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
    
│ │ │                
    
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                  
    i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │  o5 = {0 => (diff', Matrix, Matrix)                               }
│ │ │ -     {1 => (contract', Matrix, Matrix)                           }
│ │ │ +     {1 => (diff, Matrix, Matrix)                                }
│ │ │       {2 => (contract, Matrix, Matrix)                            }
│ │ │ -     {3 => (-, Matrix, Matrix)                                   }
│ │ │ -     {4 => (+, Matrix, Matrix)                                   }
│ │ │ -     {5 => (diff, Matrix, Matrix)                                }
│ │ │ +     {3 => (+, Matrix, Matrix)                                   }
│ │ │ +     {4 => (-, Matrix, Matrix)                                   }
│ │ │ +     {5 => (contract', 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)                        }
│ │ │ @@ -212,22 +212,22 @@
│ │ │       {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 => (remainder, Matrix, Matrix)                          }
│ │ │ -     {25 => (%, Matrix, Matrix)                                  }
│ │ │ +     {24 => (%, Matrix, Matrix)                                  }
│ │ │ +     {25 => (remainder, Matrix, Matrix)                          }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ -     {29 => (tensor, Matrix, Matrix)                             }
│ │ │ -     {30 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {31 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {29 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {31 => (intersect, 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 {}
│ │ │ │ @@ -16,31 +16,31 @@
│ │ │ │  
│ │ │ │  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)            }
│ │ │ │ +     {19 => (codim, BettiTally)                                   }
│ │ │ │       {20 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ -     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ -     {22 => (codim, BettiTally)                                   }
│ │ │ │ +     {21 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ +     {22 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │       {23 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │       {24 => (dual, BettiTally)                                    }
│ │ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │ │  
│ │ │ │  o1 : NumberedVerticalList
│ │ │ │  i2 : methods resolution
│ │ │ │  
│ │ │ │ @@ -86,19 +86,19 @@
│ │ │ │            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 => (diff', Matrix, Matrix)                               }
│ │ │ │ -     {1 => (contract', Matrix, Matrix)                           }
│ │ │ │ +     {1 => (diff, Matrix, Matrix)                                }
│ │ │ │       {2 => (contract, Matrix, Matrix)                            }
│ │ │ │ -     {3 => (-, Matrix, Matrix)                                   }
│ │ │ │ -     {4 => (+, Matrix, Matrix)                                   }
│ │ │ │ -     {5 => (diff, Matrix, Matrix)                                }
│ │ │ │ +     {3 => (+, Matrix, Matrix)                                   }
│ │ │ │ +     {4 => (-, Matrix, Matrix)                                   }
│ │ │ │ +     {5 => (contract', 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 => (remainder, Matrix, Matrix)                          }
│ │ │ │ -     {25 => (%, Matrix, Matrix)                                  }
│ │ │ │ +     {24 => (%, Matrix, Matrix)                                  }
│ │ │ │ +     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ │ -     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ -     {29 => (tensor, Matrix, Matrix)                             }
│ │ │ │ -     {30 => (intersect, Matrix, Matrix)                          }
│ │ │ │ -     {31 => (pullback, Matrix, Matrix)                           }
│ │ │ │ +     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ │ +     {29 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ +     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ │ +     {31 => (intersect, Matrix, Matrix)                          }
│ │ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_minimal__Betti.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.75782s elapsed
│ │ │ + -- 2.40147s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -125,15 +125,15 @@
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .720188s elapsed
│ │ │ + -- 1.04931s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │ @@ -146,15 +146,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0298402s elapsed
│ │ │ + -- .0405396s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
    
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
    
│ │ │                
    i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- 1.16354s elapsed
│ │ │ + -- 1.69433s 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
│ │ │ │ - -- 1.75782s elapsed
│ │ │ │ + -- 2.40147s 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)
│ │ │ │ - -- .720188s elapsed
│ │ │ │ + -- 1.04931s 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)
│ │ │ │ - -- .0298402s elapsed
│ │ │ │ + -- .0405396s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3  4
│ │ │ │  o7 = total: 1 35 140 189 84
│ │ │ │           0: 1  .   .   .  .
│ │ │ │           1: . 35 140 189 84
│ │ │ │  
│ │ │ │  o7 : BettiTally
│ │ │ │  i8 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ │ - -- 1.16354s elapsed
│ │ │ │ + -- 1.69433s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5
│ │ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │ │           0: 1  .   .   .   .    .
│ │ │ │           1: . 35 140 189  84    .
│ │ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_mkdir.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │          
    Description
    
│ │ │          
    Only one directory will be made, so the components of the path p other than the last must already exist.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -91,15 +91,15 @@
│ │ │  o3 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-10738-0/0/
    
│ │ │ +o1 = /tmp/M2-11388-0/0/
│ │ │              
    
│ │ │                
    i2 : mkdir p
    
│ │ │              
    
│ │ │                
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10738-0/0/foo
│ │ │ +o4 = /tmp/M2-11388-0/0/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/M2-10738-0/0/
│ │ │ │ +o1 = /tmp/M2-11388-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10738-0/0/foo
│ │ │ │ +o4 = /tmp/M2-11388-0/0/foo
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : removeFile fn
│ │ │ │  i7 : removeDirectory p
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_move__File_lp__String_cm__String_rp.html
│ │ │ @@ -81,52 +81,52 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,32 +20,32 @@
│ │ │ │            o the name of the backup file if one was created, or _n_u_l_l
│ │ │ │      * Consequences:
│ │ │ │            o the file will be moved by creating a new link to the file and
│ │ │ │              removing the old one
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10612-0/0
│ │ │ │ +o1 = /tmp/M2-11142-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10612-0/1
│ │ │ │ +o2 = /tmp/M2-11142-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10612-0/0
│ │ │ │ +o3 = /tmp/M2-11142-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-10612-0/0 -> /tmp/M2-10612-0/1
│ │ │ │ +--moving: /tmp/M2-11142-0/0 -> /tmp/M2-11142-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ │ ---backup file created: /tmp/M2-10612-0/1.bak
│ │ │ │ +--backup file created: /tmp/M2-11142-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-10612-0/1.bak
│ │ │ │ +o6 = /tmp/M2-11142-0/1.bak
│ │ │ │  i7 : removeFile bak
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_F_i_l_e
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * moveFile(String)
│ │ │ │      * _m_o_v_e_F_i_l_e_(_S_t_r_i_n_g_,_S_t_r_i_n_g_)
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_nanosleep.html
│ │ │ @@ -51,15 +51,15 @@
│ │ │        
    nanosleep -- sleep for a given number of nanoseconds
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  nanosleep n -- sleeps for n nanoseconds.        
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10612-0/0
    
│ │ │ +o1 = /tmp/M2-11142-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-10612-0/1
    
│ │ │ +o2 = /tmp/M2-11142-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-10612-0/0
│ │ │ +o3 = /tmp/M2-11142-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-10612-0/0 -> /tmp/M2-10612-0/1
    
│ │ │ +--moving: /tmp/M2-11142-0/0 -> /tmp/M2-11142-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-10612-0/1.bak
│ │ │ +--backup file created: /tmp/M2-11142-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-10612-0/1.bak
    
│ │ │ +o6 = /tmp/M2-11142-0/1.bak
│ │ │              
    
│ │ │                
    i7 : removeFile bak
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500151s elapsed
│ │ │ + -- .500167s 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
│ │ │ │ - -- .500151s elapsed
│ │ │ │ + -- .500167s 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/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html
│ │ │ @@ -72,21 +72,21 @@
│ │ │              
    		
│ │ │                
    i2 : L = random toList (1..10000);
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- 1.07356s elapsed
    
│ │ │ + -- .755934s elapsed
│ │ │              
    
│ │ │                
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .314344s elapsed
    
│ │ │ + -- .205538s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    You will have to try it on your examples to see how much they speed up.
    
│ │ │            
    Warning: Threads computing in parallel can give wrong answers if their code is not "thread safe", meaning they make modifications to memory without ensuring the modifications get safely communicated to other threads. (Thread safety can slow computations some.) Currently, modifications to Macaulay2 variables and mutable hash tables are thread safe, but not changes inside mutable lists. Also, access to external libraries such as singular, etc., may not currently be thread safe.
    
│ │ │            
    The rest of this document describes how to control parallel tasks more directly.
    
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o5 = 5
    
│ │ │              
    
│ │ │                
    i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
    
│ │ │ +o6 = 17
│ │ │              
    
│ │ │          
    
│ │ │            
    To run a function in another thread use schedule, as in the following example.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -271,15 +271,15 @@
│ │ │                
    i22 : addStartTask(F,G)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i23 : schedule F
│ │ │  
│ │ │ -o23 = <<task, result available, task done>>
│ │ │ +o23 = <<task, created>>
│ │ │  
│ │ │  o23 : Task
    
│ │ │              
    
│ │ │                
    i24 : taskResult F
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,17 +17,17 @@
│ │ │ │  big computation. If the list is long, it will be split into chunks for each
│ │ │ │  core, reducing the overhead. But the speedup is still limited by the different
│ │ │ │  threads competing for memory, including cpu caches; it is like running
│ │ │ │  Macaulay2 on a computer that is running other big programs at the same time. We
│ │ │ │  can see this using _e_l_a_p_s_e_d_T_i_m_e.
│ │ │ │  i2 : L = random toList (1..10000);
│ │ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ │ - -- 1.07356s elapsed
│ │ │ │ + -- .755934s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .314344s elapsed
│ │ │ │ + -- .205538s 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)
│ │ │ │  
│ │ │ │ @@ -116,15 +116,15 @@
│ │ │ │  
│ │ │ │  o21 = <>
│ │ │ │  
│ │ │ │  o21 : Task
│ │ │ │  i22 : addStartTask(F,G)
│ │ │ │  i23 : schedule F
│ │ │ │  
│ │ │ │ -o23 = <>
│ │ │ │ +o23 = <>
│ │ │ │  
│ │ │ │  o23 : Task
│ │ │ │  i24 : taskResult F
│ │ │ │  
│ │ │ │  o24 = result of F
│ │ │ │  i25 : taskResult G
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalBetti I
│ │ │ - -- 2.1171s elapsed
│ │ │ + -- 2.46489s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.87462s elapsed
│ │ │ + -- 2.50184s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -190,15 +190,15 @@
│ │ │  
│ │ │  o8 = 1
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.85815s elapsed
│ │ │ + -- 2.53058s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -231,15 +231,15 @@
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.14686s elapsed
│ │ │ + -- 2.84364s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
    
│ │ │ @@ -258,15 +258,15 @@
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │              
    
│ │ │                
    i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.05195s elapsed
│ │ │ + -- 2.79999s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
    
│ │ │ @@ -299,15 +299,15 @@
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │              
    
│ │ │                
    i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.98038s elapsed
│ │ │ + -- 4.52716s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -322,15 +322,15 @@
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │              
    
│ │ │                
    i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 6.54385s elapsed
│ │ │ + -- 9.18152s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -345,15 +345,15 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │              
    
│ │ │                
    i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.47844s elapsed
│ │ │ + -- 4.00098s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -396,15 +396,15 @@
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │              
    
│ │ │                
    i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.0147s elapsed
│ │ │ + -- 1.10887s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │                
    i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.17142s elapsed
│ │ │ + -- 1.77207s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -93,30 +93,30 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i3 : S = ring I
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : elapsedTime minimalBetti I
│ │ │ │ - -- 2.1171s elapsed
│ │ │ │ + -- 2.46489s 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)
│ │ │ │ - -- 1.87462s elapsed
│ │ │ │ + -- 2.50184s 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)
│ │ │ │ - -- 1.85815s elapsed
│ │ │ │ + -- 2.53058s 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)
│ │ │ │ - -- 2.14686s elapsed
│ │ │ │ + -- 2.84364s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -168,15 +168,15 @@
│ │ │ │  i14 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o14 = 1
│ │ │ │  i15 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o15 : Ideal of S
│ │ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 2.05195s elapsed
│ │ │ │ + -- 2.79999s 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");
│ │ │ │ - -- 4.98038s elapsed
│ │ │ │ + -- 4.52716s 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");
│ │ │ │ - -- 6.54385s elapsed
│ │ │ │ + -- 9.18152s 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");
│ │ │ │ - -- 4.47844s elapsed
│ │ │ │ + -- 4.00098s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o26 : Matrix S  <-- S
│ │ │ │  For Gröbner basis computation in associative algebras, ParallelizeByDegree is
│ │ │ │  not relevant. In this case, use numTBBThreads to control the amount of
│ │ │ │  parallelism.
│ │ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │ │ @@ -246,15 +246,15 @@
│ │ │ │                 2                         2                         2
│ │ │ │  o30 = ideal (5a  + 2b*c + 3c*b, 3a*c + 5b  + 2c*a, 2a*b + 3b*a + 5c )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i31 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- 1.0147s elapsed
│ │ │ │ + -- 1.10887s 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);
│ │ │ │ - -- 1.17142s elapsed
│ │ │ │ + -- 1.77207s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_i_m_a_l_B_e_t_t_i -- minimal betti numbers of (the minimal free resolution of)
│ │ │ │        a homogeneous ideal or module
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_poincare.html
│ │ │ @@ -370,36 +370,36 @@
│ │ │  
│ │ │  o27 = 3
│ │ │              
    
│ │ │                
    i28 : time poincare I
│ │ │ - -- used 0.00152129s (cpu); 1.9947e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00273319s (cpu); 1.1568e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7f2f23197540
│ │ │ +   -- registering gb 19 at 0x7fdffa32a540
│ │ │  
│ │ │     -- [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 0.0224323s (cpu); 0.0254628s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0132534s (cpu); 0.0162859s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -411,15 +411,15 @@
│ │ │                
    i30 : R = QQ[a..d];
    
│ │ │              
    
│ │ │                
    i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 20 at 0x7f2f23197380
│ │ │ +   -- registering gb 20 at 0x7fdffa32a380
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -428,24 +428,24 @@
│ │ │  o31 : Ideal of R
    
│ │ │              
    
│ │ │                
    i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 21 at 0x7f2f23197000
│ │ │ +   -- registering gb 21 at 0x7fdffa32a000
│ │ │  
│ │ │     -- [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 0.00800071s (cpu); 0.00954054s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00805159s (cpu); 0.0069751s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 22 at 0x7f2f23367e00
│ │ │ +   -- registering gb 22 at 0x7fdff9e35e00
│ │ │  
│ │ │     -- [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 0.0879724s (cpu); 0.0884082s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0520684s (cpu); 0.0555531s (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 0.00152129s (cpu); 1.9947e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00273319s (cpu); 1.1568e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o28 : ZZ[T]
│ │ │ │  i29 : time gens gb I;
│ │ │ │  
│ │ │ │ -   -- registering gb 19 at 0x7f2f23197540
│ │ │ │ +   -- registering gb 19 at 0x7fdffa32a540
│ │ │ │  
│ │ │ │     -- [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 0.0224323s (cpu); 0.0254628s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0132534s (cpu); 0.0162859s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      11
│ │ │ │  o29 : Matrix R  <-- R
│ │ │ │  In this case, the savings is minimal, but often it can be dramatic. Another
│ │ │ │  important situation is to compute a Gröbner basis using a different monomial
│ │ │ │  order.
│ │ │ │  i30 : R = QQ[a..d];
│ │ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │ │  
│ │ │ │ -   -- registering gb 20 at 0x7f2f23197380
│ │ │ │ +   -- registering gb 20 at 0x7fdffa32a380
│ │ │ │  
│ │ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │ │     -- number of monomials                = 0
│ │ │ │     -- #reduction steps = 0
│ │ │ │     -- #spairs done = 0
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │     -- nsaved = 0
│ │ │ │     --
│ │ │ │  o31 : Ideal of R
│ │ │ │  i32 : time p = poincare I
│ │ │ │  
│ │ │ │ -   -- registering gb 21 at 0x7f2f23197000
│ │ │ │ +   -- registering gb 21 at 0x7fdffa32a000
│ │ │ │  
│ │ │ │     -- [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 0.00800071s (cpu); 0.00954054s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00805159s (cpu); 0.0069751s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o32 : ZZ[T]
│ │ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ │  
│ │ │ │ @@ -281,30 +281,30 @@
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : gbTrace = 3
│ │ │ │  
│ │ │ │  o36 = 3
│ │ │ │  i37 : time gens gb J;
│ │ │ │  
│ │ │ │ -   -- registering gb 22 at 0x7f2f23367e00
│ │ │ │ +   -- registering gb 22 at 0x7fdff9e35e00
│ │ │ │  
│ │ │ │     -- [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 0.0879724s (cpu); 0.0884082s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0520684s (cpu); 0.0555531s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      39
│ │ │ │  o37 : Matrix S  <-- S
│ │ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │ │  
│ │ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_printing_spto_spa_spfile.html
│ │ │ @@ -97,22 +97,22 @@
│ │ │  o2 : File
│ │ │              
    
│ │ │                
    i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-10929-0/0
    
│ │ │ +o3 = /tmp/M2-11779-0/0
│ │ │              
    
│ │ │                
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-10929-0/0
│ │ │ +o4 = /tmp/M2-11779-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ @@ -151,15 +151,15 @@
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-10929-0/0
│ │ │ +o9 = /tmp/M2-11779-0/0
│ │ │  
│ │ │  o9 : File
    
│ │ │              
    
│ │ │                
    i10 : get fn
│ │ │ @@ -169,15 +169,15 @@
│ │ │        + 1
    
│ │ │              
    
│ │ │                
    i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-10929-0/0
│ │ │ +o11 = /tmp/M2-11779-0/0
│ │ │  
│ │ │  o11 : File
    
│ │ │              
    
│ │ │                
    i12 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,18 +36,18 @@
│ │ │ │  -- ho there --
│ │ │ │  
│ │ │ │  o2 = stdio
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-10929-0/0
│ │ │ │ +o3 = /tmp/M2-11779-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-10929-0/0
│ │ │ │ +o4 = /tmp/M2-11779-0/0
│ │ │ │  
│ │ │ │  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/M2-10929-0/0
│ │ │ │ +o9 = /tmp/M2-11779-0/0
│ │ │ │  
│ │ │ │  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/M2-10929-0/0
│ │ │ │ +o11 = /tmp/M2-11779-0/0
│ │ │ │  
│ │ │ │  o11 : File
│ │ │ │  i12 : get fn
│ │ │ │  
│ │ │ │  o12 = x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
│ │ │ │        1
│ │ │ │  i13 : value get fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_process__I__D.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : processID()
│ │ │  
│ │ │ -o1 = 10188
    
│ │ │ +o1 = 10308
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 = 10188
│ │ │ │ +o1 = 10308
│ │ │ │  ********** 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
│ │ │ @@ -90,36 +90,36 @@
│ │ │            
      Afterwards, running profileSummary and coverageSummary produces easy to read tables summarizing the accumulated data so far in different ways.
      
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o9 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -173,15 +173,15 @@
│ │ │  o14 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : profileSummary
│ │ │  
│ │ │ -o2 = #run  %time   position                         
│ │ │ -     1     94.7    ../../m2/matrix1.m2:279:4-282:58 
│ │ │ -     1     92.28   ../../m2/matrix1.m2:281:22-281:43
│ │ │ -     1     44.62   ../../m2/matrix1.m2:193:25-193:52
│ │ │ -     1     30.92   ../../m2/matrix1.m2:114:5-156:72 
│ │ │ -     1     29.74   ../../m2/matrix1.m2:140:10-155:16
│ │ │ -     1     22.38   ../../m2/matrix1.m2:181:4-181:42 
│ │ │ -     1     21.12   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ -     1     21.01   ../../m2/set.m2:127:5-127:61     
│ │ │ -     1     3.29    ../../m2/matrix1.m2:112:5-112:29 
│ │ │ -     1     2.44    ../../m2/matrix1.m2:96:5-109:11  
│ │ │ -     1     2.34    ../../m2/matrix1.m2:141:13-141:78
│ │ │ -     1     1.36    ../../m2/matrix1.m2:98:10-98:46  
│ │ │ -     1     1.33    ../../m2/matrix1.m2:281:7-281:16 
│ │ │ -     1     1.33    ../../m2/matrix1.m2:147:20-147:64
│ │ │ -     1     1.24    ../../m2/matrix1.m2:276:4-277:73 
│ │ │ -     1     1.20    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ -     1     1.08    ../../m2/matrix1.m2:182:4-184:74 
│ │ │ -     1     .74     ../../m2/modules.m2:279:4-279:52 
│ │ │ -     20    .51     ../../m2/matrix1.m2:191:14-192:67
│ │ │ -     20    .48     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ -     1     .0037s  elapsed total                    
    
│ │ │ +o2 = #run  %time  position                         
│ │ │ +     1     93.48  ../../m2/matrix1.m2:279:4-282:58 
│ │ │ +     1     91.02  ../../m2/matrix1.m2:281:22-281:43
│ │ │ +     1     45.78  ../../m2/matrix1.m2:193:25-193:52
│ │ │ +     1     31.03  ../../m2/matrix1.m2:114:5-156:72 
│ │ │ +     1     29.95  ../../m2/matrix1.m2:140:10-155:16
│ │ │ +     1     21.54  ../../m2/matrix1.m2:181:4-181:42 
│ │ │ +     1     20.7   ../../m2/matrix1.m2:45:10-49:22  
│ │ │ +     1     20.23  ../../m2/set.m2:127:5-127:61     
│ │ │ +     1     3.31   ../../m2/matrix1.m2:112:5-112:29 
│ │ │ +     1     2.72   ../../m2/matrix1.m2:96:5-109:11  
│ │ │ +     1     2.44   ../../m2/matrix1.m2:141:13-141:78
│ │ │ +     1     1.6    ../../m2/matrix1.m2:111:5-111:91 
│ │ │ +     1     1.59   ../../m2/matrix1.m2:147:20-147:64
│ │ │ +     1     1.52   ../../m2/matrix1.m2:281:7-281:16 
│ │ │ +     1     1.39   ../../m2/matrix1.m2:98:10-98:46  
│ │ │ +     1     1.34   ../../m2/matrix1.m2:276:4-277:73 
│ │ │ +     20    1.21   ../../m2/matrix1.m2:191:14-192:67
│ │ │ +     1     1.04   ../../m2/matrix1.m2:182:4-184:74 
│ │ │ +     1     .90    ../../m2/modules.m2:279:4-279:52 
│ │ │ +     20    .8     ../../m2/matrix1.m2:47:43-47:71  
│ │ │ +     1     .004s  elapsed total                    
│ │ │              
    
│ │ │                
    i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,36 +24,36 @@
│ │ │ │  
│ │ │ │                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     94.7    ../../m2/matrix1.m2:279:4-282:58
│ │ │ │ -     1     92.28   ../../m2/matrix1.m2:281:22-281:43
│ │ │ │ -     1     44.62   ../../m2/matrix1.m2:193:25-193:52
│ │ │ │ -     1     30.92   ../../m2/matrix1.m2:114:5-156:72
│ │ │ │ -     1     29.74   ../../m2/matrix1.m2:140:10-155:16
│ │ │ │ -     1     22.38   ../../m2/matrix1.m2:181:4-181:42
│ │ │ │ -     1     21.12   ../../m2/matrix1.m2:45:10-49:22
│ │ │ │ -     1     21.01   ../../m2/set.m2:127:5-127:61
│ │ │ │ -     1     3.29    ../../m2/matrix1.m2:112:5-112:29
│ │ │ │ -     1     2.44    ../../m2/matrix1.m2:96:5-109:11
│ │ │ │ -     1     2.34    ../../m2/matrix1.m2:141:13-141:78
│ │ │ │ -     1     1.36    ../../m2/matrix1.m2:98:10-98:46
│ │ │ │ -     1     1.33    ../../m2/matrix1.m2:281:7-281:16
│ │ │ │ -     1     1.33    ../../m2/matrix1.m2:147:20-147:64
│ │ │ │ -     1     1.24    ../../m2/matrix1.m2:276:4-277:73
│ │ │ │ -     1     1.20    ../../m2/matrix1.m2:111:5-111:91
│ │ │ │ -     1     1.08    ../../m2/matrix1.m2:182:4-184:74
│ │ │ │ -     1     .74     ../../m2/modules.m2:279:4-279:52
│ │ │ │ -     20    .51     ../../m2/matrix1.m2:191:14-192:67
│ │ │ │ -     20    .48     ../../m2/matrix1.m2:47:43-47:71
│ │ │ │ -     1     .0037s  elapsed total
│ │ │ │ +o2 = #run  %time  position
│ │ │ │ +     1     93.48  ../../m2/matrix1.m2:279:4-282:58
│ │ │ │ +     1     91.02  ../../m2/matrix1.m2:281:22-281:43
│ │ │ │ +     1     45.78  ../../m2/matrix1.m2:193:25-193:52
│ │ │ │ +     1     31.03  ../../m2/matrix1.m2:114:5-156:72
│ │ │ │ +     1     29.95  ../../m2/matrix1.m2:140:10-155:16
│ │ │ │ +     1     21.54  ../../m2/matrix1.m2:181:4-181:42
│ │ │ │ +     1     20.7   ../../m2/matrix1.m2:45:10-49:22
│ │ │ │ +     1     20.23  ../../m2/set.m2:127:5-127:61
│ │ │ │ +     1     3.31   ../../m2/matrix1.m2:112:5-112:29
│ │ │ │ +     1     2.72   ../../m2/matrix1.m2:96:5-109:11
│ │ │ │ +     1     2.44   ../../m2/matrix1.m2:141:13-141:78
│ │ │ │ +     1     1.6    ../../m2/matrix1.m2:111:5-111:91
│ │ │ │ +     1     1.59   ../../m2/matrix1.m2:147:20-147:64
│ │ │ │ +     1     1.52   ../../m2/matrix1.m2:281:7-281:16
│ │ │ │ +     1     1.39   ../../m2/matrix1.m2:98:10-98:46
│ │ │ │ +     1     1.34   ../../m2/matrix1.m2:276:4-277:73
│ │ │ │ +     20    1.21   ../../m2/matrix1.m2:191:14-192:67
│ │ │ │ +     1     1.04   ../../m2/matrix1.m2:182:4-184:74
│ │ │ │ +     1     .90    ../../m2/modules.m2:279:4-279:52
│ │ │ │ +     20    .8     ../../m2/matrix1.m2:47:43-47:71
│ │ │ │ +     1     .004s  elapsed total
│ │ │ │  i3 : coverageSummary
│ │ │ │  
│ │ │ │  o3 = covered lines:
│ │ │ │       ../../m2/lists.m2:145:24-145:32
│ │ │ │       ../../m2/lists.m2:145:34-145:58
│ │ │ │       ../../m2/matrix.m2:12:5-12:35
│ │ │ │       ../../m2/matrix.m2:13:5-13:46
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__K__Rational__Point.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
    
│ │ │                
    i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.168784s (cpu); 0.136714s (thread); 0s (gc)
│ │ │ + -- used 0.275836s (cpu); 0.110111s (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 0.717081s (cpu); 0.414694s (thread); 0s (gc)
│ │ │ + -- used 0.834012s (cpu); 0.328562s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │  
│ │ │  o10 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
│ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ - -- used 3.7226s (cpu); 2.08535s (thread); 0s (gc)
│ │ │ + -- used 4.74283s (cpu); 2.23073s (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 0.168784s (cpu); 0.136714s (thread); 0s (gc)
│ │ │ │ + -- used 0.275836s (cpu); 0.110111s (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 0.717081s (cpu); 0.414694s (thread); 0s (gc)
│ │ │ │ + -- used 0.834012s (cpu); 0.328562s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  The claim that $63 \%$ of the intersections contain a K-rational point can be
│ │ │ │  experimentally tested:
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  o13 : RR (of precision 53)
│ │ │ │  i14 : I=ideal random(n,R);
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c-
│ │ │ │  >degree c);
│ │ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ │ - -- used 3.7226s (cpu); 2.08535s (thread); 0s (gc)
│ │ │ │ + -- used 4.74283s (cpu); 2.23073s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 58
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommKKRRaattiioonnaallPPooiinntt:: **********
│ │ │ │      * randomKRationalPoint(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_read__Directory.html
│ │ │ @@ -68,38 +68,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11562-0/0
    
│ │ │ +o1 = /tmp/M2-13072-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11562-0/0
    
│ │ │ +o2 = /tmp/M2-13072-0/0
│ │ │              
    
│ │ │                
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11562-0/0/foo
│ │ │ +o3 = /tmp/M2-13072-0/0/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/M2-11562-0/0
│ │ │ │ +o1 = /tmp/M2-13072-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11562-0/0
│ │ │ │ +o2 = /tmp/M2-13072-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11562-0/0/foo
│ │ │ │ +o3 = /tmp/M2-13072-0/0/foo
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : readDirectory dir
│ │ │ │  
│ │ │ │ -o4 = {., .., foo}
│ │ │ │ +o4 = {.., ., foo}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : removeFile fn
│ │ │ │  i6 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_m_o_v_e_D_i_r_e_c_t_o_r_y -- remove a directory
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_reading_spfiles.html
│ │ │ @@ -52,22 +52,22 @@
│ │ │        
    
│ │ │  Sometimes a file will contain a single expression whose value you wish to have access to.  For example, it might be a polynomial produced by another program.  The function get can be used to obtain the entire contents of a file as a single string.  We illustrate this here with a file whose name is expression.        
    
│ │ │  First we create the file by writing the desired text to it.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11104-0/0
    
│ │ │ +o1 = /tmp/M2-12134-0/0
│ │ │              
    
│ │ │                
    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/M2-11104-0/0
│ │ │ +o2 = /tmp/M2-12134-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │  Now we get the contents of the file, as a single string.        
│ │ │            
│ │ │ @@ -116,15 +116,15 @@
│ │ │  Often a file will contain code written in the Macaulay2 language. Let's create such a file.        
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11104-0/0
│ │ │ +o7 = /tmp/M2-12134-0/0
│ │ │  
│ │ │  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/M2-11104-0/0
│ │ │ │ +o1 = /tmp/M2-12134-0/0
│ │ │ │  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/M2-11104-0/0
│ │ │ │ +o2 = /tmp/M2-12134-0/0
│ │ │ │  
│ │ │ │  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/M2-11104-0/0
│ │ │ │ +o7 = /tmp/M2-12134-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  Now verify that it contains the desired text with _g_e_t.
│ │ │ │  i8 : get fn
│ │ │ │  
│ │ │ │  o8 = sample = 2^100
│ │ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_readlink.html
│ │ │ @@ -68,15 +68,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the resolved path to a symbolic link, or null if the file
│ │ │ │              was not a symbolic link.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : p = temporaryFileName ()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11803-0/0
│ │ │ │ +o1 = /tmp/M2-13553-0/0
│ │ │ │  i2 : symlinkFile ("foo", p)
│ │ │ │  i3 : readlink p
│ │ │ │  
│ │ │ │  o3 = foo
│ │ │ │  i4 : removeFile p
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_l_p_a_t_h -- convert a filename to one passing through no symbolic links
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_realpath.html
│ │ │ @@ -68,57 +68,57 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-11803-0/0
    
│ │ │ +o1 = /tmp/M2-13553-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile ("foo", p)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -130,15 +130,15 @@
│ │ │          
    
│ │ │                
    i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-10188-0/86-rundir/
    
│ │ │ +o1 = /tmp/M2-10308-0/86-rundir/
│ │ │              
    
│ │ │                
    i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11822-0/0
    
│ │ │ +o2 = /tmp/M2-13592-0/0
│ │ │              
    
│ │ │                
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11822-0/1
    
│ │ │ +o3 = /tmp/M2-13592-0/1
│ │ │              
    
│ │ │                
    i4 : symlinkFile(p,q)
    
│ │ │              
    
│ │ │                
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-11822-0/0
│ │ │ +o5 = /tmp/M2-13592-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
    
│ │ │                
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-11822-0/0
    
│ │ │ +o6 = /tmp/M2-13592-0/0
│ │ │              
    
│ │ │                
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-11822-0/0
    
│ │ │ +o7 = /tmp/M2-13592-0/0
│ │ │              
    
│ │ │                
    i8 : removeFile p
    
│ │ │              
    
│ │ │          
    The empty string is interpreted as a reference to the current directory.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-10188-0/86-rundir/
    
│ │ │ +o10 = /tmp/M2-10308-0/86-rundir/
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  Every component of the path must exist in the file system and be accessible to the user. Terminal slashes will be dropped.  Warning: under most operating systems, the value returned is an absolute path (one starting at the root of the file system), but under Solaris, this system call may, in certain circumstances, return a relative path when given a relative path.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,39 +12,39 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename, or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, a pathname to fn passing through no symbolic links, and
│ │ │ │              ending with a slash if fn refers to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : realpath "."
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10188-0/86-rundir/
│ │ │ │ +o1 = /tmp/M2-10308-0/86-rundir/
│ │ │ │  i2 : p = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11822-0/0
│ │ │ │ +o2 = /tmp/M2-13592-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11822-0/1
│ │ │ │ +o3 = /tmp/M2-13592-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11822-0/0
│ │ │ │ +o5 = /tmp/M2-13592-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11822-0/0
│ │ │ │ +o6 = /tmp/M2-13592-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11822-0/0
│ │ │ │ +o7 = /tmp/M2-13592-0/0
│ │ │ │  i8 : removeFile p
│ │ │ │  i9 : removeFile q
│ │ │ │  The empty string is interpreted as a reference to the current directory.
│ │ │ │  i10 : realpath ""
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-10188-0/86-rundir/
│ │ │ │ +o10 = /tmp/M2-10308-0/86-rundir/
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Every component of the path must exist in the file system and be accessible to
│ │ │ │  the user. Terminal slashes will be dropped. Warning: under most operating
│ │ │ │  systems, the value returned is an absolute path (one starting at the root of
│ │ │ │  the file system), but under Solaris, this system call may, in certain
│ │ │ │  circumstances, return a relative path when given a relative path.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_register__Finalizer.html
│ │ │ @@ -76,23 +76,23 @@
│ │ │                  		
│ │ │                
    i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
    
│ │ │              
    
│ │ │                
    i2 : collectGarbage() 
│ │ │ ---finalization: (1)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (2)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (3)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (4)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (5)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (6)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (8)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (9)[7]: -- finalizing sequence #8 --
    
│ │ │ +--finalization: (1)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (2)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (3)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (4)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (5)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (6)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (7)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (8)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (9)[4]: -- finalizing sequence #5 --
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  This function should mainly be used for debugging.  Having a large number of finalizers might degrade the performance of the program.  Moreover, registering two or more objects that are members of a circular chain of pointers for finalization will result in a memory leak, with none of the objects in the chain being freed, even if nothing else points to any of them.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,23 +14,23 @@
│ │ │ │      * Consequences:
│ │ │ │            o A finalizer is registered with the garbage collector to print a
│ │ │ │              string when that object is collected as garbage
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-
│ │ │ │  - finalizing sequence #"|i|" --"))
│ │ │ │  i2 : collectGarbage()
│ │ │ │ ---finalization: (1)[5]: -- finalizing sequence #6 --
│ │ │ │ ---finalization: (2)[1]: -- finalizing sequence #2 --
│ │ │ │ ---finalization: (3)[8]: -- finalizing sequence #9 --
│ │ │ │ ---finalization: (4)[3]: -- finalizing sequence #4 --
│ │ │ │ ---finalization: (5)[2]: -- finalizing sequence #3 --
│ │ │ │ ---finalization: (6)[4]: -- finalizing sequence #5 --
│ │ │ │ ---finalization: (7)[6]: -- finalizing sequence #7 --
│ │ │ │ ---finalization: (8)[0]: -- finalizing sequence #1 --
│ │ │ │ ---finalization: (9)[7]: -- finalizing sequence #8 --
│ │ │ │ +--finalization: (1)[7]: -- finalizing sequence #8 --
│ │ │ │ +--finalization: (2)[2]: -- finalizing sequence #3 --
│ │ │ │ +--finalization: (3)[6]: -- finalizing sequence #7 --
│ │ │ │ +--finalization: (4)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (5)[5]: -- finalizing sequence #6 --
│ │ │ │ +--finalization: (6)[3]: -- finalizing sequence #4 --
│ │ │ │ +--finalization: (7)[8]: -- finalizing sequence #9 --
│ │ │ │ +--finalization: (8)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (9)[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
│ │ │ @@ -71,29 +71,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10776-0/0
    
│ │ │ +o1 = /tmp/M2-11466-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-10776-0/0
    
│ │ │ +o2 = /tmp/M2-11466-0/0
│ │ │              
    
│ │ │                
    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/M2-10776-0/0
│ │ │ │ +o1 = /tmp/M2-11466-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10776-0/0
│ │ │ │ +o2 = /tmp/M2-11466-0/0
│ │ │ │  i3 : readDirectory dir
│ │ │ │  
│ │ │ │ -o3 = {., ..}
│ │ │ │ +o3 = {.., .}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d_D_i_r_e_c_t_o_r_y -- read the contents of a directory
│ │ │ │      * _m_a_k_e_D_i_r_e_c_t_o_r_y -- make a directory
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__Path.html
│ │ │ @@ -65,22 +65,22 @@
│ │ │          
    Description
    
│ │ │          
    This string may be concatenated with an absolute path to get one understandable by external programs.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10280-0/0
    
│ │ │ +o1 = /tmp/M2-10470-0/0
│ │ │              
    
│ │ │                
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-10280-0/0
    
│ │ │ +o2 = /tmp/M2-10470-0/0
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    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/M2-10280-0/0
│ │ │ │ +o1 = /tmp/M2-10470-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10280-0/0
│ │ │ │ +o2 = /tmp/M2-10470-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_U_R_I
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_P_a_t_h is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2025:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__U__R__I.html
│ │ │ @@ -65,22 +65,22 @@
│ │ │          
      Description
      
│ │ │          
      This string may be concatenated with an absolute path to get one understandable by an external browser.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11505-0/0
      
│ │ │ +o1 = /tmp/M2-12955-0/0
│ │ │              
      
│ │ │                
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-11505-0/0
      
│ │ │ +o2 = file:///tmp/M2-12955-0/0
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │          
    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/M2-11505-0/0
│ │ │ │ +o1 = /tmp/M2-12955-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-11505-0/0
│ │ │ │ +o2 = file:///tmp/M2-12955-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_P_a_t_h
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_U_R_I is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2041:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_saving_sppolynomials_spand_spmatrices_spin_spfiles.html
│ │ │ @@ -90,22 +90,22 @@
│ │ │  o4 : R-module, submodule of R
│ │ │              
    
│ │ │                
    i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-11353-0/0
    
│ │ │ +o5 = /tmp/M2-12643-0/0
│ │ │              
    
│ │ │                
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11353-0/0
│ │ │ +o6 = /tmp/M2-12643-0/0
│ │ │  
│ │ │  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/M2-11353-0/0
│ │ │ │ +o5 = /tmp/M2-12643-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11353-0/0
│ │ │ │ +o6 = /tmp/M2-12643-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : get f
│ │ │ │  
│ │ │ │  o7 = (x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix {{x^2,
│ │ │ │       x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}})
│ │ │ │  i8 : (p',m',M') = value get f
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_serial__Number.html
│ │ │ @@ -68,22 +68,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1426273
    
│ │ │ +o1 = 1526273
│ │ │              
    
│ │ │                
    i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1426275
    
│ │ │ +o2 = 1526275
│ │ │              
    
│ │ │                
    i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,18 +10,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o x
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the serial number of x
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : serialNumber asdf
│ │ │ │  
│ │ │ │ -o1 = 1426273
│ │ │ │ +o1 = 1526273
│ │ │ │  i2 : serialNumber foo
│ │ │ │  
│ │ │ │ -o2 = 1426275
│ │ │ │ +o2 = 1526275
│ │ │ │  i3 : serialNumber ZZ
│ │ │ │  
│ │ │ │  o3 = 1000050
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _s_e_r_i_a_l_N_u_m_b_e_r is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_solve.html
│ │ │ @@ -366,21 +366,21 @@
│ │ │                  		
│ │ │                
    i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i30 : time X = solve(A,B);
│ │ │ - -- used 0.000215955s (cpu); 0.000209113s (thread); 0s (gc)
    
│ │ │ + -- used 0.000256235s (cpu); 0.000227845s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000155772s (cpu); 0.000155762s (thread); 0s (gc)
    
│ │ │ + -- used 0.000121023s (cpu); 0.000120607s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │ @@ -411,21 +411,21 @@
│ │ │              
    		
│ │ │                
    i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i38 : time X = solve(A,B);
│ │ │ - -- used 0.233s (cpu); 0.233s (thread); 0s (gc)
    
│ │ │ + -- used 0.143751s (cpu); 0.143759s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.492751s (cpu); 0.295313s (thread); 0s (gc)
    
│ │ │ + -- used 0.144679s (cpu); 0.144699s (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 0.000215955s (cpu); 0.000209113s (thread); 0s (gc)
│ │ │ │ + -- used 0.000256235s (cpu); 0.000227845s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.000155772s (cpu); 0.000155762s (thread); 0s (gc)
│ │ │ │ + -- used 0.000121023s (cpu); 0.000120607s (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 0.233s (cpu); 0.233s (thread); 0s (gc)
│ │ │ │ + -- used 0.143751s (cpu); 0.143759s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.492751s (cpu); 0.295313s (thread); 0s (gc)
│ │ │ │ + -- used 0.144679s (cpu); 0.144699s (thread); 0s (gc)
│ │ │ │  i40 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ │  
│ │ │ │  o40 : RR (of precision 100)
│ │ │ │  Giving the option ClosestFit=>true, in the case when the field is RR or CC,
│ │ │ │  uses a least squares algorithm to find a best fit solution.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -80,93 +80,93 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11144-0/0/
    
│ │ │ +o1 = /tmp/M2-12214-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-11144-0/1/
    
│ │ │ +o2 = /tmp/M2-12214-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-11144-0/0/a/
    
│ │ │ +o3 = /tmp/M2-12214-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-11144-0/0/b/
    
│ │ │ +o4 = /tmp/M2-12214-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-11144-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-12214-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-11144-0/0/a/f
│ │ │ +o6 = /tmp/M2-12214-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11144-0/0/a/g
│ │ │ +o7 = /tmp/M2-12214-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-11144-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-12214-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11144-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11144-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11144-0/1/a/f
    
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12214-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12214-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12214-0/1/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/M2-11144-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11144-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11144-0/1/a/f
    
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12214-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12214-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12214-0/1/b/c/g
│ │ │              
    
│ │ │  Now we remove the files and directories we created.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,53 +30,53 @@
│ │ │ │            o The directory tree rooted at src is duplicated by a directory tree
│ │ │ │              rooted at dst. The files in the source tree are represented by
│ │ │ │              relative symbolic links in the destination tree to the original
│ │ │ │              files in the source tree.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11144-0/0/
│ │ │ │ +o1 = /tmp/M2-12214-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11144-0/1/
│ │ │ │ +o2 = /tmp/M2-12214-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11144-0/0/a/
│ │ │ │ +o3 = /tmp/M2-12214-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11144-0/0/b/
│ │ │ │ +o4 = /tmp/M2-12214-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-11144-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-12214-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11144-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-12214-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11144-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-12214-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-11144-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-12214-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-11144-0/1/b/c/g
│ │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-11144-0/1/a/g
│ │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-11144-0/1/a/f
│ │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-12214-0/1/a/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-12214-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-12214-0/1/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/M2-11144-0/1/b/c/g
│ │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-11144-0/1/a/g
│ │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-11144-0/1/a/f
│ │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-12214-0/1/a/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-12214-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-12214-0/1/b/c/g
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o12 = rm
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : scan(reverse findFiles src, rm)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__File.html
│ │ │ @@ -72,15 +72,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            o dst, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o a symbolic link at the location in the directory tree specified by
│ │ │ │              dst is created, pointing to src
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11201-0/0
│ │ │ │ +o1 = /tmp/M2-12331-0/0
│ │ │ │  i2 : symlinkFile("qwert", fn)
│ │ │ │  i3 : fileExists fn
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : readlink fn
│ │ │ │  
│ │ │ │  o4 = qwert
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_temporary__File__Name.html
│ │ │ @@ -64,22 +64,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The file name is so unique that even with various suffixes appended, no collision with existing files will occur.  The files will be removed when the program terminates, unless it terminates as the result of an error.        
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11201-0/0
    
│ │ │ +o1 = /tmp/M2-12331-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile("qwert", fn)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-12166-0/0.tex
    
│ │ │ +o1 = /tmp/M2-14306-0/0.tex
│ │ │              
    
│ │ │                
    i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-12166-0/1.html
    
│ │ │ +o2 = /tmp/M2-14306-0/1.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/M2-12166-0/0.tex
│ │ │ │ +o1 = /tmp/M2-14306-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12166-0/1.html
│ │ │ │ +o2 = /tmp/M2-14306-0/1.html
│ │ │ │  This function will work under Unix, and also under Windows if you have a
│ │ │ │  directory on the same drive called /tmp.
│ │ │ │  If the name of the temporary file will be given to an external program, it may
│ │ │ │  be necessary to concatenate it with _r_o_o_t_P_a_t_h or _r_o_o_t_U_R_I to enable the external
│ │ │ │  program to find the file.
│ │ │ │  The temporary file name is derived from the value of the environment variable
│ │ │ │  TMPDIR, if it has one.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_time.html
│ │ │ @@ -59,15 +59,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value of e.  The time used by the the current thread and garbage collection during the evaluation of e is also shown.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time 3^30
│ │ │ - -- used 1.6912e-05s (cpu); 9.258e-06s (thread); 0s (gc)
│ │ │ + -- used 2.6805e-05s (cpu); 6.149e-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 1.6912e-05s (cpu); 9.258e-06s (thread); 0s (gc)
│ │ │ │ + -- used 2.6805e-05s (cpu); 6.149e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_timing.html
│ │ │ @@ -54,24 +54,24 @@
│ │ │  timing e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of cpu timing used, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000014818 seconds
│ │ │ +     -- .000024213 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000014818, 205891132094649}
    
│ │ │ +o2 = Time{.000024213, 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
│ │ │ │ -     -- .000014818 seconds
│ │ │ │ +     -- .000024213 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000014818, 205891132094649}
│ │ │ │ +o2 = Time{.000024213, 205891132094649}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_i_m_e -- the class of all timing results
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_version.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │                 "memtailor version" => 1.1
│ │ │                 "mpfi version" => 1.5.4
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.0
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.63+deb13-amd64
│ │ │ +               "operating system release" => 6.12.63+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings PruneComplex BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomPlaneCurves RandomSpaceCurves RandomGenus14Curves RandomCanonicalCurves RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing OldPolyhedra OldToricVectorBundles K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization NonminimalComplexes CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC MatrixFactorizations PathSignatures
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.13.11
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2022.3
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │                 "memtailor version" => 1.1
│ │ │ │                 "mpfi version" => 1.5.4
│ │ │ │                 "mpfr version" => 4.2.2
│ │ │ │                 "mpsolve version" => 3.2.2
│ │ │ │                 "mysql version" => not present
│ │ │ │                 "normaliz version" => 3.11.0
│ │ │ │                 "ntl version" => 11.5.1
│ │ │ │ -               "operating system release" => 6.12.63+deb13-amd64
│ │ │ │ +               "operating system release" => 6.12.63+deb13-cloud-amd64
│ │ │ │                 "operating system" => Linux
│ │ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic
│ │ │ │  Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition
│ │ │ │  FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato
│ │ │ │  ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure
│ │ │ │  HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra
│ │ │ │  Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes
│ │ ├── ./usr/share/doc/Macaulay2/Markov/example-output/___Markov.out
│ │ │ @@ -70,15 +70,15 @@
│ │ │       |   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2|   1,2,2,2 2,2,2,1    1,2,2,1 2,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │       |- p       p        + p       p       |- p       p        + p       p       |
│ │ │       |   1,1,2,1 1,2,1,1    1,1,1,1 1,2,2,1|   1,1,2,2 1,2,1,2    1,1,1,2 1,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │  
│ │ │  i8 : time netList primaryDecomposition J
│ │ │ - -- used 3.35176s (cpu); 1.65758s (thread); 0s (gc)
│ │ │ + -- used 4.57331s (cpu); 1.86854s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ ├── ./usr/share/doc/Macaulay2/Markov/html/index.html
│ │ │ @@ -161,15 +161,15 @@
│ │ │          
      
│ │ │            
      This ideal has 5 primary components.  The first is the one that has statistical significance. The significance of the other components is still poorly understood.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i8 : time netList primaryDecomposition J
│ │ │ - -- used 3.35176s (cpu); 1.65758s (thread); 0s (gc)
│ │ │ + -- used 4.57331s (cpu); 1.86854s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │ ├── html2text {}
│ │ │ │ @@ -102,15 +102,15 @@
│ │ │ │  1,2,2,2|
│ │ │ │       +-------------------------------------+-----------------------------------
│ │ │ │  --+
│ │ │ │  This ideal has 5 primary components. The first is the one that has statistical
│ │ │ │  significance. The significance of the other components is still poorly
│ │ │ │  understood.
│ │ │ │  i8 : time netList primaryDecomposition J
│ │ │ │ - -- used 3.35176s (cpu); 1.65758s (thread); 0s (gc)
│ │ │ │ + -- used 4.57331s (cpu); 1.86854s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_sp__A__S__M_spvarieties.out
│ │ │ @@ -212,17 +212,17 @@
│ │ │        | 1 -1 1 |
│ │ │        | 0 1  0 |
│ │ │  
│ │ │                 3       3
│ │ │  o22 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i23 : time schubertRegularity B
│ │ │ - -- used 0.0981507s (cpu); 0.0346871s (thread); 0s (gc)
│ │ │ + -- used 0.128136s (cpu); 0.0424302s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0158455s (cpu); 0.0158476s (thread); 0s (gc)
│ │ │ + -- used 0.018319s (cpu); 0.0183267s (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 0.000288371s (cpu); 0.000285145s (thread); 0s (gc)
│ │ │ + -- used 0.000371926s (cpu); 0.000362901s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
│ │ │  
│ │ │  i17 : time regularity comodule I
│ │ │ - -- used 0.0150993s (cpu); 0.0151029s (thread); 0s (gc)
│ │ │ + -- used 0.0178944s (cpu); 0.0179072s (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 0.00485171s (cpu); 0.00484886s (thread); 0s (gc)
│ │ │ + -- used 0.00557541s (cpu); 0.00556997s (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 0.00236407s (cpu); 0.00236466s (thread); 0s (gc)
│ │ │ + -- used 0.00291745s (cpu); 0.00291802s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_sp__A__S__M_spvarieties.html
│ │ │ @@ -383,23 +383,23 @@
│ │ │          
      
│ │ │            
      Additionally, this package facilitates investigating homological invariants of ASM ideals such as regularity (schubertRegularity) and codimension (schubertCodim). efficiently by computing the associated invariants for their antidiagonal initial ideals, which are known to be squarefree by [Wei17]. Therefore the extremal Betti numbers (which encode regularity, depth, and projective dimension) of ASM ideals coincide with those of their antidiagonal initial ideals by [CV20].
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i23 : time schubertRegularity B
│ │ │ - -- used 0.0981507s (cpu); 0.0346871s (thread); 0s (gc)
│ │ │ + -- used 0.128136s (cpu); 0.0424302s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
      
│ │ │              
      
│ │ │                
      i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0158455s (cpu); 0.0158476s (thread); 0s (gc)
│ │ │ + -- used 0.018319s (cpu); 0.0183267s (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 0.0981507s (cpu); 0.0346871s (thread); 0s (gc)
│ │ │ │ + -- used 0.128136s (cpu); 0.0424302s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.0158455s (cpu); 0.0158476s (thread); 0s (gc)
│ │ │ │ + -- used 0.018319s (cpu); 0.0183267s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = 1
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg AASSMM vvaarriieettiieess **********
│ │ │ │      * _i_s_P_a_r_t_i_a_l_A_S_M_(_M_a_t_r_i_x_) -- whether a matrix is a partial alternating sign
│ │ │ │        matrix
│ │ │ │      * _p_a_r_t_i_a_l_A_S_M_T_o_A_S_M_(_M_a_t_r_i_x_) -- extend a partial alternating sign matrix to an
│ │ │ │        alternating sign matrix
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_spmatrix_sp__Schubert_spvarieties.html
│ │ │ @@ -315,23 +315,23 @@
│ │ │          
      
│ │ │            
      Finally, this package contains functions for investigating homological invariants of matrix Schubert varieties efficiently through combinatorial algorithms produced in [PSW24] via schubertRegularityschubertCodim.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i16 : time schubertRegularity p
│ │ │ - -- used 0.000288371s (cpu); 0.000285145s (thread); 0s (gc)
│ │ │ + -- used 0.000371926s (cpu); 0.000362901s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
      
│ │ │              
      
│ │ │                
      i17 : time regularity comodule I
│ │ │ - -- used 0.0150993s (cpu); 0.0151029s (thread); 0s (gc)
│ │ │ + -- used 0.0178944s (cpu); 0.0179072s (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 0.000288371s (cpu); 0.000285145s (thread); 0s (gc)
│ │ │ │ + -- used 0.000371926s (cpu); 0.000362901s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 5
│ │ │ │  i17 : time regularity comodule I
│ │ │ │ - -- used 0.0150993s (cpu); 0.0151029s (thread); 0s (gc)
│ │ │ │ + -- used 0.0178944s (cpu); 0.0179072s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 5
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg mmaattrriixx SScchhuubbeerrtt vvaarriieettiieess **********
│ │ │ │      * _a_n_t_i_D_i_a_g_I_n_i_t_(_L_i_s_t_) -- compute the (unique) antidiagonal initial ideal of
│ │ │ │        an ASM ideal
│ │ │ │      * _r_a_n_k_T_a_b_l_e_(_L_i_s_t_) -- compute a table of rank conditions that determines the
│ │ │ │        corresponding ASM or matrix Schubert variety
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html
│ │ │ @@ -80,28 +80,28 @@
│ │ │  
│ │ │  o1 : List
      
│ │ │              
      
│ │ │                
      i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.00485171s (cpu); 0.00484886s (thread); 0s (gc)
│ │ │ + -- used 0.00557541s (cpu); 0.00556997s (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 0.00236407s (cpu); 0.00236466s (thread); 0s (gc)
│ │ │ + -- used 0.00291745s (cpu); 0.00291802s (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 0.00485171s (cpu); 0.00484886s (thread); 0s (gc)
│ │ │ │ + -- used 0.00557541s (cpu); 0.00556997s (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 0.00236407s (cpu); 0.00236466s (thread); 0s (gc)
│ │ │ │ + -- used 0.00291745s (cpu); 0.00291802s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o3 : QQ[x ..x ]
│ │ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid.out
│ │ │ @@ -51,20 +51,20 @@
│ │ │  i9 : keys R10.cache
│ │ │  
│ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time isWellDefined R10
│ │ │ - -- used 0.0519788s (cpu); 0.0541644s (thread); 0s (gc)
│ │ │ + -- used 0.0639862s (cpu); 0.065081s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.185474s (cpu); 0.0674581s (thread); 0s (gc)
│ │ │ + -- used 0.288175s (cpu); 0.101388s (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 0.000380654s (cpu); 0.000209262s (thread); 0s (gc)
│ │ │ + -- used 0.000432409s (cpu); 0.000242145s (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);
│ │ │ - -- .101577s elapsed
│ │ │ + -- .0683583s 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 0.134387s (cpu); 0.0614875s (thread); 0s (gc)
│ │ │ + -- used 0.167526s (cpu); 0.0639047s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : #autF7
│ │ │  
│ │ │  o8 = 168
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_has__Minor.out
│ │ │ @@ -9,12 +9,12 @@
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : time hasMinor(M6, M5)
│ │ │ - -- used 1.71122s (cpu); 1.27171s (thread); 0s (gc)
│ │ │ + -- used 2.35734s (cpu); 1.39914s (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 0.0158246s (cpu); 0.0128692s (thread); 0s (gc)
│ │ │ + -- used 0.0120472s (cpu); 0.0146803s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -56,15 +56,15 @@
│ │ │  i7 : N = relabel M6
│ │ │  
│ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │  
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 4.45859s (cpu); 2.94211s (thread); 0s (gc)
│ │ │ + -- used 5.43325s (cpu); 3.02341s (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 0.00121479s (cpu); 0.000619232s (thread); 0s (gc)
│ │ │ + -- used 0.00318273s (cpu); 0.000666843s (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
│ │ │ - -- .0185701s elapsed
│ │ │ + -- .018986s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html
│ │ │ @@ -148,23 +148,23 @@
│ │ │  
│ │ │  o9 : List
│ │ │              
      
│ │ │                
      i10 : time isWellDefined R10
│ │ │ - -- used 0.0519788s (cpu); 0.0541644s (thread); 0s (gc)
│ │ │ + -- used 0.0639862s (cpu); 0.065081s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
      
│ │ │              
      
│ │ │                
      i11 : time fVector R10
│ │ │ - -- used 0.185474s (cpu); 0.0674581s (thread); 0s (gc)
│ │ │ + -- used 0.288175s (cpu); 0.101388s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -182,15 +182,15 @@
│ │ │  
│ │ │  o12 : List
      
│ │ │              
      
│ │ │                
      i13 : time fVector R10
│ │ │ - -- used 0.000380654s (cpu); 0.000209262s (thread); 0s (gc)
│ │ │ + -- used 0.000432409s (cpu); 0.000242145s (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 0.0519788s (cpu); 0.0541644s (thread); 0s (gc)
│ │ │ │ + -- used 0.0639862s (cpu); 0.065081s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.185474s (cpu); 0.0674581s (thread); 0s (gc)
│ │ │ │ + -- used 0.288175s (cpu); 0.101388s (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 0.000380654s (cpu); 0.000209262s (thread); 0s (gc)
│ │ │ │ + -- used 0.000432409s (cpu); 0.000242145s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_all__Minors.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o2 : Matroid
      
│ │ │              
      
│ │ │                
      i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .101577s elapsed
      
│ │ │ + -- .0683583s 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);
│ │ │ │ - -- .101577s elapsed
│ │ │ │ + -- .0683583s elapsed
│ │ │ │  i4 : #L
│ │ │ │  
│ │ │ │  o4 = 64
│ │ │ │  i5 : netList L_{0..4}
│ │ │ │  
│ │ │ │       +----------+-------+
│ │ │ │  o5 = |set {5, 3}|set {2}|
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_get__Isos.html
│ │ │ @@ -135,15 +135,15 @@
│ │ │  
│ │ │  o6 : Matroid
│ │ │              
      
│ │ │                
      i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.134387s (cpu); 0.0614875s (thread); 0s (gc)
      
│ │ │ + -- used 0.167526s (cpu); 0.0639047s (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 0.134387s (cpu); 0.0614875s (thread); 0s (gc)
│ │ │ │ + -- used 0.167526s (cpu); 0.0639047s (thread); 0s (gc)
│ │ │ │  i8 : #autF7
│ │ │ │  
│ │ │ │  o8 = 168
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ │ │        isomorphic matroids
│ │ │ │      * _q_u_i_c_k_I_s_o_m_o_r_p_h_i_s_m_T_e_s_t -- quick checks for isomorphism between matroids
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_has__Minor.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 = false
│ │ │              
      
│ │ │                
      i3 : time hasMinor(M6, M5)
│ │ │ - -- used 1.71122s (cpu); 1.27171s (thread); 0s (gc)
│ │ │ + -- used 2.35734s (cpu); 1.39914s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
      
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │       elements, a "matroid" of rank 5 on 15 elements)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : time hasMinor(M6, M5)
│ │ │ │ - -- used 1.71122s (cpu); 1.27171s (thread); 0s (gc)
│ │ │ │ + -- used 2.35734s (cpu); 1.39914s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r -- minor of matroid
│ │ │ │      * _i_s_B_i_n_a_r_y -- whether a matroid is representable over F_2
│ │ │ │  ********** WWaayyss ttoo uussee hhaassMMiinnoorr:: **********
│ │ │ │      * hasMinor(Matroid,Matroid)
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_isomorphism_lp__Matroid_cm__Matroid_rp.html
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o4 : Matroid
    
│ │ │              
    
│ │ │                
    i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0158246s (cpu); 0.0128692s (thread); 0s (gc)
│ │ │ + -- used 0.0120472s (cpu); 0.0146803s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -164,15 +164,15 @@
│ │ │  
│ │ │  o7 : Matroid
    
│ │ │              
    
│ │ │                
    i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 4.45859s (cpu); 2.94211s (thread); 0s (gc)
│ │ │ + -- used 5.43325s (cpu); 3.02341s (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 0.0158246s (cpu); 0.0128692s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120472s (cpu); 0.0146803s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{0 => 1}
│ │ │ │                 1 => 0
│ │ │ │                 2 => 3
│ │ │ │                 3 => 2
│ │ │ │                 4 => 6
│ │ │ │                 5 => 5
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  o6 : HashTable
│ │ │ │  i7 : N = relabel M6
│ │ │ │  
│ │ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ │ - -- used 4.45859s (cpu); 2.94211s (thread); 0s (gc)
│ │ │ │ + -- used 5.43325s (cpu); 3.02341s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HashTable{0 => 11 }
│ │ │ │                 1 => 0
│ │ │ │                 2 => 1
│ │ │ │                 3 => 6
│ │ │ │                 4 => 9
│ │ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_quick__Isomorphism__Test.html
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.00121479s (cpu); 0.000619232s (thread); 0s (gc)
│ │ │ + -- used 0.00318273s (cpu); 0.000666843s (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 0.00121479s (cpu); 0.000619232s (thread); 0s (gc)
│ │ │ │ + -- used 0.00318273s (cpu); 0.000666843s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : value oo === false
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_set__Representation.html
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime fVector M
│ │ │ - -- .0185701s elapsed
│ │ │ + -- .018986s 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
│ │ │ │ - -- .0185701s elapsed
│ │ │ │ + -- .018986s elapsed
│ │ │ │  
│ │ │ │  o6 = HashTable{0 => 1 }
│ │ │ │                 1 => 6
│ │ │ │                 2 => 15
│ │ │ │                 3 => 20
│ │ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/dump/rawdocumentation.dump
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Jan 19 15:01:19 2026
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Jan 19 15:01:18 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  ZGVjb21wb3NlKElkZWFsLFN0cmF0ZWd5PT4uLi4p
│ │ ├── ./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 .000993042)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0148775)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000337994)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000020008)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .00126347)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0424773)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000432661)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000026956)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000641443
│ │ │ - -- .0244645s elapsed
│ │ │ + --  Time taken : .000871493
│ │ │ + -- .0376006s 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)
│ │ │ - -- .000455493s elapsed
│ │ │ + -- .000574959s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0122175s elapsed
│ │ │ + -- .015733s 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)
│ │ │ - -- .0670879s elapsed
│ │ │ + -- .0806032s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00171664s elapsed
│ │ │ + -- .00205325s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00127077s elapsed
│ │ │ + -- .00150585s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -72,21 +72,21 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .000993042)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0148775)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000337994)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000020008)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .00126347)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0424773)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000432661)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000026956)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000641443
│ │ │ - -- .0244645s elapsed
    
│ │ │ + --  Time taken : .000871493
│ │ │ + -- .0376006s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,24 +11,23 @@
│ │ │ │  i1 : debug MinimalPrimes
│ │ │ │  i2 : R = ZZ/101[w..z];
│ │ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid
│ │ │ │  {Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ │ -  Strategy: Linear            (time .000993042)  #primes = 0 #prunedViaCodim =
│ │ │ │ +  Strategy: Linear            (time .00126347)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Birational        (time .0424773)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Factorization     (time .000432661)  #primes = 0 #prunedViaCodim =
│ │ │ │  0
│ │ │ │ -  Strategy: Birational        (time .0148775)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Factorization     (time .000337994)  #primes = 0 #prunedViaCodim =
│ │ │ │ -0
│ │ │ │ -  Strategy: DecomposeMonomials(time .000020008)  #primes = 1 #prunedViaCodim =
│ │ │ │ +  Strategy: DecomposeMonomials(time .000026956)  #primes = 1 #prunedViaCodim =
│ │ │ │  0
│ │ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ │ - --  Time taken : .000641443
│ │ │ │ - -- .0244645s elapsed
│ │ │ │ + --  Time taken : .000871493
│ │ │ │ + -- .0376006s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n_(_._._._,_S_t_r_a_t_e_g_y_=_>_._._._)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _H_y_b_r_i_d is a _s_e_l_f_ _i_n_i_t_i_a_l_i_z_i_n_g_ _t_y_p_e, with ancestor classes _L_i_s_t <
│ │ │ │  _V_i_s_i_b_l_e_L_i_s_t < _B_a_s_i_c_L_i_s_t < _T_h_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html
│ │ │ @@ -131,25 +131,25 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000455493s elapsed
│ │ │ + -- .000574959s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0122175s elapsed
│ │ │ + -- .015733s 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)
│ │ │ │ - -- .000455493s elapsed
│ │ │ │ + -- .000574959s elapsed
│ │ │ │  
│ │ │ │  o6 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ │ - -- .0122175s elapsed
│ │ │ │ + -- .015733s elapsed
│ │ │ │  
│ │ │ │  o7 = ideal (c, b, a)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  For another example, see _P_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n.
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  Eisenbud, Huneke, Vasconcelos, Invent. Math. 110 207-235 (1992).
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html
│ │ │ @@ -125,31 +125,31 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .0670879s elapsed
│ │ │ + -- .0806032s elapsed
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00171664s elapsed
│ │ │ + -- .00205325s elapsed
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00127077s elapsed
│ │ │ + -- .00150585s 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)
│ │ │ │ - -- .0670879s elapsed
│ │ │ │ + -- .0806032s elapsed
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ │ - -- .00171664s elapsed
│ │ │ │ + -- .00205325s elapsed
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ │ - -- .00127077s elapsed
│ │ │ │ + -- .00150585s elapsed
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_d_i_c_a_l -- the radical of an ideal
│ │ │ │  ********** WWaayyss ttoo uussee rraaddiiccaallCCoonnttaaiinnmmeenntt:: **********
│ │ │ │      * radicalContainment(Ideal,Ideal)
│ │ │ │      * radicalContainment(RingElement,Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out
│ │ │ @@ -57,29 +57,29 @@
│ │ │  i9 : J = ideal vars U
│ │ │  
│ │ │  o9 = ideal (a, b, c)
│ │ │  
│ │ │  o9 : Ideal of U
│ │ │  
│ │ │  i10 : time multiReesIdeal J
│ │ │ - -- used 0.125484s (cpu); 0.0765357s (thread); 0s (gc)
│ │ │ + -- used 0.298186s (cpu); 0.107235s (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 0.0519675s (cpu); 0.0204914s (thread); 0s (gc)
│ │ │ + -- used 0.108381s (cpu); 0.0322917s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html
│ │ │ @@ -173,15 +173,15 @@
│ │ │  
│ │ │  o9 : Ideal of U
    
│ │ │              
    
│ │ │                
    i10 : time multiReesIdeal J
│ │ │ - -- used 0.125484s (cpu); 0.0765357s (thread); 0s (gc)
│ │ │ + -- used 0.298186s (cpu); 0.107235s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ @@ -190,15 +190,15 @@
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
    
│ │ │              
    
│ │ │                
    i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0519675s (cpu); 0.0204914s (thread); 0s (gc)
│ │ │ + -- used 0.108381s (cpu); 0.0322917s (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 0.125484s (cpu); 0.0765357s (thread); 0s (gc)
│ │ │ │ + -- used 0.298186s (cpu); 0.107235s (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 0.0519675s (cpu); 0.0204914s (thread); 0s (gc)
│ │ │ │ + -- used 0.108381s (cpu); 0.0322917s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  
│ │ │  o7 = image | x2 y2 |
│ │ │  
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
│ │ │  
│ │ │  i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.708937s (cpu); 0.55516s (thread); 0s (gc)
│ │ │ + -- used 0.617598s (cpu); 0.368667s (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 0.698845s (cpu); 0.573124s (thread); 0s (gc)
│ │ │ + -- used 0.822287s (cpu); 0.576794s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html
│ │ │ @@ -145,15 +145,15 @@
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
    
│ │ │              
    
│ │ │                
    i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.708937s (cpu); 0.55516s (thread); 0s (gc)
│ │ │ + -- used 0.617598s (cpu); 0.368667s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -186,15 +186,15 @@
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
    
│ │ │              
    
│ │ │                
    i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.698845s (cpu); 0.573124s (thread); 0s (gc)
│ │ │ + -- used 0.822287s (cpu); 0.576794s (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 0.708937s (cpu); 0.55516s (thread); 0s (gc)
│ │ │ │ + -- used 0.617598s (cpu); 0.368667s (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 0.698845s (cpu); 0.573124s (thread); 0s (gc)
│ │ │ │ + -- used 0.822287s (cpu); 0.576794s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │ │                      | xxi_4-y+xi_3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out
│ │ │ @@ -3,27 +3,27 @@
│ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │  
│ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │  
│ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │  
│ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ - -- .00281383s elapsed
│ │ │ - -- .00266207s elapsed
│ │ │ - -- .000291456s elapsed
│ │ │ - -- .00269586s elapsed
│ │ │ - -- .00291065s elapsed
│ │ │ - -- .000221967s elapsed
│ │ │ - -- .00275421s elapsed
│ │ │ - -- .00278305s elapsed
│ │ │ - -- .000220053s elapsed
│ │ │ - -- .00266033s elapsed
│ │ │ - -- .00269371s elapsed
│ │ │ - -- .000211407s elapsed
│ │ │ ---backup directory created: /tmp/M2-33432-0/1
│ │ │ + -- .00382851s elapsed
│ │ │ + -- .00362472s elapsed
│ │ │ + -- .000430285s elapsed
│ │ │ + -- .00337802s elapsed
│ │ │ + -- .00357111s elapsed
│ │ │ + -- .000353073s elapsed
│ │ │ + -- .0036669s elapsed
│ │ │ + -- .00373658s elapsed
│ │ │ + -- .000298425s elapsed
│ │ │ + -- .00375334s elapsed
│ │ │ + -- .00365549s elapsed
│ │ │ + -- .000342679s elapsed
│ │ │ +--backup directory created: /tmp/M2-49474-0/1
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 4
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out
│ │ │ @@ -15,128 +15,128 @@
│ │ │  
│ │ │  i7 : dLoss = diff(varMatrix, gateMatrix{{loss}});
│ │ │  
│ │ │  i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
│ │ │  
│ │ │  i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │  
│ │ │ -o9 = {{2, 0, 11, 3, 10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19,
│ │ │ +o9 = {{16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18,
│ │ │ +     4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18,
│ │ │ +     18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ +     18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8,
│ │ │ +     3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 12, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ +     2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16,
│ │ │ +     10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ +     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6,
│ │ │ +     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15,
│ │ │ +     11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5,
│ │ │ +     4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4,
│ │ │ +     14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7,
│ │ │ +     8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1,
│ │ │ +     10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6},
│ │ │ +     19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18,
│ │ │ +     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ +     {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12,
│ │ │ +     20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ +     19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0,
│ │ │ +     18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8,
│ │ │ +     16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17,
│ │ │ +     6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ +     10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4,
│ │ │ +     3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5,
│ │ │ +     11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1,
│ │ │ +     1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5,
│ │ │ +     11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19,
│ │ │ +     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3,
│ │ │ +     12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ +     11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12,
│ │ │ +     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6},
│ │ │ +     3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ +     2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18,
│ │ │ +     {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ +     19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10,
│ │ │ +     18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16,
│ │ │ +     16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 12, 17, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17,
│ │ │ +     18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 20, 16, 18, 9, 4, 6}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5,
│ │ │ +     13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 0, 10, 19, 13, 20, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 18, 16, 19, 15, 17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0,
│ │ │ +     14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 10, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1,
│ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 0, 5, 10, 2, 13, 20, 8, 18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19,
│ │ │ +     15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7,
│ │ │ +     8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9,
│ │ │ +     7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4,
│ │ │ +     11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ +     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2,
│ │ │ +     3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20},
│ │ │ +     1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18,
│ │ │ +     {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18,
│ │ │ +     6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13,
│ │ │ +     18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 9, 4}}
│ │ │ +     18, 9, 4, 6}}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html
│ │ │ @@ -96,27 +96,27 @@
│ │ │              
    		
│ │ │                
    i3 : (p0, x0) = createSeedPair polys;
    
│ │ │              
    
│ │ │                
    i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ - -- .00281383s elapsed
│ │ │ - -- .00266207s elapsed
│ │ │ - -- .000291456s elapsed
│ │ │ - -- .00269586s elapsed
│ │ │ - -- .00291065s elapsed
│ │ │ - -- .000221967s elapsed
│ │ │ - -- .00275421s elapsed
│ │ │ - -- .00278305s elapsed
│ │ │ - -- .000220053s elapsed
│ │ │ - -- .00266033s elapsed
│ │ │ - -- .00269371s elapsed
│ │ │ - -- .000211407s elapsed
│ │ │ ---backup directory created: /tmp/M2-33432-0/1
│ │ │ + -- .00382851s elapsed
│ │ │ + -- .00362472s elapsed
│ │ │ + -- .000430285s elapsed
│ │ │ + -- .00337802s elapsed
│ │ │ + -- .00357111s elapsed
│ │ │ + -- .000353073s elapsed
│ │ │ + -- .0036669s elapsed
│ │ │ + -- .00373658s elapsed
│ │ │ + -- .000298425s elapsed
│ │ │ + -- .00375334s elapsed
│ │ │ + -- .00365549s elapsed
│ │ │ + -- .000342679s elapsed
│ │ │ +--backup directory created: /tmp/M2-49474-0/1
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │    H01: 1
│ │ │    H10: 1
│ │ │  number of paths tracked: 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,27 +22,27 @@
│ │ │ │            o npaths, an _i_n_t_e_g_e_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Output is verbose. For other dynamic strategies, see _M_o_n_o_d_r_o_m_y_S_o_l_v_e_r_O_p_t_i_o_n_s.
│ │ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ │ - -- .00281383s elapsed
│ │ │ │ - -- .00266207s elapsed
│ │ │ │ - -- .000291456s elapsed
│ │ │ │ - -- .00269586s elapsed
│ │ │ │ - -- .00291065s elapsed
│ │ │ │ - -- .000221967s elapsed
│ │ │ │ - -- .00275421s elapsed
│ │ │ │ - -- .00278305s elapsed
│ │ │ │ - -- .000220053s elapsed
│ │ │ │ - -- .00266033s elapsed
│ │ │ │ - -- .00269371s elapsed
│ │ │ │ - -- .000211407s elapsed
│ │ │ │ ---backup directory created: /tmp/M2-33432-0/1
│ │ │ │ + -- .00382851s elapsed
│ │ │ │ + -- .00362472s elapsed
│ │ │ │ + -- .000430285s elapsed
│ │ │ │ + -- .00337802s elapsed
│ │ │ │ + -- .00357111s elapsed
│ │ │ │ + -- .000353073s elapsed
│ │ │ │ + -- .0036669s elapsed
│ │ │ │ + -- .00373658s elapsed
│ │ │ │ + -- .000298425s elapsed
│ │ │ │ + -- .00375334s elapsed
│ │ │ │ + -- .00365549s elapsed
│ │ │ │ + -- .000342679s elapsed
│ │ │ │ +--backup directory created: /tmp/M2-49474-0/1
│ │ │ │    H01: 1
│ │ │ │    H10: 1
│ │ │ │  number of paths tracked: 2
│ │ │ │  found 1 points in the fiber so far
│ │ │ │    H01: 1
│ │ │ │    H10: 1
│ │ │ │  number of paths tracked: 4
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_monodromy__Group.html
│ │ │ @@ -118,131 +118,131 @@
│ │ │                
    i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
    
│ │ │              
    
│ │ │                
    i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │  
│ │ │ -o9 = {{2, 0, 11, 3, 10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19,
│ │ │ +o9 = {{16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18,
│ │ │ +     4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18,
│ │ │ +     18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ +     18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8,
│ │ │ +     3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 12, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ +     2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16,
│ │ │ +     10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ +     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6,
│ │ │ +     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15,
│ │ │ +     11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5,
│ │ │ +     4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4,
│ │ │ +     14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7,
│ │ │ +     8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1,
│ │ │ +     10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6},
│ │ │ +     19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18,
│ │ │ +     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ +     {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12,
│ │ │ +     20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ +     19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0,
│ │ │ +     18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8,
│ │ │ +     16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17,
│ │ │ +     6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10,
│ │ │ +     14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ +     10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4,
│ │ │ +     3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5,
│ │ │ +     11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1,
│ │ │ +     1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5,
│ │ │ +     11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19,
│ │ │ +     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3,
│ │ │ +     12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ +     11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12,
│ │ │ +     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6},
│ │ │ +     3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ +     2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18,
│ │ │ +     {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ +     4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ +     19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10,
│ │ │ +     18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16,
│ │ │ +     16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 12, 17, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17,
│ │ │ +     18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     13, 14, 20, 16, 18, 9, 4, 6}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5,
│ │ │ +     13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 0, 10, 19, 13, 20, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11,
│ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 13, 14, 18, 16, 19, 15, 17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0,
│ │ │ +     14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 10, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1,
│ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     9, 0, 5, 10, 2, 13, 20, 8, 18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19,
│ │ │ +     15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7,
│ │ │ +     8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9,
│ │ │ +     7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4,
│ │ │ +     11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ +     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2,
│ │ │ +     3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4,
│ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20},
│ │ │ +     1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18,
│ │ │ +     {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18,
│ │ │ +     6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13,
│ │ │ +     18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     20, 6, 9, 4}}
│ │ │ +     18, 9, 4, 6}}
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,131 +32,131 @@
│ │ │ │  i4 : varMatrix = gateMatrix{{t_1,t_2}};
│ │ │ │  i5 : phi = transpose gateMatrix{{t_1^3, t_1^2*t_2, t_1*t_2^2, t_2^3}};
│ │ │ │  i6 : loss = sum for i from 0 to 3 list (u_i - phi_(i,0))^2;
│ │ │ │  i7 : dLoss = diff(varMatrix, gateMatrix{{loss}});
│ │ │ │  i8 : G = gateSystem(paramMatrix,varMatrix,transpose dLoss);
│ │ │ │  i9 : monodromyGroup(G,"msOptions" => {NumberOfEdges=>10})
│ │ │ │  
│ │ │ │ -o9 = {{2, 0, 11, 3, 10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19,
│ │ │ │ +o9 = {{16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18,
│ │ │ │ +     4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 4, 6}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ │ +     17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0, 12, 10, 15, 13, 6, 17, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18,
│ │ │ │ +     18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 20, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 19, 15, 17, 3}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
│ │ │ │ +     18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 0, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 16, 17, 18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8,
│ │ │ │ +     3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7, 15, 8, 10, 11, 17, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 15, 12, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12,
│ │ │ │ +     14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     13, 14, 15, 16, 17, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ │ +     2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4, 7, 6, 11, 1, 9, 0, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16,
│ │ │ │ +     10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 7, 8, 15, 14, 6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15,
│ │ │ │ +     7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6,
│ │ │ │ +     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4, 16, 14, 3, 5, 2, 7, 8, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     2, 13, 11, 9, 8, 10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15,
│ │ │ │ +     11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20}, {12, 16, 7, 17, 10, 9, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5,
│ │ │ │ +     4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 11, 2, 3, 10, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 7, 10, 4, 13, 11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4,
│ │ │ │ +     14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18, 19, 20}, {16, 14, 17, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 11, 14, 5, 2, 1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7,
│ │ │ │ +     8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9, 4}, {2, 0, 11, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     17, 16, 9, 0, 4, 11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1,
│ │ │ │ +     10, 4, 14, 6, 5, 1, 13, 9, 7, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 1, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6},
│ │ │ │ +     19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {17, 16, 14, 9, 8, 7, 12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18,
│ │ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18,
│ │ │ │ +     1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 4, 6}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
│ │ │ │ +     {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 19, 20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12,
│ │ │ │ +     20}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     17, 18, 19, 20}, {0, 9, 2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15,
│ │ │ │ +     19, 20}, {0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 7, 12, 13, 14, 15, 16, 17,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 17, 18, 19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0,
│ │ │ │ +     18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 19, 13, 20, 6, 9, 4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8,
│ │ │ │ +     16, 18, 9, 4, 6}, {3, 1, 12, 4, 0, 11, 13, 5, 2, 16, 10, 7, 8, 15, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     14, 15, 12, 17, 18, 19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17,
│ │ │ │ +     6, 17, 9, 18, 19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 2, 13, 20, 8, 18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10,
│ │ │ │ +     14, 20, 16, 18, 9, 4, 6}, {1, 7, 12, 19, 16, 4, 0, 6, 2, 13, 11, 9, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 5, 13, 14, 3, 16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10,
│ │ │ │ +     10, 5, 20, 14, 18, 3, 15, 17}, {1, 16, 12, 19, 11, 15, 5, 17, 2, 7, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4,
│ │ │ │ +     3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {2, 0, 1, 3, 6, 5, 9, 7, 10, 4, 13,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5,
│ │ │ │ +     11, 14, 8, 16, 15, 12, 17, 18, 19, 20}, {3, 16, 12, 4, 10, 11, 14, 5, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     8, 6, 10, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1,
│ │ │ │ +     1, 0, 7, 8, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 1, 7, 17, 16, 9, 0, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     4, 10, 2, 6, 11, 14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5,
│ │ │ │ +     11, 13, 10, 6, 5, 2, 14, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19,
│ │ │ │ +     5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {17, 16, 14, 9, 8, 7,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 11, 14, 5, 15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3,
│ │ │ │ +     12, 11, 1, 2, 0, 5, 10, 3, 13, 4, 15, 6, 20, 18, 19}, {1, 16, 12, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2,
│ │ │ │ +     11, 15, 5, 17, 2, 7, 0, 3, 8, 10, 13, 20, 14, 18, 9, 4, 6}, {0, 1, 2, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12,
│ │ │ │ +     4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {2, 0, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     1, 3, 19, 0, 11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6},
│ │ │ │ +     3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 18, 19, 20}, {0, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4,
│ │ │ │ +     2, 3, 10, 5, 14, 7, 8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     6}, {4, 1, 2, 3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18,
│ │ │ │ +     {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13, 20, 6, 9,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19, 20}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ │ +     4}, {2, 1, 0, 3, 4, 5, 6, 7, 13, 9, 10, 11, 16, 8, 14, 15, 12, 17, 18,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     18, 9, 4, 6}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ │ +     19, 20}, {12, 16, 14, 19, 4, 3, 6, 15, 1, 9, 0, 17, 10, 2, 13, 20, 8,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 15, 6, 9, 4}, {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10,
│ │ │ │ +     18, 7, 11, 5}, {0, 1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 16, 18, 9, 4, 6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16,
│ │ │ │ +     16, 15, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 12, 17, 20, 18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17,
│ │ │ │ +     18, 16, 19, 15, 17, 3}, {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     13, 14, 20, 16, 18, 9, 4, 6}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5,
│ │ │ │ +     13, 3, 8, 15, 20, 18, 19}, {0, 1, 2, 3, 9, 11, 4, 5, 8, 6, 10, 7, 12,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 0, 10, 19, 13, 20, 6, 9, 4}, {0, 1, 2, 20, 6, 5, 9, 7, 8, 4, 10, 11,
│ │ │ │ +     13, 14, 15, 16, 17, 18, 19, 20}, {16, 7, 12, 19, 9, 1, 4, 10, 2, 6, 11,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 13, 14, 18, 16, 19, 15, 17, 3}, {3, 16, 14, 4, 5, 2, 7, 8, 1, 11, 0,
│ │ │ │ +     14, 8, 0, 5, 20, 13, 18, 3, 15, 17}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 10, 15, 13, 6, 17, 9, 18, 19, 20}, {12, 16, 14, 19, 4, 7, 6, 11, 1,
│ │ │ │ +     11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 16, 3, 19, 10, 11, 14, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     9, 0, 5, 10, 2, 13, 20, 8, 18, 3, 15, 17}, {16, 1, 7, 17, 12, 18, 2, 19,
│ │ │ │ +     15, 1, 0, 7, 17, 2, 13, 20, 8, 18, 9, 4, 6}, {0, 9, 2, 3, 10, 5, 14, 7,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     11, 8, 10, 20, 5, 0, 14, 3, 13, 15, 6, 9, 4}, {0, 1, 3, 19, 12, 5, 2, 7,
│ │ │ │ +     8, 1, 4, 11, 12, 13, 6, 15, 16, 17, 18, 19, 20}, {0, 1, 2, 3, 4, 5, 6,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     15, 8, 10, 11, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {12, 16, 7, 17, 10, 9,
│ │ │ │ +     7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, {12, 1, 3, 19, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 18, 19, 20}, {12, 16, 14, 17, 4,
│ │ │ │ +     11, 13, 5, 15, 16, 10, 7, 17, 2, 14, 20, 8, 18, 9, 4, 6}, {0, 1, 3, 19,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     7, 6, 11, 1, 9, 0, 5, 10, 2, 13, 3, 8, 15, 20, 18, 19}, {0, 1, 3, 19,
│ │ │ │ +     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {4, 1, 2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0, 1, 2,
│ │ │ │ +     3, 0, 5, 13, 7, 8, 16, 10, 11, 12, 6, 14, 15, 9, 17, 18, 19, 20}, {0, 1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 6, 5, 9, 7, 8, 4, 10, 11, 12, 13, 14, 18, 16, 19, 15, 17, 3}, {4,
│ │ │ │ +     3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16, 18, 9, 4, 6}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     16, 14, 3, 5, 2, 7, 8, 1, 11, 0, 12, 10, 6, 13, 15, 9, 17, 18, 19, 20},
│ │ │ │ +     1, 7, 17, 12, 18, 2, 19, 11, 8, 10, 20, 5, 13, 14, 3, 16, 15, 6, 9, 4},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {12, 16, 7, 17, 10, 9, 14, 4, 11, 1, 0, 6, 5, 2, 13, 3, 8, 15, 20, 18,
│ │ │ │ +     {0, 14, 12, 19, 11, 15, 5, 17, 2, 7, 1, 3, 8, 13, 10, 20, 16, 18, 9, 4,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19}, {0, 11, 2, 3, 10, 4, 14, 6, 8, 1, 5, 9, 12, 13, 7, 15, 16, 17, 18,
│ │ │ │ +     6}, {2, 0, 9, 3, 10, 5, 14, 7, 4, 1, 13, 11, 6, 8, 16, 15, 12, 17, 20,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     19, 20}, {16, 14, 17, 18, 8, 7, 12, 11, 3, 2, 1, 5, 15, 0, 10, 19, 13,
│ │ │ │ +     18, 19}, {0, 1, 3, 19, 12, 11, 2, 5, 15, 8, 10, 7, 17, 13, 14, 20, 16,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     20, 6, 9, 4}}
│ │ │ │ +     18, 9, 4, 6}}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This is still somewhat experimental.
│ │ │ │  ********** WWaayyss ttoo uussee mmoonnooddrroommyyGGrroouupp:: **********
│ │ │ │      * monodromyGroup(System)
│ │ │ │      * monodromyGroup(System,AbstractPoint,List)
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/___Msolve.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : I = ideal(x, y, z)
│ │ │  
│ │ │  o2 = ideal (x, y, z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53622-0/0-in.ms -o /tmp/M2-53622-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84695-0/0-in.ms -o /tmp/M2-84695-0/0-out.ms
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -28,15 +28,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1147761463
│ │ │ +Initial prime = 1253775077
│ │ │  
│ │ │  Legend for f4 information
│ │ │  --------------------------------------------------------
│ │ │  deg       current degree of pairs selected in this round
│ │ │  sel       number of pairs selected in this round
│ │ │  pairs     total number of pairs in pair list
│ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ @@ -46,25 +46,25 @@
│ │ │  time(rd)  time of the current f4 round in seconds given
│ │ │            for real and cpu time
│ │ │  --------------------------------------------------------
│ │ │  
│ │ │  deg     sel   pairs        mat          density            new data         time(rd) in sec (real|cpu)
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │ -reduce final basis        3 x 3          33.33%        3 new       0 zero         0.06 | 0.13         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.13 sec
│ │ │ -overall(cpu)            0.29 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.07 sec  51.6%
│ │ │ -convert                 0.06 sec  48.3%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.5%
│ │ │ +update                  0.00 sec  76.8%
│ │ │ +convert                 0.00 sec   3.3%
│ │ │ +linear algebra          0.00 sec   0.9%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -78,18 +78,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1086711877
│ │ │ +New prime = 1098899071
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.04 sec
│ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -105,15 +105,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.25 sec (elapsed) /  0.63 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/html/index.html
│ │ │ @@ -78,15 +78,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-53622-0/0-in.ms -o /tmp/M2-53622-0/0-out.ms
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-84695-0/0-in.ms -o /tmp/M2-84695-0/0-out.ms
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -97,15 +97,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1147761463
│ │ │ +Initial prime = 1253775077
│ │ │  
│ │ │  Legend for f4 information
│ │ │  --------------------------------------------------------
│ │ │  deg       current degree of pairs selected in this round
│ │ │  sel       number of pairs selected in this round
│ │ │  pairs     total number of pairs in pair list
│ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ @@ -115,25 +115,25 @@
│ │ │  time(rd)  time of the current f4 round in seconds given
│ │ │            for real and cpu time
│ │ │  --------------------------------------------------------
│ │ │  
│ │ │  deg     sel   pairs        mat          density            new data         time(rd) in sec (real|cpu)
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │ -reduce final basis        3 x 3          33.33%        3 new       0 zero         0.06 | 0.13         
│ │ │ +reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -overall(elapsed)        0.13 sec
│ │ │ -overall(cpu)            0.29 sec
│ │ │ +overall(elapsed)        0.00 sec
│ │ │ +overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.07 sec  51.6%
│ │ │ -convert                 0.06 sec  48.3%
│ │ │ -linear algebra          0.00 sec   0.0%
│ │ │ +symbolic prep.          0.00 sec   0.5%
│ │ │ +update                  0.00 sec  76.8%
│ │ │ +convert                 0.00 sec   3.3%
│ │ │ +linear algebra          0.00 sec   0.9%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -147,18 +147,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1086711877
│ │ │ +New prime = 1098899071
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.04 sec
│ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -174,15 +174,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.25 sec (elapsed) /  0.63 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00857534s elapsed
│ │ │ + -- .00258617s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00905284s elapsed
│ │ │ + -- .0117182s 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
│ │ │ │ - -- .00857534s elapsed
│ │ │ │ + -- .00258617s elapsed
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 21
│ │ │ │  number of distinct multidegrees = 18
│ │ │ │ - -- .00905284s elapsed
│ │ │ │ + -- .0117182s 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
│ │ │ @@ -1,8 +1,8 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Jan 19 15:01:18 2026
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Jan 19 15:01:19 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │  #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  Z3JHcihJZGVhbCk=
│ │ ├── ./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
│ │ │ - -- .024652s elapsed
│ │ │ + -- .0287072s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : elapsedTime monjMult I
│ │ │ - -- .132531s elapsed
│ │ │ + -- .0988799s elapsed
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .172768s elapsed
│ │ │ + -- .173869s 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
│ │ │ - -- .139156s elapsed
│ │ │ + -- .0930533s 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
│ │ │ - -- .152506s elapsed
│ │ │ + -- .12245s elapsed
│ │ │  
│ │ │  o3 = 192
│ │ │  
│ │ │  i4 : elapsedTime jMult I
│ │ │ - -- 1.43501s elapsed
│ │ │ + -- 1.46012s elapsed
│ │ │  
│ │ │  o4 = 192
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html
│ │ │ @@ -88,31 +88,31 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime jMult I
│ │ │ - -- .024652s elapsed
│ │ │ + -- .0287072s elapsed
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime monjMult I
│ │ │ - -- .132531s elapsed
│ │ │ + -- .0988799s elapsed
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .172768s elapsed
│ │ │ + -- .173869s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .139156s elapsed
│ │ │ + -- .0930533s 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
│ │ │ │ - -- .139156s elapsed
│ │ │ │ + -- .0930533s elapsed
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _N_P -- the Newton polyhedron of a monomial ideal
│ │ │ │  ********** WWaayyss ttoo uussee mmoonnAAnnaallyyttiiccSSpprreeaadd:: **********
│ │ │ │      * monAnalyticSpread(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html
│ │ │ @@ -92,23 +92,23 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : elapsedTime monjMult I
│ │ │ - -- .152506s elapsed
│ │ │ + -- .12245s elapsed
│ │ │  
│ │ │  o3 = 192
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime jMult I
│ │ │ - -- 1.43501s elapsed
│ │ │ + -- 1.46012s 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
│ │ │ │ - -- .152506s elapsed
│ │ │ │ + -- .12245s elapsed
│ │ │ │  
│ │ │ │  o3 = 192
│ │ │ │  i4 : elapsedTime jMult I
│ │ │ │ - -- 1.43501s elapsed
│ │ │ │ + -- 1.46012s elapsed
│ │ │ │  
│ │ │ │  o4 = 192
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal
│ │ │ │      * _j_M_u_l_t -- the j-multiplicity of an ideal
│ │ │ │      * _m_o_n_R_e_d_u_c_t_i_o_n -- the minimal monomial reduction of a monomial ideal
│ │ │ │      * _N_P -- the Newton polyhedron of a monomial ideal
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │  o4 : ProjectiveVariety, curve in PP^8
│ │ │  
│ │ │  i5 : ? X
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 
│ │ │  
│ │ │  i6 : time f = X ===> Y;
│ │ │ - -- used 3.01286s (cpu); 1.87175s (thread); 0s (gc)
│ │ │ + -- used 3.98532s (cpu); 2.24454s (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 3.35764s (cpu); 2.18666s (thread); 0s (gc)
│ │ │ + -- used 4.35559s (cpu); 2.40335s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
│ │ │  
│ │ │  i12 : g||W
│ │ │  
│ │ │  o12 = multi-rational map consisting of one single rational map
│ │ │        source variety: 6-dimensional subvariety of PP^8 cut out by 2 hypersurfaces of degrees 1^1 2^1 
│ │ │ @@ -129,15 +129,15 @@
│ │ │  o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9
│ │ │  
│ │ │  i16 : ? Z
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │  
│ │ │  i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 7.61692s (cpu); 4.97834s (thread); 0s (gc)
│ │ │ + -- used 7.17956s (cpu); 4.95993s (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 5.10141s (cpu); 3.8222s (thread); 0s (gc)
│ │ │ + -- used 4.90249s (cpu); 3.92059s (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 0.0946816s (cpu); 0.0919773s (thread); 0s (gc)
│ │ │ + -- used 0.134079s (cpu); 0.116376s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
│ │ │  
│ │ │  i6 : dim oo, degree oo, degrees oo
│ │ │  
│ │ │  o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)})
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 1.75371s (cpu); 1.28035s (thread); 0s (gc)
│ │ │ + -- used 2.45168s (cpu); 1.52315s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 
│ │ │  
│ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
│ │ │  
│ │ │  i8 : dim oo, degree oo, degrees oo
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out
│ │ │ @@ -11,22 +11,22 @@
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of
│ │ │       target variety: hypersurface in PP^4 defined by a form of degree 2
│ │ │       ------------------------------------------------------------------------
│ │ │       multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1
│ │ │  
│ │ │  i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 3.25061s (cpu); 2.42411s (thread); 0s (gc)
│ │ │ + -- used 5.82311s (cpu); 2.9377s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.316779s (cpu); 0.246267s (thread); 0s (gc)
│ │ │ + -- used 0.419345s (cpu); 0.317152s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time degree Phi
│ │ │ - -- used 0.324508s (cpu); 0.248777s (thread); 0s (gc)
│ │ │ + -- used 0.405473s (cpu); 0.320322s (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 0.450555s (cpu); 0.380926s (thread); 0s (gc)
│ │ │ + -- used 0.595598s (cpu); 0.398325s (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 0.00107435s (cpu); 0.000182493s (thread); 0s (gc)
│ │ │ + -- used 0.00236422s (cpu); 0.000204867s (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 0.0016233s (cpu); 0.00028307s (thread); 0s (gc)
│ │ │ + -- used 0.00214176s (cpu); 0.000251269s (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 1.32992s (cpu); 1.04628s (thread); 0s (gc)
│ │ │ + -- used 1.37896s (cpu); 1.08422s (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 0.000417253s (cpu); 0.00034102s (thread); 0s (gc)
│ │ │ + -- used 0.000169325s (cpu); 0.000485814s (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 0.131659s (cpu); 0.0835875s (thread); 0s (gc)
│ │ │ + -- used 0.0394744s (cpu); 0.0264633s (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 0.0332021s (cpu); 0.0347035s (thread); 0s (gc)
│ │ │ + -- used 0.0548089s (cpu); 0.043545s (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 0.224535s (cpu); 0.145378s (thread); 0s (gc)
│ │ │ + -- used 0.320831s (cpu); 0.172977s (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 0.228772s (cpu); 0.174179s (thread); 0s (gc)
│ │ │ + -- used 0.140047s (cpu); 0.127551s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │  
│ │ │  i6 : dim Z, degree Z, degrees Z
│ │ │  
│ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 4.83484s (cpu); 2.73189s (thread); 0s (gc)
│ │ │ + -- used 10.8739s (cpu); 2.90681s (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 0.332534s (cpu); 0.282691s (thread); 0s (gc)
│ │ │ + -- used 0.495127s (cpu); 0.377117s (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 1.32156s (cpu); 0.976s (thread); 0s (gc)
│ │ │ + -- used 1.33999s (cpu); 1.10063s (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 0.131138s (cpu); 0.0656896s (thread); 0s (gc)
│ │ │ + -- used 0.0806611s (cpu); 0.0593844s (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 0.117479s (cpu); 0.0823356s (thread); 0s (gc)
│ │ │ + -- used 0.291373s (cpu); 0.126059s (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 0.414769s (cpu); 0.278602s (thread); 0s (gc)
│ │ │ + -- used 0.626448s (cpu); 0.347774s (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 0.00327872s (cpu); 1.3305e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00261607s (cpu); 9.931e-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 0.374874s (cpu); 0.248351s (thread); 0s (gc)
│ │ │ + -- used 0.509708s (cpu); 0.228625s (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 1.73767s (cpu); 1.02066s (thread); 0s (gc)
│ │ │ + -- used 2.12698s (cpu); 1.01396s (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 0.365441s (cpu); 0.257759s (thread); 0s (gc)
│ │ │ + -- used 0.604681s (cpu); 0.300965s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.162613s (cpu); 0.0913151s (thread); 0s (gc)
│ │ │ + -- used 0.133132s (cpu); 0.0738748s (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 4.02211s (cpu); 3.01219s (thread); 0s (gc)
│ │ │ + -- used 4.23696s (cpu); 3.40102s (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 0.00994984s (cpu); 0.0095208s (thread); 0s (gc)
│ │ │ + -- used 0.130328s (cpu); 0.0391214s (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 0.463885s (cpu); 0.384217s (thread); 0s (gc)
│ │ │ + -- used 0.585092s (cpu); 0.487927s (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 0.834357s (cpu); 0.684192s (thread); 0s (gc)
│ │ │ + -- used 0.6682s (cpu); 0.401967s (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 0.000930625s (cpu); 0.00115266s (thread); 0s (gc)
│ │ │ - -- used 0.245778s (cpu); 0.14314s (thread); 0s (gc)
│ │ │ - -- used 0.246905s (cpu); 0.189409s (thread); 0s (gc)
│ │ │ - -- used 0.246127s (cpu); 0.200789s (thread); 0s (gc)
│ │ │ - -- used 0.239025s (cpu); 0.178577s (thread); 0s (gc)
│ │ │ + -- used 0.00216402s (cpu); 0.00135701s (thread); 0s (gc)
│ │ │ + -- used 0.253558s (cpu); 0.157205s (thread); 0s (gc)
│ │ │ + -- used 0.285527s (cpu); 0.183699s (thread); 0s (gc)
│ │ │ + -- used 0.275403s (cpu); 0.175187s (thread); 0s (gc)
│ │ │ + -- used 0.249151s (cpu); 0.143155s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.194473s (cpu); 0.12171s (thread); 0s (gc)
│ │ │ + -- used 0.216222s (cpu); 0.0986321s (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 0.0168798s (cpu); 0.0151997s (thread); 0s (gc)
│ │ │ + -- used 0.0826583s (cpu); 0.0275925s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i4 : Y = random({2,1,2},X);
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i5 : time q = point Y
│ │ │ - -- used 1.97559s (cpu); 1.22084s (thread); 0s (gc)
│ │ │ + -- used 1.85125s (cpu); 1.15934s (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 0.694773s (cpu); 0.535072s (thread); 0s (gc)
│ │ │ + -- used 1.61766s (cpu); 0.732759s (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 0.328036s (cpu); 0.20466s (thread); 0s (gc)
│ │ │ + -- used 0.276889s (cpu); 0.168118s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1)
│ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2 
│ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ │       dominance: true
│ │ │       multidegree: {10, 14, 19, 25}
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15
    │ │ │
    │ │ │ 
    i6 : time f = X ===> Y;
│ │ │ - -- used 3.01286s (cpu); 1.87175s (thread); 0s (gc)
│ │ │ + -- used 3.98532s (cpu); 2.24454s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
    │ │ │
    │ │ │ 
    i7 : f X
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
    │ │ │
    │ │ │ 
    i11 : time g = V ===> W;
│ │ │ - -- used 3.35764s (cpu); 2.18666s (thread); 0s (gc)
│ │ │ + -- used 4.35559s (cpu); 2.40335s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
    │ │ │
    │ │ │ 
    i12 : g||W
│ │ │ @@ -252,15 +252,15 @@
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
    │ │ │
    │ │ │ 
    i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 7.61692s (cpu); 4.97834s (thread); 0s (gc)
│ │ │ + -- used 7.17956s (cpu); 4.95993s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
    │ │ │
    │ │ │ 
    i4 : time X = Phi^* Y;
│ │ │ - -- used 5.10141s (cpu); 3.8222s (thread); 0s (gc)
│ │ │ + -- used 4.90249s (cpu); 3.92059s (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 5.10141s (cpu); 3.8222s (thread); 0s (gc)
│ │ │ │ + -- used 4.90249s (cpu); 3.92059s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension
│ │ │ │  2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-
│ │ │ │  degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
│ │ │ │  i5 : dim X, degree X, degrees X
│ │ │ │  
│ │ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
    │ │ │
    │ │ │ 
    i5 : time Phi Z;
│ │ │ - -- used 0.0946816s (cpu); 0.0919773s (thread); 0s (gc)
│ │ │ + -- used 0.134079s (cpu); 0.116376s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
    │ │ │
    │ │ │ 
    i6 : dim oo, degree oo, degrees oo
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o6 : Sequence
    │ │ │
    │ │ │ 
    i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 1.75371s (cpu); 1.28035s (thread); 0s (gc)
│ │ │ + -- used 2.45168s (cpu); 1.52315s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 
│ │ │  
│ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
    │ │ │
    │ │ │ 
    i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 3.25061s (cpu); 2.42411s (thread); 0s (gc)
│ │ │ + -- used 5.82311s (cpu); 2.9377s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    │ │ │
    │ │ │ 
    i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.316779s (cpu); 0.246267s (thread); 0s (gc)
│ │ │ + -- used 0.419345s (cpu); 0.317152s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    │ │ │
    │ │ │ 
    i6 : time degree Phi
│ │ │ - -- used 0.324508s (cpu); 0.248777s (thread); 0s (gc)
│ │ │ + -- used 0.405473s (cpu); 0.320322s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 2
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -27,23 +27,23 @@ │ │ │ │ │ │ │ │ o3 = multi-rational map consisting of one single rational map │ │ │ │ source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of │ │ │ │ target variety: hypersurface in PP^4 defined by a form of degree 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1 │ │ │ │ i4 : time degree(Phi,Strategy=>"random point") │ │ │ │ - -- used 3.25061s (cpu); 2.42411s (thread); 0s (gc) │ │ │ │ + -- used 5.82311s (cpu); 2.9377s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ │ - -- used 0.316779s (cpu); 0.246267s (thread); 0s (gc) │ │ │ │ + -- used 0.419345s (cpu); 0.317152s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 2 │ │ │ │ i6 : time degree Phi │ │ │ │ - -- used 0.324508s (cpu); 0.248777s (thread); 0s (gc) │ │ │ │ + -- used 0.405473s (cpu); 0.320322s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = 2 │ │ │ │ Note, as in the example above, that calculation times may vary depending on the │ │ │ │ strategy used. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- degree of a multi-rational map │ │ │ │ * _d_e_g_r_e_e_M_a_p_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_rp.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time degree Phi
│ │ │ - -- used 0.450555s (cpu); 0.380926s (thread); 0s (gc)
│ │ │ + -- used 0.595598s (cpu); 0.398325s (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 0.450555s (cpu); 0.380926s (thread); 0s (gc) │ │ │ │ + -- used 0.595598s (cpu); 0.398325s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = 1 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_,_O_p_t_i_o_n_) -- degree of a multi-rational map using a │ │ │ │ probabilistic approach │ │ │ │ * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ │ │ * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_describe_lp__Multirational__Map_rp.html │ │ │ @@ -77,15 +77,15 @@ │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time ? Phi
│ │ │ - -- used 0.00107435s (cpu); 0.000182493s (thread); 0s (gc)
│ │ │ + -- used 0.00236422s (cpu); 0.000204867s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │       target variety: PP^4 x PP^5
│ │ │       ------------------------------------------------------------------------
│ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
    │ │ │ │ │ │ @@ -96,27 +96,27 @@ │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time ? Phi
│ │ │ - -- used 0.0016233s (cpu); 0.00028307s (thread); 0s (gc)
│ │ │ + -- used 0.00214176s (cpu); 0.000251269s (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 1.32992s (cpu); 1.04628s (thread); 0s (gc)
│ │ │ + -- used 1.37896s (cpu); 1.08422s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │ @@ -126,15 +126,15 @@
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time ? Phi
│ │ │ - -- used 0.000417253s (cpu); 0.00034102s (thread); 0s (gc)
│ │ │ + -- used 0.000169325s (cpu); 0.000485814s (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 0.00107435s (cpu); 0.000182493s (thread); 0s (gc)
│ │ │ │ + -- used 0.00236422s (cpu); 0.000204867s (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 0.0016233s (cpu); 0.00028307s (thread); 0s (gc)
│ │ │ │ + -- used 0.00214176s (cpu); 0.000251269s (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 1.32992s (cpu); 1.04628s (thread); 0s (gc)
│ │ │ │ + -- used 1.37896s (cpu); 1.08422s (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 0.000417253s (cpu); 0.00034102s (thread); 0s (gc)
│ │ │ │ + -- used 0.000169325s (cpu); 0.000485814s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_graph_lp__Multirational__Map_rp.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.131659s (cpu); 0.0835875s (thread); 0s (gc)
│ │ │ + -- used 0.0394744s (cpu); 0.0264633s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -107,15 +107,15 @@ │ │ │ │ │ │ o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ - -- used 0.0332021s (cpu); 0.0347035s (thread); 0s (gc)
│ │ │ + -- used 0.0548089s (cpu); 0.043545s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -131,15 +131,15 @@ │ │ │ │ │ │ o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ - -- used 0.224535s (cpu); 0.145378s (thread); 0s (gc)
│ │ │ + -- used 0.320831s (cpu); 0.172977s (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 0.131659s (cpu); 0.0835875s (thread); 0s (gc) │ │ │ │ + -- used 0.0394744s (cpu); 0.0264633s (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 0.0332021s (cpu); 0.0347035s (thread); 0s (gc) │ │ │ │ + -- used 0.0548089s (cpu); 0.043545s (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 0.224535s (cpu); 0.145378s (thread); 0s (gc) │ │ │ │ + -- used 0.320831s (cpu); 0.172977s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = (Phi211, Phi212) │ │ │ │ │ │ │ │ o8 : Sequence │ │ │ │ i9 : Phi211; │ │ │ │ │ │ │ │ o9 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_image_lp__Multirational__Map_rp.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ │ │ │ o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time Z = image Phi;
│ │ │ - -- used 0.228772s (cpu); 0.174179s (thread); 0s (gc)
│ │ │ + -- used 0.140047s (cpu); 0.127551s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : dim Z, degree Z, degrees Z
│ │ │ @@ -115,15 +115,15 @@
│ │ │            
│ │ │          
│ │ │          
    Alternatively, the calculation can be performed using the Segre embedding as follows:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3,
│ │ │ │  x_3^2};
│ │ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : time multidegree Phi
│ │ │ │ - -- used 0.834357s (cpu); 0.684192s (thread); 0s (gc)
│ │ │ │ + -- used 0.6682s (cpu); 0.401967s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = {66, 46, 31, 20}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (degree source Phi,degree image Phi)
│ │ │ │  
│ │ │ │  o4 = (66, 20)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.html
│ │ │ @@ -77,29 +77,29 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 4.83484s (cpu); 2.73189s (thread); 0s (gc)
│ │ │ + -- used 10.8739s (cpu); 2.90681s (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 0.228772s (cpu); 0.174179s (thread); 0s (gc)
│ │ │ │ + -- used 0.140047s (cpu); 0.127551s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i6 : dim Z, degree Z, degrees Z
│ │ │ │  
│ │ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  Alternatively, the calculation can be performed using the Segre embedding as
│ │ │ │  follows:
│ │ │ │  i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ │ - -- used 4.83484s (cpu); 2.73189s (thread); 0s (gc)
│ │ │ │ + -- used 10.8739s (cpu); 2.90681s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i8 : assert(Z == Z')
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y -- direct image via a multi-
│ │ │ │        rational map
│ │ │ │      * _i_m_a_g_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- closure of the image of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse2.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │              
    
│ │ │                
    i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.332534s (cpu); 0.282691s (thread); 0s (gc)
│ │ │ + -- used 0.495127s (cpu); 0.377117s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
    
│ │ │              
    
│ │ │                
    i4 : assert(Phi * Psi == 1)
    
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │              
    
│ │ │                
    i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.32156s (cpu); 0.976s (thread); 0s (gc)
│ │ │ + -- used 1.33999s (cpu); 1.10063s (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 0.332534s (cpu); 0.282691s (thread); 0s (gc)
│ │ │ │ + -- used 0.495127s (cpu); 0.377117s (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 1.32156s (cpu); 0.976s (thread); 0s (gc)
│ │ │ │ + -- used 1.33999s (cpu); 1.10063s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
│ │ │ │  i7 : assert(Phi' * Psi' == 1)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _<_=_=_>_ _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p -- equality of multi-rational maps
│ │ │ │        with checks on internal data
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse_lp__Multirational__Map_rp.html
│ │ │ @@ -88,45 +88,45 @@
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
    
│ │ │                
    i3 : time inverse Phi;
│ │ │ - -- used 0.131138s (cpu); 0.0656896s (thread); 0s (gc)
│ │ │ + -- used 0.0806611s (cpu); 0.0593844s (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 0.117479s (cpu); 0.0823356s (thread); 0s (gc)
│ │ │ + -- used 0.291373s (cpu); 0.126059s (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 0.414769s (cpu); 0.278602s (thread); 0s (gc)
│ │ │ + -- used 0.626448s (cpu); 0.347774s (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 0.131138s (cpu); 0.0656896s (thread); 0s (gc)
│ │ │ │ + -- used 0.0806611s (cpu); 0.0593844s (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 0.117479s (cpu); 0.0823356s (thread); 0s (gc)
│ │ │ │ + -- used 0.291373s (cpu); 0.126059s (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 0.414769s (cpu); 0.278602s (thread); 0s (gc)
│ │ │ │ + -- used 0.626448s (cpu); 0.347774s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
│ │ │ │  i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
│ │ │ │  i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1)
│ │ │ │  i10 : assert(Eta * Eta^-1 == 1 and Eta^-1 * Eta == 1)
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Isomorphism_lp__Multirational__Map_rp.html
│ │ │ @@ -83,45 +83,45 @@
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │              
    
│ │ │                
    i4 : time isIsomorphism Phi
│ │ │ - -- used 0.00327872s (cpu); 1.3305e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00261607s (cpu); 9.931e-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 0.374874s (cpu); 0.248351s (thread); 0s (gc)
│ │ │ + -- used 0.509708s (cpu); 0.228625s (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 1.73767s (cpu); 1.02066s (thread); 0s (gc)
│ │ │ + -- used 2.12698s (cpu); 1.01396s (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 0.00327872s (cpu); 1.3305e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00261607s (cpu); 9.931e-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 0.374874s (cpu); 0.248351s (thread); 0s (gc)
│ │ │ │ + -- used 0.509708s (cpu); 0.228625s (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 1.73767s (cpu); 1.02066s (thread); 0s (gc)
│ │ │ │ + -- used 2.12698s (cpu); 1.01396s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : assert(o8 and (not o6) and (not o4))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a multi-rational map is a
│ │ │ │        morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Morphism_lp__Multirational__Map_rp.html
│ │ │ @@ -80,31 +80,31 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2)
│ │ │              
    
│ │ │                
    i3 : time isMorphism Phi
│ │ │ - -- used 0.365441s (cpu); 0.257759s (thread); 0s (gc)
│ │ │ + -- used 0.604681s (cpu); 0.300965s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
    
│ │ │              
    
│ │ │                
    i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.162613s (cpu); 0.0913151s (thread); 0s (gc)
│ │ │ + -- used 0.133132s (cpu); 0.0738748s (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 4.02211s (cpu); 3.01219s (thread); 0s (gc)
│ │ │ + -- used 4.23696s (cpu); 3.40102s (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 0.365441s (cpu); 0.257759s (thread); 0s (gc)
│ │ │ │ + -- used 0.604681s (cpu); 0.300965s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.162613s (cpu); 0.0913151s (thread); 0s (gc)
│ │ │ │ + -- used 0.133132s (cpu); 0.0738748s (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 4.02211s (cpu); 3.01219s (thread); 0s (gc)
│ │ │ │ + -- used 4.23696s (cpu); 3.40102s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : assert((not o3) and o5)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_I_s_o_m_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a birational map is an
│ │ │ │        isomorphism
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_R_a_t_i_o_n_a_l_M_a_p_) -- whether a rational map is a morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_linearly__Normal__Embedding.html
│ │ │ @@ -79,30 +79,30 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │              
    
│ │ │                
    i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.00994984s (cpu); 0.0095208s (thread); 0s (gc)
│ │ │ + -- used 0.130328s (cpu); 0.0391214s (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 0.463885s (cpu); 0.384217s (thread); 0s (gc)
│ │ │ + -- used 0.585092s (cpu); 0.487927s (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 0.00994984s (cpu); 0.0095208s (thread); 0s (gc)
│ │ │ │ + -- used 0.130328s (cpu); 0.0391214s (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 0.463885s (cpu); 0.384217s (thread); 0s (gc)
│ │ │ │ + -- used 0.585092s (cpu); 0.487927s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
│ │ │ │  i6 : assert(isIsomorphism g)
│ │ │ │  i7 : describe g
│ │ │ │  
│ │ │ │  o7 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: curve in PP^3 cut out by 6 hypersurfaces of degree 4
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Multirational__Map_rp.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │              
    
│ │ │                
    i3 : time multidegree Phi
│ │ │ - -- used 0.834357s (cpu); 0.684192s (thread); 0s (gc)
│ │ │ + -- used 0.6682s (cpu); 0.401967s (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 0.000930625s (cpu); 0.00115266s (thread); 0s (gc)
│ │ │ - -- used 0.245778s (cpu); 0.14314s (thread); 0s (gc)
│ │ │ - -- used 0.246905s (cpu); 0.189409s (thread); 0s (gc)
│ │ │ - -- used 0.246127s (cpu); 0.200789s (thread); 0s (gc)
│ │ │ - -- used 0.239025s (cpu); 0.178577s (thread); 0s (gc)
│ │ │ + -- used 0.00216402s (cpu); 0.00135701s (thread); 0s (gc)
│ │ │ + -- used 0.253558s (cpu); 0.157205s (thread); 0s (gc)
│ │ │ + -- used 0.285527s (cpu); 0.183699s (thread); 0s (gc)
│ │ │ + -- used 0.275403s (cpu); 0.175187s (thread); 0s (gc)
│ │ │ + -- used 0.249151s (cpu); 0.143155s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.194473s (cpu); 0.12171s (thread); 0s (gc)
    
│ │ │ + -- used 0.216222s (cpu); 0.0986321s (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 0.000930625s (cpu); 0.00115266s (thread); 0s (gc) │ │ │ │ - -- used 0.245778s (cpu); 0.14314s (thread); 0s (gc) │ │ │ │ - -- used 0.246905s (cpu); 0.189409s (thread); 0s (gc) │ │ │ │ - -- used 0.246127s (cpu); 0.200789s (thread); 0s (gc) │ │ │ │ - -- used 0.239025s (cpu); 0.178577s (thread); 0s (gc) │ │ │ │ + -- used 0.00216402s (cpu); 0.00135701s (thread); 0s (gc) │ │ │ │ + -- used 0.253558s (cpu); 0.157205s (thread); 0s (gc) │ │ │ │ + -- used 0.285527s (cpu); 0.183699s (thread); 0s (gc) │ │ │ │ + -- used 0.275403s (cpu); 0.175187s (thread); 0s (gc) │ │ │ │ + -- used 0.249151s (cpu); 0.143155s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = {51, 28, 14, 6, 2} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : time assert(oo == multidegree Phi) │ │ │ │ - -- used 0.194473s (cpu); 0.12171s (thread); 0s (gc) │ │ │ │ + -- used 0.216222s (cpu); 0.0986321s (thread); 0s (gc) │ │ │ │ ********** RReeffeerreenncceess ********** │ │ │ │ ArXiv preprint: _C_o_m_p_u_t_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _m_a_p_s_ _b_e_t_w_e_e_n_ _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e │ │ │ │ _v_a_r_i_e_t_i_e_s. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational │ │ │ │ map │ │ │ │ * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s_(_R_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_point_lp__Multiprojective__Variety_rp.html │ │ │ @@ -80,15 +80,15 @@ │ │ │ │ │ │ o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time p := point X
│ │ │ - -- used 0.0168798s (cpu); 0.0151997s (thread); 0s (gc)
│ │ │ + -- used 0.0826583s (cpu); 0.0275925s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -97,15 +97,15 @@ │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time q = point Y
│ │ │ - -- used 1.97559s (cpu); 1.22084s (thread); 0s (gc)
│ │ │ + -- used 1.85125s (cpu); 1.15934s (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 0.0168798s (cpu); 0.0151997s (thread); 0s (gc) │ │ │ │ + -- used 0.0826583s (cpu); 0.0275925s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1], │ │ │ │ [3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1]) │ │ │ │ │ │ │ │ o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ i4 : Y = random({2,1,2},X); │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9 │ │ │ │ i5 : time q = point Y │ │ │ │ - -- used 1.97559s (cpu); 1.22084s (thread); 0s (gc) │ │ │ │ + -- used 1.85125s (cpu); 1.15934s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = q │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ i6 : assert(isSubset(p,X) and isSubset(q,Y)) │ │ │ │ The list of homogeneous coordinates can be obtained with the operator |-. │ │ │ │ i7 : |- p │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html │ │ │ @@ -101,15 +101,15 @@ │ │ │ │ │ │ o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time segre Phi;
│ │ │ - -- used 0.694773s (cpu); 0.535072s (thread); 0s (gc)
│ │ │ + -- used 1.61766s (cpu); 0.732759s (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 0.694773s (cpu); 0.535072s (thread); 0s (gc)
│ │ │ │ + -- used 1.61766s (cpu); 0.732759s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
│ │ │ │  i7 : describe segre Phi
│ │ │ │  
│ │ │ │  o7 = rational map defined by forms of degree 6
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^149
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time describe Phi
│ │ │ - -- used 0.328036s (cpu); 0.20466s (thread); 0s (gc)
│ │ │ + -- used 0.276889s (cpu); 0.168118s (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 0.328036s (cpu); 0.20466s (thread); 0s (gc)
│ │ │ │ + -- used 0.276889s (cpu); 0.168118s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of
│ │ │ │  multi-degree (1,1)
│ │ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces
│ │ │ │  of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2
│ │ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out
│ │ │ @@ -26,22 +26,22 @@
│ │ │  
│ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true};
│ │ │  
│ │ │  i8 : prob = n -> log(n)/n;
│ │ │  
│ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (66, 81, 84, 94, 96, 95, 95, 95, 97, 95, 98, 99, 96, 97, 98, 93, 96, 97,
│ │ │ +o9 = (72, 78, 93, 94, 94, 96, 95, 95, 98, 94, 97, 100, 99, 96, 98, 98, 100,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     96, 96, 99, 99, 98, 100, 99, 100, 97, 98, 98)
│ │ │ +     100, 98, 96, 97, 98, 99, 97, 99, 100, 99, 98, 99)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (15, 8, 9, 4, 4, 2, 4, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1,
│ │ │ +o10 = (13, 6, 11, 1, 5, 3, 6, 4, 1, 1, 1, 1, 0, 4, 1, 1, 0, 1, 1, 0, 2, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 0, 0, 0, 0)
│ │ │ +      0, 0, 0, 0, 1, 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 = {DLw, Dfw, DIO, DfG, D{K}
│ │ │ +o2 = {DBG, DS_, Duc, DMo, DKW}
│ │ │  
│ │ │  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}, {a, d}, {b, d}, {b, e}, {c, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, 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 0.000534563s (cpu); 0.000419045s (thread); 0s (gc)
│ │ │ + -- used 0.000788656s (cpu); 0.00062459s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.158681s (cpu); 0.0791909s (thread); 0s (gc)
│ │ │ + -- used 0.20193s (cpu); 0.0879606s (thread); 0s (gc)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -117,28 +117,28 @@
│ │ │                
    i8 : prob = n -> log(n)/n;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (66, 81, 84, 94, 96, 95, 95, 95, 97, 95, 98, 99, 96, 97, 98, 93, 96, 97,
│ │ │ +o9 = (72, 78, 93, 94, 94, 96, 95, 95, 98, 94, 97, 100, 99, 96, 98, 98, 100,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     96, 96, 99, 99, 98, 100, 99, 100, 97, 98, 98)
│ │ │ +     100, 98, 96, 97, 98, 99, 97, 99, 100, 99, 98, 99)
│ │ │  
│ │ │  o9 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (15, 8, 9, 4, 4, 2, 4, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1,
│ │ │ +o10 = (13, 6, 11, 1, 5, 3, 6, 4, 1, 1, 1, 1, 0, 4, 1, 1, 0, 1, 1, 0, 2, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 0, 0, 0, 0)
│ │ │ +      0, 0, 0, 0, 1, 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 = (66, 81, 84, 94, 96, 95, 95, 95, 97, 95, 98, 99, 96, 97, 98, 93, 96, 97, │ │ │ │ +o9 = (72, 78, 93, 94, 94, 96, 95, 95, 98, 94, 97, 100, 99, 96, 98, 98, 100, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 96, 96, 99, 99, 98, 100, 99, 100, 97, 98, 98) │ │ │ │ + 100, 98, 96, 97, 98, 99, 97, 99, 100, 99, 98, 99) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (15, 8, 9, 4, 4, 2, 4, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, │ │ │ │ +o10 = (13, 6, 11, 1, 5, 3, 6, 4, 1, 1, 1, 1, 0, 4, 1, 1, 0, 1, 1, 0, 2, 0, 0, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 0, 0, 0, 0, 0, 0) │ │ │ │ + 0, 0, 0, 0, 1, 0) │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with │ │ │ │ filterGraphs and countGraphs │ │ │ │ * _f_i_l_t_e_r_G_r_a_p_h_s -- filters (i.e., selects) graphs in a list for given │ │ │ │ properties │ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Graphs.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DLw, Dfw, DIO, DfG, D{K}
│ │ │ +o2 = {DBG, DS_, Duc, DMo, DKW}
│ │ │  
│ │ │  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 = {DLw, Dfw, DIO, DfG, D{K}
│ │ │ │ +o2 = {DBG, DS_, Duc, DMo, DKW}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -87,23 +87,23 @@
│ │ │                
    i1 : R = QQ[a..e];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{a, c}, {a, d}, {b, d}, {b, e}, {c, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, 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}, {a, d}, {b, d}, {b, e}, {c, e}}},
│ │ │ │ +o2 = {Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}},
│ │ │ │              "ring" => R
│ │ │ │              "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{b, c}, {a, d}, {b, d}, {a, e}, {c, e}}},
│ │ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}}}
│ │ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}}
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The number of vertices $n$ must be positive as nauty cannot handle graphs with
│ │ │ │  zero vertices.
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html
│ │ │ @@ -116,21 +116,21 @@
│ │ │              
│ │ │                
    i4 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time graphComplement G;
│ │ │ - -- used 0.000534563s (cpu); 0.000419045s (thread); 0s (gc)
    
│ │ │ + -- used 0.000788656s (cpu); 0.00062459s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time (graphComplement \ G);
│ │ │ - -- used 0.158681s (cpu); 0.0791909s (thread); 0s (gc)
    │ │ │ + -- used 0.20193s (cpu); 0.0879606s (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 0.000534563s (cpu); 0.000419045s (thread); 0s (gc) │ │ │ │ + -- used 0.000788656s (cpu); 0.00062459s (thread); 0s (gc) │ │ │ │ i6 : time (graphComplement \ G); │ │ │ │ - -- used 0.158681s (cpu); 0.0791909s (thread); 0s (gc) │ │ │ │ + -- used 0.20193s (cpu); 0.0879606s (thread); 0s (gc) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_o_m_p_l_e_m_e_n_t_G_r_a_p_h -- returns the complement of a graph or hypergraph │ │ │ │ ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: ********** │ │ │ │ * graphComplement(Graph) │ │ │ │ * graphComplement(List) │ │ │ │ * graphComplement(String) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ @@ -26,22 +26,22 @@ │ │ │ │ │ │ i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true}; │ │ │ │ │ │ i8 : prob = n -> log(n)/n; │ │ │ │ │ │ i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected)) │ │ │ │ │ │ -o9 = (73, 79, 90, 93, 91, 96, 96, 97, 97, 97, 100, 94, 98, 96, 99, 98, 95, │ │ │ +o9 = (75, 83, 86, 91, 93, 96, 98, 95, 96, 94, 99, 96, 93, 98, 96, 98, 98, 98, │ │ │ ------------------------------------------------------------------------ │ │ │ - 98, 98, 98, 98, 100, 98, 96, 98, 97, 99, 99, 100) │ │ │ + 98, 97, 98, 99, 99, 99, 98, 98, 98, 98, 99) │ │ │ │ │ │ o9 : Sequence │ │ │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected)) │ │ │ │ │ │ -o10 = (9, 11, 6, 3, 2, 5, 1, 3, 2, 2, 1, 2, 0, 0, 2, 0, 1, 1, 0, 1, 0, 1, 0, │ │ │ +o10 = (17, 8, 3, 5, 2, 0, 2, 0, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, │ │ │ ----------------------------------------------------------------------- │ │ │ - 0, 0, 0, 0, 0, 0) │ │ │ + 1, 0, 0, 0, 1, 0) │ │ │ │ │ │ o10 : Sequence │ │ │ │ │ │ i11 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ │ │ │ o1 = {DDO, Dx_, Dlw, Dx{, D_K} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : generateRandomGraphs(5, 5) │ │ │ │ │ │ -o2 = {DcS, DGW, DeG, DoK, DpC} │ │ │ +o2 = {D^c, DsS, DM{, DC?, Dwk} │ │ │ │ │ │ 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 = {DpS, Dhc, DkK} │ │ │ +o1 = {DUW, D[S, D[S} │ │ │ │ │ │ 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 0.000680426s (cpu); 0.000518453s (thread); 0s (gc) │ │ │ + -- used 0.000867606s (cpu); 0.000602696s (thread); 0s (gc) │ │ │ │ │ │ i5 : time (graphComplement \ G); │ │ │ - -- used 0.137002s (cpu); 0.0728931s (thread); 0s (gc) │ │ │ + -- used 0.268156s (cpu); 0.164631s (thread); 0s (gc) │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ @@ -117,28 +117,28 @@ │ │ │ 
      i8 : prob = n -> log(n)/n;
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (73, 79, 90, 93, 91, 96, 96, 97, 97, 97, 100, 94, 98, 96, 99, 98, 95,
│ │ │ +o9 = (75, 83, 86, 91, 93, 96, 98, 95, 96, 94, 99, 96, 93, 98, 96, 98, 98, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     98, 98, 98, 98, 100, 98, 96, 98, 97, 99, 99, 100)
│ │ │ +     98, 97, 98, 99, 99, 99, 98, 98, 98, 98, 99)
│ │ │  
│ │ │  o9 : Sequence
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (9, 11, 6, 3, 2, 5, 1, 3, 2, 2, 1, 2, 0, 0, 2, 0, 1, 1, 0, 1, 0, 1, 0,
│ │ │ +o10 = (17, 8, 3, 5, 2, 0, 2, 0, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 0, 0, 0, 0, 0)
│ │ │ +      1, 0, 0, 0, 1, 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 = (73, 79, 90, 93, 91, 96, 96, 97, 97, 97, 100, 94, 98, 96, 99, 98, 95, │ │ │ │ +o9 = (75, 83, 86, 91, 93, 96, 98, 95, 96, 94, 99, 96, 93, 98, 96, 98, 98, 98, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 98, 98, 98, 98, 100, 98, 96, 98, 97, 99, 99, 100) │ │ │ │ + 98, 97, 98, 99, 99, 99, 98, 98, 98, 98, 99) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (9, 11, 6, 3, 2, 5, 1, 3, 2, 2, 1, 2, 0, 0, 2, 0, 1, 1, 0, 1, 0, 1, 0, │ │ │ │ +o10 = (17, 8, 3, 5, 2, 0, 2, 0, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 0, 0, 0, 0, 0, 0) │ │ │ │ + 1, 0, 0, 0, 1, 0) │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with │ │ │ │ filterGraphs and countGraphs │ │ │ │ * _f_i_l_t_e_r_G_r_a_p_h_s -- filters (i.e., selects) graphs in a list for given │ │ │ │ properties │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Graphs.html │ │ │ @@ -93,15 +93,15 @@ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DcS, DGW, DeG, DoK, DpC}
│ │ │ +o2 = {D^c, DsS, DM{, DC?, Dwk}
│ │ │  
│ │ │  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 = {DcS, DGW, DeG, DoK, DpC}
│ │ │ │ +o2 = {D^c, DsS, DM{, DC?, Dwk}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │            
    This method generates a specified number of random graphs on a given number of vertices with a given regularity. Note that some graphs may be isomorphic.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │  
│ │ │ -o1 = {DpS, Dhc, DkK}
│ │ │ +o1 = {DUW, D[S, D[S}
│ │ │  
│ │ │  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 = {DpS, Dhc, DkK} │ │ │ │ +o1 = {DUW, D[S, D[S} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ The number of vertices $n$ must be positive as nauty cannot handle graphs with │ │ │ │ zero vertices. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_a_t_e_R_a_n_d_o_m_G_r_a_p_h_s -- generates random graphs on a given number of │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ @@ -110,21 +110,21 @@ │ │ │ │ │ │ 
    i3 : G = generateBipartiteGraphs 7;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time graphComplement G;
│ │ │ - -- used 0.000680426s (cpu); 0.000518453s (thread); 0s (gc)
    │ │ │ + -- used 0.000867606s (cpu); 0.000602696s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time (graphComplement \ G);
│ │ │ - -- used 0.137002s (cpu); 0.0728931s (thread); 0s (gc)
    │ │ │ + -- used 0.268156s (cpu); 0.164631s (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 0.000680426s (cpu); 0.000518453s (thread); 0s (gc) │ │ │ │ + -- used 0.000867606s (cpu); 0.000602696s (thread); 0s (gc) │ │ │ │ i5 : time (graphComplement \ G); │ │ │ │ - -- used 0.137002s (cpu); 0.0728931s (thread); 0s (gc) │ │ │ │ + -- used 0.268156s (cpu); 0.164631s (thread); 0s (gc) │ │ │ │ ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: ********** │ │ │ │ * graphComplement(Graph) │ │ │ │ * graphComplement(List) │ │ │ │ * graphComplement(String) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _g_r_a_p_h_C_o_m_p_l_e_m_e_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ @@ -47,15 +47,15 @@ │ │ │ o4 : Ideal of R │ │ │ │ │ │ i5 : isPrimary Q │ │ │ │ │ │ o5 = true │ │ │ │ │ │ i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot") │ │ │ - -- .112449s elapsed │ │ │ + -- .0992123s elapsed │ │ │ │ │ │ o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, | │ │ │ ------------------------------------------------------------------------ │ │ │ 2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |} │ │ │ │ │ │ o6 : List │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ @@ -120,15 +120,15 @@ │ │ │ │ │ │ o5 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .112449s elapsed
│ │ │ + -- .0992123s 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") │ │ │ │ - -- .112449s elapsed │ │ │ │ + -- .0992123s elapsed │ │ │ │ │ │ │ │ o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ @@ -78,15 +78,15 @@ │ │ │ i13 : for i to dim X list hilbertFunction (i, A1) │ │ │ │ │ │ o13 = {1, 2, 3, 3, 2, 1} │ │ │ │ │ │ o13 : List │ │ │ │ │ │ i14 : Y = time smoothFanoToricVariety(5,100); │ │ │ - -- used 0.230349s (cpu); 0.228441s (thread); 0s (gc) │ │ │ + -- used 0.267923s (cpu); 0.266508s (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 0.000654147s (cpu); 0.00121668s (thread); 0s (gc) │ │ │ - -- used 2.4616e-05s (cpu); 8.2975e-05s (thread); 0s (gc) │ │ │ - -- used 9.318e-06s (cpu); 7.2305e-05s (thread); 0s (gc) │ │ │ - -- used 8.797e-06s (cpu); 7.5291e-05s (thread); 0s (gc) │ │ │ - -- used 8.867e-06s (cpu); 7.0392e-05s (thread); 0s (gc) │ │ │ - -- used 8.726e-06s (cpu); 7.2306e-05s (thread); 0s (gc) │ │ │ + -- used 0.0039447s (cpu); 0.00251108s (thread); 0s (gc) │ │ │ + -- used 5.4573e-05s (cpu); 0.000163569s (thread); 0s (gc) │ │ │ + -- used 1.6645e-05s (cpu); 8.7056e-05s (thread); 0s (gc) │ │ │ + -- used 1.0951e-05s (cpu); 8.7017e-05s (thread); 0s (gc) │ │ │ + -- used 1.0901e-05s (cpu); 8.9751e-05s (thread); 0s (gc) │ │ │ + -- used 1.0778e-05s (cpu); 8.3416e-05s (thread); 0s (gc) │ │ │ │ │ │ o19 = {1, 6, 13, 13, 6, 1} │ │ │ │ │ │ o19 : List │ │ │ │ │ │ i20 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_is__Well__Defined_lp__Normal__Toric__Variety_rp.out │ │ │ @@ -1,29 +1,29 @@ │ │ │ -- -*- M2-comint -*- hash: 16408385764843695632 │ │ │ │ │ │ i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1)) │ │ │ │ │ │ i2 : setRandomSeed (currentTime ()); │ │ │ - -- setting random seed to 1768844604 │ │ │ + -- setting random seed to 1769302525 │ │ │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ -o3 = {0, 2, 5} │ │ │ +o3 = {5, 6, 6} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : assert isWellDefined kleinschmidt (4,a) │ │ │ │ │ │ i5 : q = sort apply (5, j -> random (1,9)); │ │ │ │ │ │ i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9)); │ │ │ │ │ │ i7 : q │ │ │ │ │ │ -o7 = {1, 2, 4, 4, 5} │ │ │ +o7 = {1, 4, 6, 7, 8} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : assert isWellDefined weightedProjectiveSpace q │ │ │ │ │ │ i9 : X = new MutableHashTable; │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ @@ -6,61 +6,61 @@ │ │ │ │ │ │ o2 = 5*PP2 │ │ │ 0 │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │ │ │ │ i3 : M1 = elapsedTime monomials D1 │ │ │ - -- .0278162s elapsed │ │ │ + -- .0412586s 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)) │ │ │ - -- .0011212s elapsed │ │ │ + -- .00141443s 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 │ │ │ - -- .0290341s elapsed │ │ │ + -- .043058s 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)) │ │ │ - -- .000958015s elapsed │ │ │ + -- .00122144s 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 │ │ │ - -- 41.7392s elapsed │ │ │ + -- 31.7754s elapsed │ │ │ │ │ │ o11 = 7909 │ │ │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3)) │ │ │ - -- .0373902s elapsed │ │ │ + -- .0304072s elapsed │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ @@ -24,19 +24,19 @@ │ │ │ o3 : List │ │ │ │ │ │ i4 : X = normalToricVariety F; │ │ │ │ │ │ i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F) │ │ │ │ │ │ i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}}) │ │ │ - -- used 2.7692e-05s (cpu); 2.1681e-05s (thread); 0s (gc) │ │ │ + -- used 3.8344e-05s (cpu); 2.4789e-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 0.0440063s (cpu); 0.0440101s (thread); 0s (gc) │ │ │ + -- used 0.0525118s (cpu); 0.0525229s (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 0.024071s (cpu); 0.0240699s (thread); 0s (gc) │ │ │ + -- used 0.0293367s (cpu); 0.0293359s (thread); 0s (gc) │ │ │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ - -- used 0.00249395s (cpu); 0.00249453s (thread); 0s (gc) │ │ │ + -- used 0.0032601s (cpu); 0.00326571s (thread); 0s (gc) │ │ │ │ │ │ i21 : assert (set rays X2 === set rays X1 and max X1 === max X2) │ │ │ │ │ │ i22 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ @@ -207,15 +207,15 @@ │ │ │
    │ │ │

    We end with a slightly larger example.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -254,20 +254,20 @@ │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.230349s (cpu); 0.228441s (thread); 0s (gc)
    │ │ │ + -- used 0.267923s (cpu); 0.266508s (thread); 0s (gc) │ │ │
    │ │ │ 
    i15 : A2 = intersectionRing Y;
    │ │ │
    │ │ │ 
    i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.000654147s (cpu); 0.00121668s (thread); 0s (gc)
│ │ │ - -- used 2.4616e-05s (cpu); 8.2975e-05s (thread); 0s (gc)
│ │ │ - -- used 9.318e-06s (cpu); 7.2305e-05s (thread); 0s (gc)
│ │ │ - -- used 8.797e-06s (cpu); 7.5291e-05s (thread); 0s (gc)
│ │ │ - -- used 8.867e-06s (cpu); 7.0392e-05s (thread); 0s (gc)
│ │ │ - -- used 8.726e-06s (cpu); 7.2306e-05s (thread); 0s (gc)
│ │ │ + -- used 0.0039447s (cpu); 0.00251108s (thread); 0s (gc)
│ │ │ + -- used 5.4573e-05s (cpu); 0.000163569s (thread); 0s (gc)
│ │ │ + -- used 1.6645e-05s (cpu); 8.7056e-05s (thread); 0s (gc)
│ │ │ + -- used 1.0951e-05s (cpu); 8.7017e-05s (thread); 0s (gc)
│ │ │ + -- used 1.0901e-05s (cpu); 8.9751e-05s (thread); 0s (gc)
│ │ │ + -- used 1.0778e-05s (cpu); 8.3416e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = {1, 6, 13, 13, 6, 1}
│ │ │  
│ │ │  o19 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -96,15 +96,15 @@ │ │ │ │ i13 : for i to dim X list hilbertFunction (i, A1) │ │ │ │ │ │ │ │ o13 = {1, 2, 3, 3, 2, 1} │ │ │ │ │ │ │ │ o13 : List │ │ │ │ We end with a slightly larger example. │ │ │ │ i14 : Y = time smoothFanoToricVariety(5,100); │ │ │ │ - -- used 0.230349s (cpu); 0.228441s (thread); 0s (gc) │ │ │ │ + -- used 0.267923s (cpu); 0.266508s (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 0.000654147s (cpu); 0.00121668s (thread); 0s (gc) │ │ │ │ - -- used 2.4616e-05s (cpu); 8.2975e-05s (thread); 0s (gc) │ │ │ │ - -- used 9.318e-06s (cpu); 7.2305e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.797e-06s (cpu); 7.5291e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.867e-06s (cpu); 7.0392e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.726e-06s (cpu); 7.2306e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.0039447s (cpu); 0.00251108s (thread); 0s (gc) │ │ │ │ + -- used 5.4573e-05s (cpu); 0.000163569s (thread); 0s (gc) │ │ │ │ + -- used 1.6645e-05s (cpu); 8.7056e-05s (thread); 0s (gc) │ │ │ │ + -- used 1.0951e-05s (cpu); 8.7017e-05s (thread); 0s (gc) │ │ │ │ + -- used 1.0901e-05s (cpu); 8.9751e-05s (thread); 0s (gc) │ │ │ │ + -- used 1.0778e-05s (cpu); 8.3416e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o19 = {1, 6, 13, 13, 6, 1} │ │ │ │ │ │ │ │ o19 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _s_h_e_a_v_e_s -- information about coherent sheaves and total │ │ │ │ coordinate rings (a.k.a. Cox rings) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_is__Well__Defined_lp__Normal__Toric__Variety_rp.html │ │ │ @@ -93,22 +93,22 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i2 : setRandomSeed (currentTime ());
│ │ │ - -- setting random seed to 1768844604
    │ │ │ + -- setting random seed to 1769302525 │ │ │
    │ │ │ 
    i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {0, 2, 5}
│ │ │ +o3 = {5, 6, 6}
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ 
    i4 : assert isWellDefined kleinschmidt (4,a)
    │ │ │ @@ -126,15 +126,15 @@ │ │ │ 
    i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9));
    │ │ │
    │ │ │ 
    i7 : q
│ │ │  
│ │ │ -o7 = {1, 2, 4, 4, 5}
│ │ │ +o7 = {1, 4, 6, 7, 8}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ 
    i8 : assert isWellDefined weightedProjectiveSpace q
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,27 +28,27 @@ │ │ │ │ * the intersection of the cones associated to two elements of coneList is a │ │ │ │ face of each cone. │ │ │ │ The first examples illustrate that small projective spaces are well-defined. │ │ │ │ i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1)) │ │ │ │ The second examples show that a randomly selected Kleinschmidt toric variety │ │ │ │ and a weighted projective space are also well-defined. │ │ │ │ i2 : setRandomSeed (currentTime ()); │ │ │ │ - -- setting random seed to 1768844604 │ │ │ │ + -- setting random seed to 1769302525 │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ │ │ -o3 = {0, 2, 5} │ │ │ │ +o3 = {5, 6, 6} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : assert isWellDefined kleinschmidt (4,a) │ │ │ │ i5 : q = sort apply (5, j -> random (1,9)); │ │ │ │ i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j │ │ │ │ -> random (1,9)); │ │ │ │ i7 : q │ │ │ │ │ │ │ │ -o7 = {1, 2, 4, 4, 5} │ │ │ │ +o7 = {1, 4, 6, 7, 8} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : assert isWellDefined weightedProjectiveSpace q │ │ │ │ The next ten examples illustrate various ways that two lists can fail to define │ │ │ │ a normal toric variety. By making the current debugging level greater than one, │ │ │ │ one gets some addition information about the nature of the failure. │ │ │ │ i9 : X = new MutableHashTable; │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ @@ -96,15 +96,15 @@ │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │
    │ │ │ 
    i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .0278162s elapsed
│ │ │ + -- .0412586s elapsed
│ │ │  
│ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4   
│ │ │  o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
│ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2 
│ │ │       ------------------------------------------------------------------------
│ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ 
    i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .0011212s elapsed
    │ │ │ + -- .00141443s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ @@ -138,27 +138,27 @@ │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -171,23 +171,23 @@ │ │ │ │ │ │ o10 : ToricDivisor on X │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .0290341s elapsed
│ │ │ + -- .043058s 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))
│ │ │ - -- .000958015s elapsed
    │ │ │ + -- .00122144s elapsed │ │ │
    │ │ │ 
    i9 : X = kleinschmidt (5, {1,2,3});
    │ │ │
    │ │ │ 
    i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 41.7392s elapsed
│ │ │ + -- 31.7754s elapsed
│ │ │  
│ │ │  o11 = 7909
    │ │ │
    │ │ │ 
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .0373902s elapsed
    │ │ │ + -- .0304072s 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 │ │ │ │ - -- .0278162s elapsed │ │ │ │ + -- .0412586s 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)) │ │ │ │ - -- .0011212s elapsed │ │ │ │ + -- .00141443s 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 │ │ │ │ - -- .0290341s elapsed │ │ │ │ + -- .043058s 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)) │ │ │ │ - -- .000958015s elapsed │ │ │ │ + -- .00122144s 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 │ │ │ │ - -- 41.7392s elapsed │ │ │ │ + -- 31.7754s elapsed │ │ │ │ │ │ │ │ o11 = 7909 │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety │ │ │ │ D3)) │ │ │ │ - -- .0373902s elapsed │ │ │ │ + -- .0304072s elapsed │ │ │ │ By exploiting _l_a_t_t_i_c_e_P_o_i_n_t_s, this method function avoids using the _b_a_s_i_s │ │ │ │ function. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _d_i_v_i_s_o_r_s -- information about toric divisors and their │ │ │ │ related groups │ │ │ │ * _r_i_n_g_(_N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_) -- make the total coordinate ring (a.k.a. Cox │ │ │ │ ring) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ @@ -125,25 +125,25 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -48,22 +48,22 @@ │ │ │ │ i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F) │ │ │ │ The recommended method for creating a _N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y from a fan is │ │ │ │ _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_L_i_s_t_,_L_i_s_t_). In fact, this package avoids using objects from │ │ │ │ the _P_o_l_y_h_e_d_r_a package whenever possible. Here is a trivial example, namely │ │ │ │ projective 2-space, illustrating the substantial increase in time resulting │ │ │ │ from the use of a _P_o_l_y_h_e_d_r_a fan. │ │ │ │ i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}}) │ │ │ │ - -- used 2.7692e-05s (cpu); 2.1681e-05s (thread); 0s (gc) │ │ │ │ + -- used 3.8344e-05s (cpu); 2.4789e-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 0.0440063s (cpu); 0.0440101s (thread); 0s (gc) │ │ │ │ + -- used 0.0525118s (cpu); 0.0525229s (thread); 0s (gc) │ │ │ │ i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y -- make a normal toric variety │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_F_a_n_) -- make a normal toric variety from a 'Polyhedra' │ │ │ │ fan │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ @@ -233,21 +233,21 @@ │ │ │ 2 3 │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -102,17 +102,17 @@ │ │ │ │ │ │ │ │ o18 = | 0 1 0 | │ │ │ │ | 0 0 1 | │ │ │ │ │ │ │ │ 2 3 │ │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ i19 : X1 = time normalToricVariety convexHull (vertMatrix); │ │ │ │ - -- used 0.024071s (cpu); 0.0240699s (thread); 0s (gc) │ │ │ │ + -- used 0.0293367s (cpu); 0.0293359s (thread); 0s (gc) │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ │ - -- used 0.00249395s (cpu); 0.00249453s (thread); 0s (gc) │ │ │ │ + -- used 0.0032601s (cpu); 0.00326571s (thread); 0s (gc) │ │ │ │ i21 : assert (set rays X2 === set rays X1 and max X1 === max X2) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_M_a_t_r_i_x_) -- make a normal toric variety from a polytope │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_P_o_l_y_h_e_d_r_o_n_) -- make a normal toric variety from a │ │ │ │ 'Polyhedra' polyhedron │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ @@ -21,21 +21,21 @@ │ │ │ │ │ │ i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue == 0 │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ Sampling image points ... │ │ │ - -- used .0896332 seconds │ │ │ + -- used .111717 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .00617284 seconds │ │ │ + -- used .00611409 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00287754 seconds │ │ │ + -- used .00305823 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .00046879 seconds │ │ │ + -- used .000465784 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 .00393121 seconds │ │ │ + -- used .00435217 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0029037 seconds │ │ │ + -- used .00322697 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00114173 seconds │ │ │ + -- used .00125522 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000312997 seconds │ │ │ + -- used .000292997 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 .0130679 seconds │ │ │ + -- used .0157352 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0117076 seconds │ │ │ + -- used .0135033 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00731237 seconds │ │ │ + -- used .00784566 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000777268 seconds │ │ │ + -- used .000953385 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 .00322951 seconds │ │ │ + -- used .00380118 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00852724 seconds │ │ │ + -- used .0093833 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000909917 seconds │ │ │ + -- used .00115813 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 0.0678731s (cpu); 0.0678708s (thread); 0s (gc) │ │ │ + -- used 0.0765464s (cpu); 0.0765368s (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) │ │ │ - -- .614736s elapsed │ │ │ + -- .605363s elapsed │ │ │ │ │ │ o7 = p │ │ │ │ │ │ o7 : Point │ │ │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html │ │ │ @@ -100,21 +100,21 @@ │ │ │ o4 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -33,21 +33,21 @@ │ │ │ │ o3 : Ideal of R │ │ │ │ i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue │ │ │ │ == 0 │ │ │ │ │ │ │ │ o4 = true │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .0896332 seconds │ │ │ │ + -- used .111717 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .00617284 seconds │ │ │ │ + -- used .00611409 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00287754 seconds │ │ │ │ + -- used .00305823 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .00046879 seconds │ │ │ │ + -- used .000465784 seconds │ │ │ │ │ │ │ │ o5 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 3 forms in the ideal of the image has dimension 3 │ │ │ │ │ │ │ │ o5 : NumericalInterpolationTable │ │ │ │ i6 : extractImageEquations(T, AttemptZZ => true) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html │ │ │ @@ -102,21 +102,21 @@ │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,21 +38,21 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .00393121 seconds │ │ │ │ + -- used .00435217 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .0029037 seconds │ │ │ │ + -- used .00322697 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00114173 seconds │ │ │ │ + -- used .00125522 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000312997 seconds │ │ │ │ + -- used .000292997 seconds │ │ │ │ │ │ │ │ o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 | │ │ │ │ │ │ │ │ 1 3 │ │ │ │ o3 : Matrix (CC [y ..y ]) <-- (CC [y ..y ]) │ │ │ │ 53 0 3 53 0 3 │ │ │ │ Here is how to do the same computation symbolically. │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html │ │ │ @@ -107,21 +107,21 @@ │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -147,19 +147,19 @@ │ │ │ 
    i6 : S = numericalImageSample(F, ideal 0_R, 60);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -57,39 +57,39 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : numericalHilbertFunction(F, ideal 0_R, 4) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .0130679 seconds │ │ │ │ + -- used .0157352 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .0117076 seconds │ │ │ │ + -- used .0135033 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00731237 seconds │ │ │ │ + -- used .00784566 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000777268 seconds │ │ │ │ + -- used .000953385 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 .00322951 seconds │ │ │ │ + -- used .00380118 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00852724 seconds │ │ │ │ + -- used .0093833 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000909917 seconds │ │ │ │ + -- used .00115813 seconds │ │ │ │ │ │ │ │ o7 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 2 forms in the ideal of the image has dimension 1 │ │ │ │ │ │ │ │ o7 : NumericalInterpolationTable │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _N_u_m_e_r_i_c_a_l_I_n_t_e_r_p_o_l_a_t_i_o_n_T_a_b_l_e -- the class of all │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ @@ -140,15 +140,15 @@ │ │ │ 1 70 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 2.7692e-05s (cpu); 2.1681e-05s (thread); 0s (gc)
│ │ │ + -- used 3.8344e-05s (cpu); 2.4789e-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 0.0440063s (cpu); 0.0440101s (thread); 0s (gc)
    │ │ │ + -- used 0.0525118s (cpu); 0.0525229s (thread); 0s (gc) │ │ │
    │ │ │ 
    i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
    │ │ │
    │ │ │ 
    i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.024071s (cpu); 0.0240699s (thread); 0s (gc)
    │ │ │ + -- used 0.0293367s (cpu); 0.0293359s (thread); 0s (gc) │ │ │
    │ │ │ 
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00249395s (cpu); 0.00249453s (thread); 0s (gc)
    │ │ │ + -- used 0.0032601s (cpu); 0.00326571s (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 .0896332 seconds
│ │ │ +     -- used .111717 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00617284 seconds
│ │ │ +     -- used .00611409 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00287754 seconds
│ │ │ +     -- used .00305823 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .00046879 seconds
│ │ │ +     -- used .000465784 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 .00393121 seconds
│ │ │ +     -- used .00435217 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0029037 seconds
│ │ │ +     -- used .00322697 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00114173 seconds
│ │ │ +     -- used .00125522 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000312997 seconds
│ │ │ +     -- used .000292997 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 .0130679 seconds
│ │ │ +     -- used .0157352 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0117076 seconds
│ │ │ +     -- used .0135033 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00731237 seconds
│ │ │ +     -- used .00784566 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000777268 seconds
│ │ │ +     -- used .000953385 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 .00322951 seconds
│ │ │ +     -- used .00380118 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00852724 seconds
│ │ │ +     -- used .0093833 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000909917 seconds
│ │ │ +     -- used .00115813 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 0.0678731s (cpu); 0.0678708s (thread); 0s (gc)
│ │ │ + -- used 0.0765464s (cpu); 0.0765368s (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 0.0678731s (cpu); 0.0678708s (thread); 0s (gc) │ │ │ │ + -- used 0.0765464s (cpu); 0.0765368s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 = 69 │ │ │ │ ********** WWaayyss ttoo uussee nnuummeerriiccaallIImmaaggeeDDiimm:: ********** │ │ │ │ * numericalImageDim(List,Ideal) │ │ │ │ * numericalImageDim(List,Ideal,Point) │ │ │ │ * numericalImageDim(Matrix,Ideal) │ │ │ │ * numericalImageDim(Matrix,Ideal,Point) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html │ │ │ @@ -132,15 +132,15 @@ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .614736s elapsed
│ │ │ + -- .605363s 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) │ │ │ │ - -- .614736s elapsed │ │ │ │ + -- .605363s elapsed │ │ │ │ │ │ │ │ o7 = p │ │ │ │ │ │ │ │ o7 : Point │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ │ │ │ │ │ │ o8 = | .722359 .289465 -.295808 .591752 -.454678 | │ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/example-output/_set__Verbose__Level.out │ │ │ @@ -52,92 +52,92 @@ │ │ │ │ │ │ i4 : assert all(S,s->checkIncidenceSolution(s,SchPblm)) │ │ │ │ │ │ i5 : setVerboseLevel 1; │ │ │ │ │ │ i6 : S = solveSchubertProblem(SchPblm,2,4) │ │ │ -- playCheckers │ │ │ --- cpu time = .100804 │ │ │ +-- cpu time = .0120396 │ │ │ -- making a recursive call to resolveNode │ │ │ -- playCheckers │ │ │ --- cpu time = .00399912 │ │ │ +-- cpu time = .00795362 │ │ │ -- making a recursive call to resolveNode │ │ │ -- playCheckers │ │ │ -- cpu time = 0 │ │ │ resolveNode reached node of no remaining conditions │ │ │ --- time to make equations: .00504761 │ │ │ +-- time to make equations: .00806788 │ │ │ Setup time: 0 │ │ │ Computing time:0 │ │ │ - -- trackHomotopy time = .044435 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}] │ │ │ --- time of performing one checker move: .0789155 │ │ │ --- time of performing one checker move: .00400182 │ │ │ + -- trackHomotopy time = .00854832 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}] │ │ │ +-- time of performing one checker move: .023931 │ │ │ +-- time of performing one checker move: .00399611 │ │ │ -- time of performing one checker move: 0 │ │ │ --- time to make equations: .00800052 │ │ │ +-- time to make equations: .00797147 │ │ │ Setup time: 0 │ │ │ Computing time:0 │ │ │ - -- trackHomotopy time = .00689276 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}] │ │ │ --- time of performing one checker move: .0200011 │ │ │ --- time to make equations: .0972349 │ │ │ + -- trackHomotopy time = .00742244 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}] │ │ │ +-- time of performing one checker move: .0201014 │ │ │ +-- time to make equations: .00800177 │ │ │ Setup time: 0 │ │ │ Computing time:0 │ │ │ - -- trackHomotopy time = .0072978 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}] │ │ │ --- time of performing one checker move: .112973 │ │ │ --- time to make equations: .00399856 │ │ │ + -- trackHomotopy time = .0330732 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}] │ │ │ +-- time of performing one checker move: .142606 │ │ │ +-- time to make equations: .00808383 │ │ │ Setup time: 0 │ │ │ Computing time:0 │ │ │ - -- trackHomotopy time = .00724033 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}] │ │ │ --- time of performing one checker move: .121158 │ │ │ --- time to make equations: .0123239 │ │ │ + -- trackHomotopy time = .00864067 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}] │ │ │ +-- time of performing one checker move: .0240247 │ │ │ +-- time to make equations: .139921 │ │ │ Setup time: 0 │ │ │ Computing time:0 │ │ │ - -- trackHomotopy time = .00797456 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}] │ │ │ --- time of performing one checker move: .027106 │ │ │ --- time to make equations: .0750451 │ │ │ + -- trackHomotopy time = .00997745 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}] │ │ │ +-- time of performing one checker move: .155873 │ │ │ +-- time to make equations: .0160292 │ │ │ Setup time: 0 │ │ │ Computing time:0 │ │ │ - -- trackHomotopy time = .00796739 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}] │ │ │ --- time of performing one checker move: .0880204 │ │ │ + -- trackHomotopy time = .0371607 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}] │ │ │ +-- time of performing one checker move: .153219 │ │ │ +-- time of performing one checker move: .0039078 │ │ │ -- time of performing one checker move: 0 │ │ │ --- time of performing one checker move: .00400093 │ │ │ --- time to make equations: .108205 │ │ │ +-- time to make equations: .0159416 │ │ │ Setup time: 0 │ │ │ Computing time:0 │ │ │ - -- trackHomotopy time = .00796862 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}] │ │ │ --- time of performing one checker move: .124193 │ │ │ + -- trackHomotopy time = .0382822 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}] │ │ │ +-- time of performing one checker move: .143473 │ │ │ -- making a recursive call to resolveNode │ │ │ -- playCheckers │ │ │ --- cpu time = .00800028 │ │ │ +-- cpu time = .00397309 │ │ │ -- making a recursive call to resolveNode │ │ │ -- playCheckers │ │ │ --- cpu time = 0 │ │ │ +-- cpu time = .00404597 │ │ │ resolveNode reached node of no remaining conditions │ │ │ --- time to make equations: .00705985 │ │ │ +-- time to make equations: .00805531 │ │ │ Setup time: 0 │ │ │ Computing time:0 │ │ │ - -- trackHomotopy time = .0062539 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}] │ │ │ --- time of performing one checker move: .073368 │ │ │ --- time of performing one checker move: .00400236 │ │ │ --- time to make equations: .00800006 │ │ │ + -- trackHomotopy time = .00737808 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}] │ │ │ +-- time of performing one checker move: .143768 │ │ │ +-- time of performing one checker move: .00396881 │ │ │ +-- time to make equations: .00463389 │ │ │ Setup time: 0 │ │ │ Computing time:0 │ │ │ - -- trackHomotopy time = .0657139 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}] │ │ │ --- time of performing one checker move: .0970197 │ │ │ --- time of performing one checker move: 0 │ │ │ + -- trackHomotopy time = .00870738 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}] │ │ │ +-- time of performing one checker move: .0237647 │ │ │ +-- time of performing one checker move: .124621 │ │ │ -- time of performing one checker move: 0 │ │ │ +-- time of performing one checker move: .00404595 │ │ │ +-- time of performing one checker move: .00397077 │ │ │ -- time of performing one checker move: 0 │ │ │ +-- time of performing one checker move: .00401083 │ │ │ -- time of performing one checker move: 0 │ │ │ --- time of performing one checker move: .00400019 │ │ │ --- time of performing one checker move: 0 │ │ │ --- time of performing one checker move: .00400058 │ │ │ --- time to make equations: .087017 │ │ │ +-- time to make equations: .012004 │ │ │ Setup time: 0 │ │ │ Computing time:0 │ │ │ - -- trackHomotopy time = .00760701 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}] │ │ │ --- time of performing one checker move: .0990156 │ │ │ --- time of performing one checker move: 0 │ │ │ + -- trackHomotopy time = .050622 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}] │ │ │ +-- time of performing one checker move: .160807 │ │ │ +-- time of performing one checker move: .00403988 │ │ │ │ │ │ o6 = {| -1.65573-.600637ii .0201935+.0437095ii |, | -.154703+.175591ii │ │ │ | -1.23037-1.66989ii -.0308057-.00120618ii | | -.801221-.0354303ii │ │ │ | 1.35971-.743988ii -.0713133-.049047ii | | .325581-2.08048ii │ │ │ | -.397038-1.8974ii .0102261-.024397ii | | -.475895-.209388ii │ │ │ ------------------------------------------------------------------------ │ │ │ .0376857+.0683239ii |} │ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/html/_set__Verbose__Level.html │ │ │ @@ -147,92 +147,92 @@ │ │ │ 
    i5 : setVerboseLevel 1;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │  -- playCheckers
│ │ │ --- cpu time = .100804
│ │ │ +-- cpu time = .0120396
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00399912
│ │ │ +-- cpu time = .00795362
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │  -- cpu time = 0
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00504761
│ │ │ +-- time to make equations: .00806788
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .044435 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .0789155
│ │ │ --- time of performing one checker move: .00400182
│ │ │ + -- trackHomotopy time = .00854832 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .023931
│ │ │ +-- time of performing one checker move: .00399611
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time to make equations: .00800052
│ │ │ +-- time to make equations: .00797147
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00689276 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0200011
│ │ │ --- time to make equations: .0972349
│ │ │ + -- trackHomotopy time = .00742244 sec. for [{1, 2, 3, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0201014
│ │ │ +-- time to make equations: .00800177
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0072978 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .112973
│ │ │ --- time to make equations: .00399856
│ │ │ + -- trackHomotopy time = .0330732 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .142606
│ │ │ +-- time to make equations: .00808383
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00724033 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ --- time of performing one checker move: .121158
│ │ │ --- time to make equations: .0123239
│ │ │ + -- trackHomotopy time = .00864067 sec. for [{2, 3, 1, 0}, {2, infinity, infinity, 1}]
│ │ │ +-- time of performing one checker move: .0240247
│ │ │ +-- time to make equations: .139921
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00797456 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ --- time of performing one checker move: .027106
│ │ │ --- time to make equations: .0750451
│ │ │ + -- trackHomotopy time = .00997745 sec. for [{0, 1, 2, 3}, {infinity, 1, 2, infinity}]
│ │ │ +-- time of performing one checker move: .155873
│ │ │ +-- time to make equations: .0160292
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00796739 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ --- time of performing one checker move: .0880204
│ │ │ + -- trackHomotopy time = .0371607 sec. for [{0, 1, 3, 2}, {infinity, 1, infinity, 2}]
│ │ │ +-- time of performing one checker move: .153219
│ │ │ +-- time of performing one checker move: .0039078
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00400093
│ │ │ --- time to make equations: .108205
│ │ │ +-- time to make equations: .0159416
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00796862 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ --- time of performing one checker move: .124193
│ │ │ + -- trackHomotopy time = .0382822 sec. for [{1, 3, 2, 0}, {infinity, 3, infinity, 1}]
│ │ │ +-- time of performing one checker move: .143473
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = .00800028
│ │ │ +-- cpu time = .00397309
│ │ │  -- making a recursive call to resolveNode
│ │ │  -- playCheckers
│ │ │ --- cpu time = 0
│ │ │ +-- cpu time = .00404597
│ │ │  resolveNode reached node of no remaining conditions
│ │ │ --- time to make equations: .00705985
│ │ │ +-- time to make equations: .00805531
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0062539 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ --- time of performing one checker move: .073368
│ │ │ --- time of performing one checker move: .00400236
│ │ │ --- time to make equations: .00800006
│ │ │ + -- trackHomotopy time = .00737808 sec. for [{0, 1, 2, 3}, {0, infinity, 2, infinity}]
│ │ │ +-- time of performing one checker move: .143768
│ │ │ +-- time of performing one checker move: .00396881
│ │ │ +-- time to make equations: .00463389
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .0657139 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ --- time of performing one checker move: .0970197
│ │ │ --- time of performing one checker move: 0
│ │ │ + -- trackHomotopy time = .00870738 sec. for [{0, 2, 3, 1}, {0, infinity, infinity, 2}]
│ │ │ +-- time of performing one checker move: .0237647
│ │ │ +-- time of performing one checker move: .124621
│ │ │  -- time of performing one checker move: 0
│ │ │ +-- time of performing one checker move: .00404595
│ │ │ +-- time of performing one checker move: .00397077
│ │ │  -- time of performing one checker move: 0
│ │ │ +-- time of performing one checker move: .00401083
│ │ │  -- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00400019
│ │ │ --- time of performing one checker move: 0
│ │ │ --- time of performing one checker move: .00400058
│ │ │ --- time to make equations: .087017
│ │ │ +-- time to make equations: .012004
│ │ │  Setup time: 0
│ │ │  Computing time:0
│ │ │ - -- trackHomotopy time = .00760701 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ --- time of performing one checker move: .0990156
│ │ │ --- time of performing one checker move: 0
│ │ │ + -- trackHomotopy time = .050622 sec. for [{1, 3, 2, 0}, {1, infinity, infinity, 3}]
│ │ │ +-- time of performing one checker move: .160807
│ │ │ +-- time of performing one checker move: .00403988
│ │ │  
│ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii 
│ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii  
│ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii 
│ │ │       ------------------------------------------------------------------------
│ │ │       .0376857+.0683239ii   |}
│ │ │ ├── html2text {}
│ │ │ │ @@ -65,102 +65,102 @@
│ │ │ │       -.0336427+.0141017ii  |
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : assert all(S,s->checkIncidenceSolution(s,SchPblm))
│ │ │ │  i5 : setVerboseLevel 1;
│ │ │ │  i6 : S = solveSchubertProblem(SchPblm,2,4)
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .100804
│ │ │ │ +-- cpu time = .0120396
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00399912
│ │ │ │ +-- cpu time = .00795362
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │  -- cpu time = 0
│ │ │ │  resolveNode reached node of no remaining conditions
│ │ │ │ --- time to make equations: .00504761
│ │ │ │ +-- time to make equations: .00806788
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .044435 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │ + -- trackHomotopy time = .00854832 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .0789155
│ │ │ │ --- time of performing one checker move: .00400182
│ │ │ │ +-- time of performing one checker move: .023931
│ │ │ │ +-- time of performing one checker move: .00399611
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time to make equations: .00800052
│ │ │ │ +-- time to make equations: .00797147
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00689276 sec. for [{1, 2, 3, 0}, {1, infinity,
│ │ │ │ + -- trackHomotopy time = .00742244 sec. for [{1, 2, 3, 0}, {1, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .0200011
│ │ │ │ --- time to make equations: .0972349
│ │ │ │ +-- time of performing one checker move: .0201014
│ │ │ │ +-- time to make equations: .00800177
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0072978 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │ + -- trackHomotopy time = .0330732 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .112973
│ │ │ │ --- time to make equations: .00399856
│ │ │ │ +-- time of performing one checker move: .142606
│ │ │ │ +-- time to make equations: .00808383
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00724033 sec. for [{2, 3, 1, 0}, {2, infinity,
│ │ │ │ + -- trackHomotopy time = .00864067 sec. for [{2, 3, 1, 0}, {2, infinity,
│ │ │ │  infinity, 1}]
│ │ │ │ --- time of performing one checker move: .121158
│ │ │ │ --- time to make equations: .0123239
│ │ │ │ +-- time of performing one checker move: .0240247
│ │ │ │ +-- time to make equations: .139921
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00797456 sec. for [{0, 1, 2, 3}, {infinity, 1, 2,
│ │ │ │ + -- trackHomotopy time = .00997745 sec. for [{0, 1, 2, 3}, {infinity, 1, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .027106
│ │ │ │ --- time to make equations: .0750451
│ │ │ │ +-- time of performing one checker move: .155873
│ │ │ │ +-- time to make equations: .0160292
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00796739 sec. for [{0, 1, 3, 2}, {infinity, 1,
│ │ │ │ + -- trackHomotopy time = .0371607 sec. for [{0, 1, 3, 2}, {infinity, 1,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .0880204
│ │ │ │ +-- time of performing one checker move: .153219
│ │ │ │ +-- time of performing one checker move: .0039078
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .00400093
│ │ │ │ --- time to make equations: .108205
│ │ │ │ +-- time to make equations: .0159416
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00796862 sec. for [{1, 3, 2, 0}, {infinity, 3,
│ │ │ │ + -- trackHomotopy time = .0382822 sec. for [{1, 3, 2, 0}, {infinity, 3,
│ │ │ │  infinity, 1}]
│ │ │ │ --- time of performing one checker move: .124193
│ │ │ │ +-- time of performing one checker move: .143473
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = .00800028
│ │ │ │ +-- cpu time = .00397309
│ │ │ │  -- making a recursive call to resolveNode
│ │ │ │  -- playCheckers
│ │ │ │ --- cpu time = 0
│ │ │ │ +-- cpu time = .00404597
│ │ │ │  resolveNode reached node of no remaining conditions
│ │ │ │ --- time to make equations: .00705985
│ │ │ │ +-- time to make equations: .00805531
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0062539 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │ + -- trackHomotopy time = .00737808 sec. for [{0, 1, 2, 3}, {0, infinity, 2,
│ │ │ │  infinity}]
│ │ │ │ --- time of performing one checker move: .073368
│ │ │ │ --- time of performing one checker move: .00400236
│ │ │ │ --- time to make equations: .00800006
│ │ │ │ +-- time of performing one checker move: .143768
│ │ │ │ +-- time of performing one checker move: .00396881
│ │ │ │ +-- time to make equations: .00463389
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .0657139 sec. for [{0, 2, 3, 1}, {0, infinity,
│ │ │ │ + -- trackHomotopy time = .00870738 sec. for [{0, 2, 3, 1}, {0, infinity,
│ │ │ │  infinity, 2}]
│ │ │ │ --- time of performing one checker move: .0970197
│ │ │ │ --- time of performing one checker move: 0
│ │ │ │ +-- time of performing one checker move: .0237647
│ │ │ │ +-- time of performing one checker move: .124621
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ +-- time of performing one checker move: .00404595
│ │ │ │ +-- time of performing one checker move: .00397077
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ +-- time of performing one checker move: .00401083
│ │ │ │  -- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .00400019
│ │ │ │ --- time of performing one checker move: 0
│ │ │ │ --- time of performing one checker move: .00400058
│ │ │ │ --- time to make equations: .087017
│ │ │ │ +-- time to make equations: .012004
│ │ │ │  Setup time: 0
│ │ │ │  Computing time:0
│ │ │ │ - -- trackHomotopy time = .00760701 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │ + -- trackHomotopy time = .050622 sec. for [{1, 3, 2, 0}, {1, infinity,
│ │ │ │  infinity, 3}]
│ │ │ │ --- time of performing one checker move: .0990156
│ │ │ │ --- time of performing one checker move: 0
│ │ │ │ +-- time of performing one checker move: .160807
│ │ │ │ +-- time of performing one checker move: .00403988
│ │ │ │  
│ │ │ │  o6 = {| -1.65573-.600637ii .0201935+.0437095ii   |, | -.154703+.175591ii
│ │ │ │        | -1.23037-1.66989ii -.0308057-.00120618ii |  | -.801221-.0354303ii
│ │ │ │        | 1.35971-.743988ii  -.0713133-.049047ii   |  | .325581-2.08048ii
│ │ │ │        | -.397038-1.8974ii  .0102261-.024397ii    |  | -.475895-.209388ii
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .0376857+.0683239ii   |}
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/___Lab__Book__Protocol.out
│ │ │ @@ -14,35 +14,35 @@
│ │ │  
│ │ │  i4 : LL7a=select(LL7,L->not knownExample L);#LL7a
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0,Verbose=>true))
│ │ │  unfolding
│ │ │ - -- .12405s elapsed
│ │ │ + -- .124767s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .137827s elapsed
│ │ │ + -- .101688s elapsed
│ │ │  next gb
│ │ │ - -- .000839943s elapsed
│ │ │ + -- .000890446s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .140656s elapsed
│ │ │ + -- .107431s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .131511s elapsed
│ │ │ + -- .104326s elapsed
│ │ │  next gb
│ │ │ - -- .000606166s elapsed
│ │ │ + -- .000661159s elapsed
│ │ │  true
│ │ │ - -- 1.52544s elapsed
│ │ │ + -- 1.3174s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.45726s elapsed
│ │ │ + -- 1.36603s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : LL7b=={}
│ │ │  
│ │ │ @@ -75,23 +75,23 @@
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .234139s elapsed
│ │ │ + -- .201195s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .136291s elapsed
│ │ │ + -- .124437s elapsed
│ │ │  next gb
│ │ │ - -- .000789879s elapsed
│ │ │ + -- .000957374s elapsed
│ │ │  true
│ │ │ - -- .661417s elapsed
│ │ │ + -- .588375s elapsed
│ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ - -- .691981s elapsed
│ │ │ + -- .629862s elapsed
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : L={6,8,9,11}
│ │ │  
│ │ │ @@ -100,22 +100,22 @@
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : genus L
│ │ │  
│ │ │  o13 = 8
│ │ │  
│ │ │  i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .124617s elapsed
│ │ │ + -- .0876054s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .3134s elapsed
│ │ │ + -- .310543s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(1, 1), (16, 3)}
│ │ │  {0, -1}
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_heuristic__Smoothness.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │   -- setting random seed to 1644814534404491274313411285186041988099567563905780374824086062516559438
│ │ │  
│ │ │  i4 : elapsedTime tally apply(10,i-> (
│ │ │               c=minors(2,random(S^2,S^{3:-2}));
│ │ │               c=sub(c,x_0=>1);
│ │ │               R=kk[support c];c=sub(c,R);
│ │ │               heuristicSmoothness c))
│ │ │ - -- 3.60044s elapsed
│ │ │ + -- 3.4544s elapsed
│ │ │  
│ │ │  o4 = Tally{false => 6}
│ │ │             true => 4
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_is__Smoothable__Semigroup.out
│ │ │ @@ -7,17 +7,17 @@
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : genus L
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ - -- 1.09521s elapsed
│ │ │ + -- .993194s elapsed
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.0809s elapsed
│ │ │ + -- 3.72999s elapsed
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_is__Weierstrass__Semigroup.out
│ │ │ @@ -7,12 +7,12 @@
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : genus L
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ - -- 3.7123s elapsed
│ │ │ + -- 3.50193s elapsed
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_non__Weierstrass__Semigroups.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 6860996532851631556
│ │ │  
│ │ │  i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │  (6, 7,  all semigroups are smoothable)
│ │ │ - -- 1.34606s elapsed
│ │ │ + -- 1.31762s elapsed
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : LLdifficult={{6, 8, 9, 11}}
│ │ │  
│ │ │ @@ -14,61 +14,61 @@
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .399715s elapsed
│ │ │ + -- .377839s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .147261s elapsed
│ │ │ + -- .297098s elapsed
│ │ │  next gb
│ │ │ - -- .00181989s elapsed
│ │ │ + -- .00201069s elapsed
│ │ │  true
│ │ │ - -- .962213s elapsed
│ │ │ + -- 1.04364s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .372433s elapsed
│ │ │ + -- .346302s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .215016s elapsed
│ │ │ + -- .180267s elapsed
│ │ │  next gb
│ │ │ - -- .00264601s elapsed
│ │ │ + -- .00284077s elapsed
│ │ │  decompose
│ │ │ - -- .19227s elapsed
│ │ │ + -- .140676s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 3.17778s elapsed
│ │ │ + -- 2.68191s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .144377s elapsed
│ │ │ + -- .128577s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .203021s elapsed
│ │ │ + -- .102517s elapsed
│ │ │  next gb
│ │ │ - -- .000519653s elapsed
│ │ │ + -- .00055482s elapsed
│ │ │  true
│ │ │ - -- .836809s elapsed
│ │ │ + -- .651686s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .549379s elapsed
│ │ │ + -- .502636s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .245197s elapsed
│ │ │ + -- .214285s elapsed
│ │ │  next gb
│ │ │ - -- .00518358s elapsed
│ │ │ + -- .0057598s elapsed
│ │ │  decompose
│ │ │ - -- 1.17294s elapsed
│ │ │ + -- .848059s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 3.27777s elapsed
│ │ │ - -- 8.25469s elapsed
│ │ │ + -- 2.57056s elapsed
│ │ │ + -- 6.94795s elapsed
│ │ │  0
│ │ │  
│ │ │  {}
│ │ │ - -- .000003747s elapsed
│ │ │ - -- 8.29109s elapsed
│ │ │ + -- .000006249s elapsed
│ │ │ + -- 6.993s elapsed
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/___Lab__Book__Protocol.html
│ │ │ @@ -96,38 +96,38 @@
│ │ │  o5 = 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0,Verbose=>true))
│ │ │  unfolding
│ │ │ - -- .12405s elapsed
│ │ │ + -- .124767s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .137827s elapsed
│ │ │ + -- .101688s elapsed
│ │ │  next gb
│ │ │ - -- .000839943s elapsed
│ │ │ + -- .000890446s elapsed
│ │ │  true
│ │ │  unfolding
│ │ │ - -- .140656s elapsed
│ │ │ + -- .107431s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .131511s elapsed
│ │ │ + -- .104326s elapsed
│ │ │  next gb
│ │ │ - -- .000606166s elapsed
│ │ │ + -- .000661159s elapsed
│ │ │  true
│ │ │ - -- 1.52544s elapsed
│ │ │ + -- 1.3174s elapsed
│ │ │  
│ │ │  o6 = {}
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0))
│ │ │ - -- 1.45726s elapsed
│ │ │ + -- 1.36603s elapsed
│ │ │  
│ │ │  o7 = {}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -184,23 +184,23 @@ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true)
│ │ │  (13, 1)
│ │ │  {5, 8, 11, 12}
│ │ │  unfolding
│ │ │ - -- .234139s elapsed
│ │ │ + -- .201195s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .136291s elapsed
│ │ │ + -- .124437s elapsed
│ │ │  next gb
│ │ │ - -- .000789879s elapsed
│ │ │ + -- .000957374s elapsed
│ │ │  true
│ │ │ - -- .661417s elapsed
│ │ │ + -- .588375s elapsed
│ │ │  (5, 8,  all semigroups are smoothable)
│ │ │ - -- .691981s elapsed
│ │ │ + -- .629862s elapsed
│ │ │  
│ │ │  o11 = {}
│ │ │  
│ │ │  o11 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -223,22 +223,22 @@ │ │ │ │ │ │ o13 = 8 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true)
│ │ │ - -- .124617s elapsed
│ │ │ + -- .0876054s elapsed
│ │ │  6
│ │ │  false
│ │ │  5
│ │ │  false
│ │ │  4
│ │ │  decompose
│ │ │ - -- .3134s elapsed
│ │ │ + -- .310543s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(1, 1), (16, 3)}
│ │ │  {0, -1}
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -26,34 +26,34 @@ │ │ │ │ o3 = 39 │ │ │ │ i4 : LL7a=select(LL7,L->not knownExample L);#LL7a │ │ │ │ │ │ │ │ o5 = 2 │ │ │ │ i6 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup │ │ │ │ (L,0.25,0,Verbose=>true)) │ │ │ │ unfolding │ │ │ │ - -- .12405s elapsed │ │ │ │ + -- .124767s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .137827s elapsed │ │ │ │ + -- .101688s elapsed │ │ │ │ next gb │ │ │ │ - -- .000839943s elapsed │ │ │ │ + -- .000890446s elapsed │ │ │ │ true │ │ │ │ unfolding │ │ │ │ - -- .140656s elapsed │ │ │ │ + -- .107431s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .131511s elapsed │ │ │ │ + -- .104326s elapsed │ │ │ │ next gb │ │ │ │ - -- .000606166s elapsed │ │ │ │ + -- .000661159s elapsed │ │ │ │ true │ │ │ │ - -- 1.52544s elapsed │ │ │ │ + -- 1.3174s elapsed │ │ │ │ │ │ │ │ o6 = {} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime LL7b=select(LL7a,L->not isSmoothableSemigroup(L,0.25,0)) │ │ │ │ - -- 1.45726s elapsed │ │ │ │ + -- 1.36603s elapsed │ │ │ │ │ │ │ │ o7 = {} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : LL7b=={} │ │ │ │ │ │ │ │ o8 = true │ │ │ │ @@ -92,23 +92,23 @@ │ │ │ │ o10 = (5, 8) │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ i11 : elapsedTime nonWeierstrassSemigroups(m,g,Verbose=>true) │ │ │ │ (13, 1) │ │ │ │ {5, 8, 11, 12} │ │ │ │ unfolding │ │ │ │ - -- .234139s elapsed │ │ │ │ + -- .201195s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .136291s elapsed │ │ │ │ + -- .124437s elapsed │ │ │ │ next gb │ │ │ │ - -- .000789879s elapsed │ │ │ │ + -- .000957374s elapsed │ │ │ │ true │ │ │ │ - -- .661417s elapsed │ │ │ │ + -- .588375s elapsed │ │ │ │ (5, 8, all semigroups are smoothable) │ │ │ │ - -- .691981s elapsed │ │ │ │ + -- .629862s elapsed │ │ │ │ │ │ │ │ o11 = {} │ │ │ │ │ │ │ │ o11 : List │ │ │ │ In the verbose mode we get timings of various computation steps and further │ │ │ │ information. The first line, (13,1), indicates that there 13 semigroups of │ │ │ │ multiplicity 5 and genus 8 of which only 1 is not flagged as smoothable by the │ │ │ │ @@ -120,22 +120,22 @@ │ │ │ │ o12 = {6, 8, 9, 11} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ i13 : genus L │ │ │ │ │ │ │ │ o13 = 8 │ │ │ │ i14 : isWeierstrassSemigroup(L,0.2,Verbose=>true) │ │ │ │ - -- .124617s elapsed │ │ │ │ + -- .0876054s elapsed │ │ │ │ 6 │ │ │ │ false │ │ │ │ 5 │ │ │ │ false │ │ │ │ 4 │ │ │ │ decompose │ │ │ │ - -- .3134s elapsed │ │ │ │ + -- .310543s elapsed │ │ │ │ number of components: 2 │ │ │ │ support c, codim c: {(1, 1), (16, 3)} │ │ │ │ {0, -1} │ │ │ │ │ │ │ │ o14 = true │ │ │ │ The first integer, 6, tells that in this attempt deformation parameters of │ │ │ │ degree >= 6 were used and no smooth fiber was found. Finally with all │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_heuristic__Smoothness.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime tally apply(10,i-> (
│ │ │               c=minors(2,random(S^2,S^{3:-2}));
│ │ │               c=sub(c,x_0=>1);
│ │ │               R=kk[support c];c=sub(c,R);
│ │ │               heuristicSmoothness c))
│ │ │ - -- 3.60044s elapsed
│ │ │ + -- 3.4544s elapsed
│ │ │  
│ │ │  o4 = Tally{false => 6}
│ │ │             true => 4
│ │ │  
│ │ │  o4 : Tally
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -27,15 +27,15 @@ │ │ │ │ -- setting random seed to │ │ │ │ 1644814534404491274313411285186041988099567563905780374824086062516559438 │ │ │ │ i4 : elapsedTime tally apply(10,i-> ( │ │ │ │ c=minors(2,random(S^2,S^{3:-2})); │ │ │ │ c=sub(c,x_0=>1); │ │ │ │ R=kk[support c];c=sub(c,R); │ │ │ │ heuristicSmoothness c)) │ │ │ │ - -- 3.60044s elapsed │ │ │ │ + -- 3.4544s elapsed │ │ │ │ │ │ │ │ o4 = Tally{false => 6} │ │ │ │ true => 4 │ │ │ │ │ │ │ │ o4 : Tally │ │ │ │ ********** WWaayyss ttoo uussee hheeuurriissttiiccSSmmooootthhnneessss:: ********** │ │ │ │ * heuristicSmoothness(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_is__Smoothable__Semigroup.html │ │ │ @@ -95,23 +95,23 @@ │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime isSmoothableSemigroup(L,0.30,0)
│ │ │ - -- 1.09521s elapsed
│ │ │ + -- .993194s elapsed
│ │ │  
│ │ │  o3 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime isSmoothableSemigroup(L,0.14,0)
│ │ │ - -- 4.0809s elapsed
│ │ │ + -- 3.72999s elapsed
│ │ │  
│ │ │  o4 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,19 +29,19 @@ │ │ │ │ o1 = {6, 8, 9, 11} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : genus L │ │ │ │ │ │ │ │ o2 = 8 │ │ │ │ i3 : elapsedTime isSmoothableSemigroup(L,0.30,0) │ │ │ │ - -- 1.09521s elapsed │ │ │ │ + -- .993194s elapsed │ │ │ │ │ │ │ │ o3 = false │ │ │ │ i4 : elapsedTime isSmoothableSemigroup(L,0.14,0) │ │ │ │ - -- 4.0809s elapsed │ │ │ │ + -- 3.72999s elapsed │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_e_U_n_f_o_l_d_i_n_g -- Makes the universal homogeneous unfolding of an ideal │ │ │ │ with positive degree parameters │ │ │ │ * _f_l_a_t_t_e_n_i_n_g_R_e_l_a_t_i_o_n_s -- Compute the flattening relations of an unfolding │ │ │ │ * _g_e_t_F_l_a_t_F_a_m_i_l_y -- Compute the flat family depending on a subset of │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_is__Weierstrass__Semigroup.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ │ │ o2 = 8 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime isWeierstrassSemigroup(L,0.15)
│ │ │ - -- 3.7123s elapsed
│ │ │ + -- 3.50193s elapsed
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o1 = {6, 8, 9, 11} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : genus L │ │ │ │ │ │ │ │ o2 = 8 │ │ │ │ i3 : elapsedTime isWeierstrassSemigroup(L,0.15) │ │ │ │ - -- 3.7123s elapsed │ │ │ │ + -- 3.50193s elapsed │ │ │ │ │ │ │ │ o3 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_e_U_n_f_o_l_d_i_n_g -- Makes the universal homogeneous unfolding of an ideal │ │ │ │ with positive degree parameters │ │ │ │ * _f_l_a_t_t_e_n_i_n_g_R_e_l_a_t_i_o_n_s -- Compute the flattening relations of an unfolding │ │ │ │ * _g_e_t_F_l_a_t_F_a_m_i_l_y -- Compute the flat family depending on a subset of │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/html/_non__Weierstrass__Semigroups.html │ │ │ @@ -79,15 +79,15 @@ │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -101,62 +101,62 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : elapsedTime nonWeierstrassSemigroups(6,7)
│ │ │  (6, 7,  all semigroups are smoothable)
│ │ │ - -- 1.34606s elapsed
│ │ │ + -- 1.31762s elapsed
│ │ │  
│ │ │  o1 = {}
│ │ │  
│ │ │  o1 : List
    │ │ │
    │ │ │ 
    i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
│ │ │  (17, 5)
│ │ │  {6, 7, 8, 17}
│ │ │  unfolding
│ │ │ - -- .399715s elapsed
│ │ │ + -- .377839s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .147261s elapsed
│ │ │ + -- .297098s elapsed
│ │ │  next gb
│ │ │ - -- .00181989s elapsed
│ │ │ + -- .00201069s elapsed
│ │ │  true
│ │ │ - -- .962213s elapsed
│ │ │ + -- 1.04364s elapsed
│ │ │  {6, 7, 9, 17}
│ │ │  unfolding
│ │ │ - -- .372433s elapsed
│ │ │ + -- .346302s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .215016s elapsed
│ │ │ + -- .180267s elapsed
│ │ │  next gb
│ │ │ - -- .00264601s elapsed
│ │ │ + -- .00284077s elapsed
│ │ │  decompose
│ │ │ - -- .19227s elapsed
│ │ │ + -- .140676s elapsed
│ │ │  number of components: 2
│ │ │  support c, codim c: {(2, 2), (5, 2)}
│ │ │  {0, -1}
│ │ │ - -- 3.17778s elapsed
│ │ │ + -- 2.68191s elapsed
│ │ │  {6, 8, 9, 10}
│ │ │  unfolding
│ │ │ - -- .144377s elapsed
│ │ │ + -- .128577s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .203021s elapsed
│ │ │ + -- .102517s elapsed
│ │ │  next gb
│ │ │ - -- .000519653s elapsed
│ │ │ + -- .00055482s elapsed
│ │ │  true
│ │ │ - -- .836809s elapsed
│ │ │ + -- .651686s elapsed
│ │ │  {6, 8, 10, 11, 13}
│ │ │  unfolding
│ │ │ - -- .549379s elapsed
│ │ │ + -- .502636s elapsed
│ │ │  flatteningRelations
│ │ │ - -- .245197s elapsed
│ │ │ + -- .214285s elapsed
│ │ │  next gb
│ │ │ - -- .00518358s elapsed
│ │ │ + -- .0057598s elapsed
│ │ │  decompose
│ │ │ - -- 1.17294s elapsed
│ │ │ + -- .848059s elapsed
│ │ │  number of components: 1
│ │ │  support c, codim c: {(5, 1)}
│ │ │  {-1}
│ │ │ - -- 3.27777s elapsed
│ │ │ - -- 8.25469s elapsed
│ │ │ + -- 2.57056s elapsed
│ │ │ + -- 6.94795s elapsed
│ │ │  0
│ │ │  
│ │ │  {}
│ │ │ - -- .000003747s elapsed
│ │ │ - -- 8.29109s elapsed
│ │ │ + -- .000006249s elapsed
│ │ │ + -- 6.993s elapsed
│ │ │  
│ │ │  o3 = {{6, 8, 9, 11}}
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,76 +22,76 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ We test which semigroups of multiplicity m and genus g are smoothable. If no │ │ │ │ smoothing was found then L is a candidate for a non Weierstrass semigroup. In │ │ │ │ this search certain semigroups L in LLdifficult, where the computation is │ │ │ │ particular heavy are excluded. │ │ │ │ i1 : elapsedTime nonWeierstrassSemigroups(6,7) │ │ │ │ (6, 7, all semigroups are smoothable) │ │ │ │ - -- 1.34606s elapsed │ │ │ │ + -- 1.31762s elapsed │ │ │ │ │ │ │ │ o1 = {} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : LLdifficult={{6, 8, 9, 11}} │ │ │ │ │ │ │ │ o2 = {{6, 8, 9, 11}} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : elapsedTime nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true) │ │ │ │ (17, 5) │ │ │ │ {6, 7, 8, 17} │ │ │ │ unfolding │ │ │ │ - -- .399715s elapsed │ │ │ │ + -- .377839s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .147261s elapsed │ │ │ │ + -- .297098s elapsed │ │ │ │ next gb │ │ │ │ - -- .00181989s elapsed │ │ │ │ + -- .00201069s elapsed │ │ │ │ true │ │ │ │ - -- .962213s elapsed │ │ │ │ + -- 1.04364s elapsed │ │ │ │ {6, 7, 9, 17} │ │ │ │ unfolding │ │ │ │ - -- .372433s elapsed │ │ │ │ + -- .346302s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .215016s elapsed │ │ │ │ + -- .180267s elapsed │ │ │ │ next gb │ │ │ │ - -- .00264601s elapsed │ │ │ │ + -- .00284077s elapsed │ │ │ │ decompose │ │ │ │ - -- .19227s elapsed │ │ │ │ + -- .140676s elapsed │ │ │ │ number of components: 2 │ │ │ │ support c, codim c: {(2, 2), (5, 2)} │ │ │ │ {0, -1} │ │ │ │ - -- 3.17778s elapsed │ │ │ │ + -- 2.68191s elapsed │ │ │ │ {6, 8, 9, 10} │ │ │ │ unfolding │ │ │ │ - -- .144377s elapsed │ │ │ │ + -- .128577s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .203021s elapsed │ │ │ │ + -- .102517s elapsed │ │ │ │ next gb │ │ │ │ - -- .000519653s elapsed │ │ │ │ + -- .00055482s elapsed │ │ │ │ true │ │ │ │ - -- .836809s elapsed │ │ │ │ + -- .651686s elapsed │ │ │ │ {6, 8, 10, 11, 13} │ │ │ │ unfolding │ │ │ │ - -- .549379s elapsed │ │ │ │ + -- .502636s elapsed │ │ │ │ flatteningRelations │ │ │ │ - -- .245197s elapsed │ │ │ │ + -- .214285s elapsed │ │ │ │ next gb │ │ │ │ - -- .00518358s elapsed │ │ │ │ + -- .0057598s elapsed │ │ │ │ decompose │ │ │ │ - -- 1.17294s elapsed │ │ │ │ + -- .848059s elapsed │ │ │ │ number of components: 1 │ │ │ │ support c, codim c: {(5, 1)} │ │ │ │ {-1} │ │ │ │ - -- 3.27777s elapsed │ │ │ │ - -- 8.25469s elapsed │ │ │ │ + -- 2.57056s elapsed │ │ │ │ + -- 6.94795s elapsed │ │ │ │ 0 │ │ │ │ │ │ │ │ {} │ │ │ │ - -- .000003747s elapsed │ │ │ │ - -- 8.29109s elapsed │ │ │ │ + -- .000006249s elapsed │ │ │ │ + -- 6.993s elapsed │ │ │ │ │ │ │ │ o3 = {{6, 8, 9, 11}} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ In the verbose mode we get timings of various computation steps and further │ │ │ │ information. The first line, (17,5), indicates that there 17 semigroups of │ │ │ │ multiplicity 6 and genus 8 of which only 5 is not flagged as smoothable by the │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map.out │ │ │ @@ -8,15 +8,15 @@ │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2); │ │ │ │ │ │ i5 : C = oiRes({b}, 2) │ │ │ │ │ │ o5 = 0: (e0, {3}, {-2}) │ │ │ 1: (e1, {5, 5}, {-3, -4}) │ │ │ - 2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -5, -5, -2, -4, -4, -3, -3, -4}) │ │ │ + 2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -5, -4, -3, -5, -4, -5, -3, -4}) │ │ │ │ │ │ o5 : OIResolution │ │ │ │ │ │ i6 : phi = C.dd_1 │ │ │ │ │ │ o6 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2}) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map_sp__Vector__In__Width.out │ │ │ @@ -8,15 +8,15 @@ │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2); │ │ │ │ │ │ i5 : C = oiRes({b}, 2) │ │ │ │ │ │ o5 = 0: (e0, {3}, {-2}) │ │ │ 1: (e1, {5, 5}, {-4, -3}) │ │ │ - 2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -4, -5, -4, -5, -5, -3, -2}) │ │ │ + 2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -4, -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 0.0838715s (cpu); 0.0838719s (thread); 0s (gc) │ │ │ + -- used 0.100828s (cpu); 0.100829s (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 0.216887s (cpu); 0.116314s (thread); 0s (gc) │ │ │ + -- used 0.283177s (cpu); 0.143792s (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 0.48843s (cpu); 0.310297s (thread); 0s (gc) │ │ │ + -- used 0.595167s (cpu); 0.324251s (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 0.255799s (cpu); 0.175309s (thread); 0s (gc) │ │ │ + -- used 0.118515s (cpu); 0.118513s (thread); 0s (gc) │ │ │ │ │ │ i6 : describeFull C │ │ │ │ │ │ o6 = 0: Module: Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__Free__O__I__Module__Map_rp.out │ │ │ @@ -10,18 +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 0.0907428s (cpu); 0.0907408s (thread); 0s (gc) │ │ │ + -- used 0.107596s (cpu); 0.107596s (thread); 0s (gc) │ │ │ │ │ │ i6 : describe C │ │ │ │ │ │ o6 = 0: Module: Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_get__Schreyer__Map.out │ │ │ @@ -17,21 +17,21 @@ │ │ │ 2,3 2,1 1,2 3,{1},2 │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : G' = oiSyz(G, d) │ │ │ │ │ │ o6 = {x d - x d + 1d , x d │ │ │ - 1,2 3,{1, 3},1 1,1 3,{2, 3},1 3,{1, 2, 3},2 1,2 4,{1, 3, 4},2 │ │ │ + 1,2 3,{1, 3},1 1,1 3,{2, 3},1 3,{1, 2, 3},2 2,4 4,{1, 2, 3},2 │ │ │ ------------------------------------------------------------------------ │ │ │ - - x d - x d , x d - │ │ │ - 1,1 4,{2, 3, 4},2 1,3 4,{1, 2, 4},2 2,4 4,{1, 2, 3},2 │ │ │ + - x d , x d - x d - │ │ │ + 2,3 4,{1, 2, 4},2 1,2 4,{1, 3, 4},2 1,1 4,{2, 3, 4},2 │ │ │ ------------------------------------------------------------------------ │ │ │ x d } │ │ │ - 2,3 4,{1, 2, 4},2 │ │ │ + 1,3 4,{1, 2, 4},2 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : H = getFreeOIModule G'#0 │ │ │ │ │ │ o7 = Basis symbol: d │ │ │ Basis element widths: {2, 3} │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_image_lp__Free__O__I__Module__Map_rp.out │ │ │ @@ -10,19 +10,19 @@ │ │ │ │ │ │ i5 : C = oiRes({b}, 2); │ │ │ │ │ │ i6 : phi = C.dd_1; │ │ │ │ │ │ i7 : image phi │ │ │ │ │ │ -o7 = {-x 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 0.328549s (cpu); 0.256907s (thread); 0s (gc) │ │ │ + -- used 0.471386s (cpu); 0.353819s (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 0.0235505s (cpu); 0.0235504s (thread); 0s (gc) │ │ │ + -- used 0.029428s (cpu); 0.0294314s (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 0.028037s (cpu); 0.0280374s (thread); 0s (gc) │ │ │ + -- used 0.0337702s (cpu); 0.0337726s (thread); 0s (gc) │ │ │ │ │ │ o10 = {x e + x e , x x e + x x e , │ │ │ 1,1 1,{1},1 2,1 1,{1},2 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ ----------------------------------------------------------------------- │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ @@ -41,17 +41,18 @@ │ │ │ - x x e } │ │ │ 3},3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ o13 : List │ │ │ │ │ │ i14 : minimizeOIGB C -- an element gets removed │ │ │ │ │ │ - 2 │ │ │ -o14 = {x e + x e , x x x e - x x e │ │ │ - 1,1 1,{1},1 2,1 1,{1},2 2,3 2,2 1,1 3,{2, 3},3 2,1 1,2 3,{1, │ │ │ + │ │ │ +o14 = {x x e + x x e , x x x e - │ │ │ + 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 2,3 2,2 1,1 3,{2, 3},3 │ │ │ ----------------------------------------------------------------------- │ │ │ - , x x e + x x e } │ │ │ - 3},3 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ + 2 │ │ │ + x x e , x e + x e } │ │ │ + 2,1 1,2 3,{1, 3},3 1,1 1,{1},1 2,1 1,{1},2 │ │ │ │ │ │ o14 : List │ │ │ │ │ │ i15 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Free__O__I__Module__Map_rp.out │ │ │ @@ -10,10 +10,10 @@ │ │ │ │ │ │ i5 : C = oiRes({b}, 2); │ │ │ │ │ │ i6 : phi = C.dd_1; │ │ │ │ │ │ i7 : net phi │ │ │ │ │ │ -o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2}) │ │ │ +o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2}) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.221228s (cpu); 0.126662s (thread); 0s (gc) │ │ │ + -- used 0.30398s (cpu); 0.14814s (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 0.0274955s (cpu); 0.0274953s (thread); 0s (gc) │ │ │ + -- used 0.0346491s (cpu); 0.0346529s (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 0.494297s (cpu); 0.290279s (thread); 0s (gc) │ │ │ + -- used 0.602363s (cpu); 0.331233s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Syz.out │ │ │ @@ -17,18 +17,18 @@ │ │ │ 2,3 2,1 1,2 3,{1},2 │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : oiSyz(G, d) │ │ │ │ │ │ o6 = {x d - x d + 1d , x d │ │ │ - 1,2 3,{1, 3},1 1,1 3,{2, 3},1 3,{1, 2, 3},2 1,2 4,{1, 3, 4},2 │ │ │ + 1,2 3,{1, 3},1 1,1 3,{2, 3},1 3,{1, 2, 3},2 2,4 4,{1, 2, 3},2 │ │ │ ------------------------------------------------------------------------ │ │ │ - - x d - x d , x d - │ │ │ - 1,1 4,{2, 3, 4},2 1,3 4,{1, 2, 4},2 2,4 4,{1, 2, 3},2 │ │ │ + - x d , x d - x d - │ │ │ + 2,3 4,{1, 2, 4},2 1,2 4,{1, 3, 4},2 1,1 4,{2, 3, 4},2 │ │ │ ------------------------------------------------------------------------ │ │ │ x d } │ │ │ - 2,3 4,{1, 2, 4},2 │ │ │ + 1,3 4,{1, 2, 4},2 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i4 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2); │ │ │ │ │ │ i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2); │ │ │ │ │ │ i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal) │ │ │ - -- used 0.275582s (cpu); 0.210169s (thread); 0s (gc) │ │ │ + -- used 0.145488s (cpu); 0.145487s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ o9 = {x e + x e , x x e + x x e , │ │ │ 2,1 1,{1},2 1,1 1,{1},2 1,2 1,1 2,{2},1 2,2 1,2 2,{2},2 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 │ │ │ x x e - x x e } │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module__Map.html │ │ │ @@ -79,15 +79,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-3, -4})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -5, -5, -2, -4, -4, -3, -3, -4})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -5, -4, -3, -5, -4, -5, -3, -4})
│ │ │  
│ │ │  o5 : OIResolution
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : phi = C.dd_1
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 3);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2);
│ │ │ │  i5 : C = oiRes({b}, 2)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │ │       1: (e1, {5, 5}, {-3, -4})
│ │ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-5, -5, -5, -2, -4, -4, -3, -3, -4})
│ │ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-2, -5, -4, -3, -5, -4, -5, -3, -4})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : phi = C.dd_1
│ │ │ │  
│ │ │ │  o6 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │ │  
│ │ │ │  o6 : FreeOIModuleMap
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module__Map_sp__Vector__In__Width.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : C = oiRes({b}, 2)
│ │ │  
│ │ │  o5 = 0: (e0, {3}, {-2})
│ │ │       1: (e1, {5, 5}, {-4, -3})
│ │ │ -     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -4, -5, -4, -5, -5, -3, -2})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -4, -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}, {-3, -4, -4, -5, -4, -5, -5, -3, -2})
│ │ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -4, -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
│ │ │ @@ -74,15 +74,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1)
│ │ │ - -- used 0.0838715s (cpu); 0.0838719s (thread); 0s (gc)
│ │ │ + -- used 0.100828s (cpu); 0.100829s (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 0.0838715s (cpu); 0.0838719s (thread); 0s (gc)
│ │ │ │ + -- used 0.100828s (cpu); 0.100829s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : C.dd_0
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution_sp_us_sp__Z__Z.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.216887s (cpu); 0.116314s (thread); 0s (gc)
    
│ │ │ + -- used 0.283177s (cpu); 0.143792s (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 0.216887s (cpu); 0.116314s (thread); 0s (gc)
│ │ │ │ + -- used 0.283177s (cpu); 0.143792s (thread); 0s (gc)
│ │ │ │  i6 : C_0
│ │ │ │  
│ │ │ │  o6 = Basis symbol: e0
│ │ │ │       Basis element widths: {2}
│ │ │ │       Degree shifts: {-2}
│ │ │ │       Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │       Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Top__Nonminimal.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.48843s (cpu); 0.310297s (thread); 0s (gc)
│ │ │ + -- used 0.595167s (cpu); 0.324251s (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 0.48843s (cpu); 0.310297s (thread); 0s (gc)
│ │ │ │ + -- used 0.595167s (cpu); 0.324251s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd TTooppNNoonnmmiinniimmaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe__Full.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.255799s (cpu); 0.175309s (thread); 0s (gc)
    
│ │ │ + -- used 0.118515s (cpu); 0.118513s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : describeFull C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │  Displays the free OI-modules and describes the differentials of an OI-
│ │ │ │  resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.255799s (cpu); 0.175309s (thread); 0s (gc)
│ │ │ │ + -- used 0.118515s (cpu); 0.118513s (thread); 0s (gc)
│ │ │ │  i6 : describeFull C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__Free__O__I__Module__Map_rp.html
│ │ │ @@ -102,23 +102,26 @@
│ │ │                
    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- │ │ │ │ 1.25.11+ds/M2/Macaulay2/packages/OIGroebnerBases.m2:1979:0. │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__O__I__Resolution_rp.html │ │ │ @@ -91,15 +91,15 @@ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0907428s (cpu); 0.0907408s (thread); 0s (gc)
    │ │ │ + -- used 0.107596s (cpu); 0.107596s (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 0.0907428s (cpu); 0.0907408s (thread); 0s (gc)
│ │ │ │ + -- used 0.107596s (cpu); 0.107596s (thread); 0s (gc)
│ │ │ │  i6 : describe C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_get__Schreyer__Map.html
│ │ │ @@ -105,21 +105,21 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : G' = oiSyz(G, d)
│ │ │  
│ │ │  o6 = {x   d           - x   d           + 1d             , x   d             
│ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - x   d              - x   d             , x   d              -
│ │ │ -        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2  
│ │ │ +     - x   d             , x   d              - x   d              -
│ │ │ +        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   d             }
│ │ │ -      2,3 4,{1, 2, 4},2
│ │ │ +      1,3 4,{1, 2, 4},2
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : H = getFreeOIModule G'#0
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,21 +29,21 @@
│ │ │ │       x   x   x   e       }
│ │ │ │        2,3 2,1 1,2 3,{1},2
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : G' = oiSyz(G, d)
│ │ │ │  
│ │ │ │  o6 = {x   d           - x   d           + 1d             , x   d
│ │ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     - x   d              - x   d             , x   d              -
│ │ │ │ -        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2
│ │ │ │ +     - x   d             , x   d              - x   d              -
│ │ │ │ +        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   d             }
│ │ │ │ -      2,3 4,{1, 2, 4},2
│ │ │ │ +      1,3 4,{1, 2, 4},2
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : H = getFreeOIModule G'#0
│ │ │ │  
│ │ │ │  o7 = Basis symbol: d
│ │ │ │       Basis element widths: {2, 3}
│ │ │ │       Degree shifts: {-2, -3}
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_image_lp__Free__O__I__Module__Map_rp.html
│ │ │ @@ -102,22 +102,22 @@
│ │ │                
    i6 : phi = C.dd_1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : image phi
│ │ │  
│ │ │ -o7 = {-x   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 │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.328549s (cpu); 0.256907s (thread); 0s (gc)
│ │ │ + -- used 0.471386s (cpu); 0.353819s (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 0.328549s (cpu); 0.256907s (thread); 0s (gc) │ │ │ │ + -- used 0.471386s (cpu); 0.353819s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ │ 1: (e1, {4}, {-4}) │ │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ │ │ o5 : OIResolution │ │ │ │ i6 : isComplex C │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__O__I__G__B.html │ │ │ @@ -116,15 +116,15 @@ │ │ │ │ │ │ o10 = false     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0235505s (cpu); 0.0235504s (thread); 0s (gc)
│ │ │ + -- used 0.029428s (cpu); 0.0294314s (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 0.0235505s (cpu); 0.0235504s (thread); 0s (gc)
│ │ │ │ + -- used 0.029428s (cpu); 0.0294314s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │              
│ │ │                
    i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.028037s (cpu); 0.0280374s (thread); 0s (gc)
│ │ │ + -- used 0.0337702s (cpu); 0.0337726s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │  
│ │ │ @@ -148,20 +148,21 @@
│ │ │  o13 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : minimizeOIGB C -- an element gets removed
│ │ │  
│ │ │ -                                                                   2
│ │ │ -o14 = {x   e        + x   e       , x   x   x   e           - x   x   e     
│ │ │ -        1,1 1,{1},1    2,1 1,{1},2   2,3 2,2 1,1 3,{2, 3},3    2,1 1,2 3,{1,
│ │ │ +                                                                        
│ │ │ +o14 = {x   x   e        + x   x   e          , x   x   x   e           -
│ │ │ +        1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3   2,3 2,2 1,1 3,{2, 3},3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -          , x   x   e        + x   x   e          }
│ │ │ -      3},3   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ +           2
│ │ │ +      x   x   e          , x   e        + x   e       }
│ │ │ +       2,1 1,2 3,{1, 3},3   1,1 1,{1},1    2,1 1,{1},2
│ │ │  
│ │ │  o14 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -21,15 +21,15 @@ │ │ │ │ i2 : F = makeFreeOIModule(e, {1,1,2}, P); │ │ │ │ i3 : installGeneratorsInWidth(F, 1); │ │ │ │ i4 : installGeneratorsInWidth(F, 2); │ │ │ │ i5 : installGeneratorsInWidth(F, 3); │ │ │ │ i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2); │ │ │ │ i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3); │ │ │ │ i10 : time B = oiGB {b1, b2} │ │ │ │ - -- used 0.028037s (cpu); 0.0280374s (thread); 0s (gc) │ │ │ │ + -- used 0.0337702s (cpu); 0.0337726s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 = {x e + x e , x x e + x x e , │ │ │ │ 1,1 1,{1},1 2,1 1,{1},2 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ x x x e - x x x e } │ │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ │ │ @@ -49,20 +49,21 @@ │ │ │ │ 2 │ │ │ │ - x x e } │ │ │ │ 3},3 2,1 1,2 3,{1, 3},3 │ │ │ │ │ │ │ │ o13 : List │ │ │ │ i14 : minimizeOIGB C -- an element gets removed │ │ │ │ │ │ │ │ - 2 │ │ │ │ -o14 = {x e + x e , x x x e - x x e │ │ │ │ - 1,1 1,{1},1 2,1 1,{1},2 2,3 2,2 1,1 3,{2, 3},3 2,1 1,2 3,{1, │ │ │ │ + │ │ │ │ +o14 = {x x e + x x e , x x x e - │ │ │ │ + 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 2,3 2,2 1,1 3,{2, 3},3 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - , x x e + x x e } │ │ │ │ - 3},3 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ │ + 2 │ │ │ │ + x x e , x e + x e } │ │ │ │ + 2,1 1,2 3,{1, 3},3 1,1 1,{1},1 2,1 1,{1},2 │ │ │ │ │ │ │ │ o14 : List │ │ │ │ ********** WWaayyss ttoo uussee mmiinniimmiizzeeOOIIGGBB:: ********** │ │ │ │ * minimizeOIGB(List) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _m_i_n_i_m_i_z_e_O_I_G_B is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Free__O__I__Module__Map_rp.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ 
    i6 : phi = C.dd_1;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : net phi
│ │ │  
│ │ │ -o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
    │ │ │ +o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})     │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -18,13 +18,13 @@ │ │ │ │ i2 : F = makeFreeOIModule(e, {1,2}, P); │ │ │ │ i3 : installGeneratorsInWidth(F, 3); │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(3,{2},1)+x_(2,2)*x_(2,1)*e_(3,{1,3},2); │ │ │ │ i5 : C = oiRes({b}, 2); │ │ │ │ i6 : phi = C.dd_1; │ │ │ │ i7 : net phi │ │ │ │ │ │ │ │ -o7 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2}) │ │ │ │ +o7 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2}) │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_e_t_(_F_r_e_e_O_I_M_o_d_u_l_e_M_a_p_) -- display a free OI-module map source and target │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.25.11+ds/M2/Macaulay2/packages/OIGroebnerBases.m2:1931:0. │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html │ │ │ @@ -91,15 +91,15 @@ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.221228s (cpu); 0.126662s (thread); 0s (gc)
    │ │ │ + -- used 0.30398s (cpu); 0.14814s (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 0.221228s (cpu); 0.126662s (thread); 0s (gc)
│ │ │ │ + -- used 0.30398s (cpu); 0.14814s (thread); 0s (gc)
│ │ │ │  i6 : net C
│ │ │ │  
│ │ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_e_t_(_O_I_R_e_s_o_l_u_t_i_o_n_) -- display an OI-resolution
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time oiGB {b1, b2}
│ │ │ - -- used 0.0274955s (cpu); 0.0274953s (thread); 0s (gc)
│ │ │ + -- used 0.0346491s (cpu); 0.0346529s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   x   e           - x   x   x   e          }
│ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i9 : time oiGB {b1, b2}
│ │ │ │ - -- used 0.0274955s (cpu); 0.0274953s (thread); 0s (gc)
│ │ │ │ + -- used 0.0346491s (cpu); 0.0346529s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   x   x   e           - x   x   x   e          }
│ │ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.494297s (cpu); 0.290279s (thread); 0s (gc)
│ │ │ + -- used 0.602363s (cpu); 0.331233s (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 0.494297s (cpu); 0.290279s (thread); 0s (gc)
│ │ │ │ + -- used 0.602363s (cpu); 0.331233s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** WWaayyss ttoo uussee ooiiRReess:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Syz.html
│ │ │ @@ -119,21 +119,21 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : oiSyz(G, d)
│ │ │  
│ │ │  o6 = {x   d           - x   d           + 1d             , x   d             
│ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - x   d              - x   d             , x   d              -
│ │ │ -        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2  
│ │ │ +     - x   d             , x   d              - x   d              -
│ │ │ +        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   d             }
│ │ │ -      2,3 4,{1, 2, 4},2
│ │ │ +      1,3 4,{1, 2, 4},2
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    References:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,21 +48,21 @@
│ │ │ │       x   x   x   e       }
│ │ │ │        2,3 2,1 1,2 3,{1},2
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : oiSyz(G, d)
│ │ │ │  
│ │ │ │  o6 = {x   d           - x   d           + 1d             , x   d
│ │ │ │ -       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   1,2 4,{1, 3, 4},2
│ │ │ │ +       1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     - x   d              - x   d             , x   d              -
│ │ │ │ -        1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2   2,4 4,{1, 2, 3},2
│ │ │ │ +     - x   d             , x   d              - x   d              -
│ │ │ │ +        2,3 4,{1, 2, 4},2   1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   d             }
│ │ │ │ -      2,3 4,{1, 2, 4},2
│ │ │ │ +      1,3 4,{1, 2, 4},2
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  RReeffeerreenncceess::
│ │ │ │  [1] M. Morrow and U. Nagel, Computing Gröbner Bases and Free Resolutions of
│ │ │ │  OI-Modules, Preprint, arXiv:2303.06725, 2023.
│ │ │ │  ********** WWaayyss ttoo uussee ooiiSSyyzz:: **********
│ │ │ │      * oiSyz(List,Symbol)
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ - -- used 0.275582s (cpu); 0.210169s (thread); 0s (gc)
│ │ │ + -- used 0.145488s (cpu); 0.145487s (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 0.275582s (cpu); 0.210169s (thread); 0s (gc)
│ │ │ │ + -- used 0.145488s (cpu); 0.145487s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2                  2
│ │ │ │       x   x   e        - x   x   e       }
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Fast__Nonminimal.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.00696s elapsed
│ │ │ + -- 2.64716s 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_*)
│ │ │ - -- 1.40686s elapsed
│ │ │ + -- 1.63368s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.01037s elapsed
│ │ │ + -- 2.61521s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out
│ │ │ @@ -36,16 +36,16 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.863282s (cpu); 0.702465s (thread); 0s (gc)
│ │ │ - -- used 0.689555s (cpu); 0.620226s (thread); 0s (gc)
│ │ │ + -- used 1.06882s (cpu); 0.972865s (thread); 0s (gc)
│ │ │ + -- used 1.16244s (cpu); 0.961163s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Fast__Nonminimal.html
│ │ │ @@ -89,28 +89,28 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.00696s elapsed
│ │ │ + -- 2.64716s 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_*)
│ │ │ - -- 1.40686s elapsed
│ │ │ + -- 1.63368s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ │ - -- 2.00696s elapsed
│ │ │ │ + -- 2.64716s 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_*)
│ │ │ │ - -- 1.40686s elapsed
│ │ │ │ + -- 1.63368s elapsed
│ │ │ │  
│ │ │ │        1      35      140      385      819      1080      819      385      140
│ │ │ │  35      1
│ │ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S
│ │ │ │  <-- S   <-- S  <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 2.01037s elapsed
│ │ │ + -- 2.61521s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ │ - -- 2.01037s elapsed
│ │ │ │ + -- 2.61521s elapsed
│ │ │ │  
│ │ │ │        1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S
│ │ │ │  <-- S    <-- S   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html
│ │ │ @@ -112,16 +112,16 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.863282s (cpu); 0.702465s (thread); 0s (gc)
│ │ │ - -- used 0.689555s (cpu); 0.620226s (thread); 0s (gc)
│ │ │ + -- used 1.06882s (cpu); 0.972865s (thread); 0s (gc)
│ │ │ + -- used 1.16244s (cpu); 0.961163s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,16 +50,16 @@
│ │ │ │            << res M << endl << endl;
│ │ │ │            break;
│ │ │ │            ) else (
│ │ │ │            << "-- computation interrupted" << endl;
│ │ │ │            status M.cache.resolution;
│ │ │ │            << "-- continuing the computation" << endl;
│ │ │ │            ))
│ │ │ │ - -- used 0.863282s (cpu); 0.702465s (thread); 0s (gc)
│ │ │ │ - -- used 0.689555s (cpu); 0.620226s (thread); 0s (gc)
│ │ │ │ + -- used 1.06882s (cpu); 0.972865s (thread); 0s (gc)
│ │ │ │ + -- used 1.16244s (cpu); 0.961163s (thread); 0s (gc)
│ │ │ │  -- computation started:
│ │ │ │  -- computation interrupted
│ │ │ │  -- continuing the computation
│ │ │ │  -- computation complete
│ │ │ │   4      11      89      122      40
│ │ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/example-output/___Working_spwith_spfans_sp-_sp__Part_sp2.out
│ │ │ @@ -39,15 +39,15 @@
│ │ │  o10 : Fan
│ │ │  
│ │ │  i11 : isComplete F
│ │ │  
│ │ │  o11 = true
│ │ │  
│ │ │  i12 : isPolytopal F
│ │ │ -{({ambient dimension => 3           }, {1, 2, 5}), ({ambient dimension => 3           }, {1, 2, 4}), ({ambient dimension => 3           }, {2, 3, 5}), ({ambient dimension => 3           }, {2, 3, 4}), ({ambient dimension => 3           }, {1, 3, 5}), ({ambient dimension => 3           }, {1, 3, 4})}
│ │ │ +{({ambient dimension => 3           }, {2, 3, 5}), ({ambient dimension => 3           }, {2, 3, 4}), ({ambient dimension => 3           }, {1, 3, 5}), ({ambient dimension => 3           }, {1, 3, 4}), ({ambient dimension => 3           }, {1, 2, 5}), ({ambient dimension => 3           }, {1, 2, 4})}
│ │ │     dimension of lineality space => 0                 dimension of lineality space => 0                 dimension of lineality space => 0                 dimension of lineality space => 0                 dimension of lineality space => 0                 dimension of lineality space => 0
│ │ │     dimension of the cone => 1                        dimension of the cone => 1                        dimension of the cone => 1                        dimension of the cone => 1                        dimension of the cone => 1                        dimension of the cone => 1
│ │ │     number of facets => 1                             number of facets => 1                             number of facets => 1                             number of facets => 1                             number of facets => 1                             number of facets => 1
│ │ │     number of rays => 1                               number of rays => 1                               number of rays => 1                               number of rays => 1                               number of rays => 1                               number of rays => 1
│ │ │  v: {{ambient dimension => 3           }, {ambient dimension => 3           }}
│ │ │       dimension of lineality space => 0    dimension of lineality space => 0
│ │ │       dimension of the cone => 3           dimension of the cone => 3
│ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/example-output/_max__Cones.out
│ │ │ @@ -35,18 +35,18 @@
│ │ │        number of facets => 4
│ │ │        number of rays => 4
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : apply(L,rays)
│ │ │  
│ │ │ -o3 = {| -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 -1 1  1 |  | 1 
│ │ │ -      | -1 -1 -1 -1 |  | -1 -1 1  1  |  | -1 -1 1  1  |  | 1  1  1  1 |  | -1
│ │ │ +o3 = {| -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  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  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 -1 -1 -1 |
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/example-output/_normal__Fan.out
│ │ │ @@ -18,13 +18,13 @@
│ │ │        number of rays => 3
│ │ │        top dimension of the cones => 2
│ │ │  
│ │ │  o2 : Fan
│ │ │  
│ │ │  i3 : apply(maxCones F,rays)
│ │ │  
│ │ │ -o3 = {| 1 -1 |, | 1 0 |, | -1 0 |}
│ │ │ -      | 0 -1 |  | 0 1 |  | -1 1 |
│ │ │ +o3 = {| 1 0 |, | -1 0 |, | 1 -1 |}
│ │ │ +      | 0 1 |  | -1 1 |  | 0 -1 |
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/example-output/_skeleton.out
│ │ │ @@ -51,18 +51,18 @@
│ │ │        number of generating polyhedra => 6
│ │ │        top dimension of the polyhedra => 2
│ │ │  
│ │ │  o6 : PolyhedralComplex
│ │ │  
│ │ │  i7 : apply(maxPolyhedra PC1,vertices)
│ │ │  
│ │ │ -o7 = {| -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 1  -1 1 |  | -1
│ │ │ -      | -1 -1 -1 -1 |  | -1 -1 1  1  |  | -1 -1 1  1 |  | -1 -1 1  1 |  | -1
│ │ │ +o7 = {| -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 -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 -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 -1 -1 -1 |
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/html/___Working_spwith_spfans_sp-_sp__Part_sp2.html
│ │ │ @@ -134,15 +134,15 @@
│ │ │            
│ │ │          
│ │ │          
    
│ │ │  For a complete fan we can check if it is projective:        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : isPolytopal F
│ │ │ -{({ambient dimension => 3           }, {1, 2, 5}), ({ambient dimension => 3           }, {1, 2, 4}), ({ambient dimension => 3           }, {2, 3, 5}), ({ambient dimension => 3           }, {2, 3, 4}), ({ambient dimension => 3           }, {1, 3, 5}), ({ambient dimension => 3           }, {1, 3, 4})}
│ │ │ +{({ambient dimension => 3           }, {2, 3, 5}), ({ambient dimension => 3           }, {2, 3, 4}), ({ambient dimension => 3           }, {1, 3, 5}), ({ambient dimension => 3           }, {1, 3, 4}), ({ambient dimension => 3           }, {1, 2, 5}), ({ambient dimension => 3           }, {1, 2, 4})}
│ │ │     dimension of lineality space => 0                 dimension of lineality space => 0                 dimension of lineality space => 0                 dimension of lineality space => 0                 dimension of lineality space => 0                 dimension of lineality space => 0
│ │ │     dimension of the cone => 1                        dimension of the cone => 1                        dimension of the cone => 1                        dimension of the cone => 1                        dimension of the cone => 1                        dimension of the cone => 1
│ │ │     number of facets => 1                             number of facets => 1                             number of facets => 1                             number of facets => 1                             number of facets => 1                             number of facets => 1
│ │ │     number of rays => 1                               number of rays => 1                               number of rays => 1                               number of rays => 1                               number of rays => 1                               number of rays => 1
│ │ │  v: {{ambient dimension => 3           }, {ambient dimension => 3           }}
│ │ │       dimension of lineality space => 0    dimension of lineality space => 0
│ │ │       dimension of the cone => 3           dimension of the cone => 3
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,18 +37,18 @@
│ │ │ │  
│ │ │ │  o10 : Fan
│ │ │ │  i11 : isComplete F
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  For a complete fan we can check if it is projective:
│ │ │ │  i12 : isPolytopal F
│ │ │ │ -{({ambient dimension => 3           }, {1, 2, 5}), ({ambient dimension => 3
│ │ │ │ -}, {1, 2, 4}), ({ambient dimension => 3           }, {2, 3, 5}), ({ambient
│ │ │ │ -dimension => 3           }, {2, 3, 4}), ({ambient dimension => 3           },
│ │ │ │ -{1, 3, 5}), ({ambient dimension => 3           }, {1, 3, 4})}
│ │ │ │ +{({ambient dimension => 3           }, {2, 3, 5}), ({ambient dimension => 3
│ │ │ │ +}, {2, 3, 4}), ({ambient dimension => 3           }, {1, 3, 5}), ({ambient
│ │ │ │ +dimension => 3           }, {1, 3, 4}), ({ambient dimension => 3           },
│ │ │ │ +{1, 2, 5}), ({ambient dimension => 3           }, {1, 2, 4})}
│ │ │ │     dimension of lineality space => 0                 dimension of lineality
│ │ │ │  space => 0                 dimension of lineality space => 0
│ │ │ │  dimension of lineality space => 0                 dimension of lineality space
│ │ │ │  => 0                 dimension of lineality space => 0
│ │ │ │     dimension of the cone => 1                        dimension of the cone => 1
│ │ │ │  dimension of the cone => 1                        dimension of the cone => 1
│ │ │ │  dimension of the cone => 1                        dimension of the cone => 1
│ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/html/_max__Cones.html
│ │ │ @@ -112,21 +112,21 @@
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : apply(L,rays)
│ │ │  
│ │ │ -o3 = {| -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 -1 1  1 |  | 1 
│ │ │ -      | -1 -1 -1 -1 |  | -1 -1 1  1  |  | -1 -1 1  1  |  | 1  1  1  1 |  | -1
│ │ │ +o3 = {| -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  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  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 -1 -1 -1 |
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -47,21 +47,21 @@ │ │ │ │ dimension of the cone => 3 │ │ │ │ number of facets => 4 │ │ │ │ number of rays => 4 │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : apply(L,rays) │ │ │ │ │ │ │ │ -o3 = {| -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 -1 1 1 | | 1 │ │ │ │ - | -1 -1 -1 -1 | | -1 -1 1 1 | | -1 -1 1 1 | | 1 1 1 1 | | -1 │ │ │ │ +o3 = {| -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 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 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 -1 -1 -1 | │ │ │ │ │ │ │ │ o3 : List │ │ │ │ ********** WWaayyss ttoo uussee mmaaxxCCoonneess:: ********** │ │ │ │ * maxCones(Fan) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _m_a_x_C_o_n_e_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/html/_normal__Fan.html │ │ │ @@ -95,16 +95,16 @@ │ │ │ o2 : Fan │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : apply(maxCones F,rays)
│ │ │  
│ │ │ -o3 = {| 1 -1 |, | 1 0 |, | -1 0 |}
│ │ │ -      | 0 -1 |  | 0 1 |  | -1 1 |
│ │ │ +o3 = {| 1 0 |, | -1 0 |, | 1 -1 |}
│ │ │ +      | 0 1 |  | -1 1 |  | 0 -1 |
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -30,16 +30,16 @@ │ │ │ │ number of generating cones => 3 │ │ │ │ number of rays => 3 │ │ │ │ top dimension of the cones => 2 │ │ │ │ │ │ │ │ o2 : Fan │ │ │ │ i3 : apply(maxCones F,rays) │ │ │ │ │ │ │ │ -o3 = {| 1 -1 |, | 1 0 |, | -1 0 |} │ │ │ │ - | 0 -1 | | 0 1 | | -1 1 | │ │ │ │ +o3 = {| 1 0 |, | -1 0 |, | 1 -1 |} │ │ │ │ + | 0 1 | | -1 1 | | 0 -1 | │ │ │ │ │ │ │ │ o3 : List │ │ │ │ ********** WWaayyss ttoo uussee nnoorrmmaallFFaann:: ********** │ │ │ │ * normalFan(Polyhedron) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _n_o_r_m_a_l_F_a_n is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/OldPolyhedra/html/_skeleton.html │ │ │ @@ -147,21 +147,21 @@ │ │ │ o6 : PolyhedralComplex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : apply(maxPolyhedra PC1,vertices)
│ │ │  
│ │ │ -o7 = {| -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 1  -1 1 |  | -1
│ │ │ -      | -1 -1 -1 -1 |  | -1 -1 1  1  |  | -1 -1 1  1 |  | -1 -1 1  1 |  | -1
│ │ │ +o7 = {| -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 -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 -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 -1 -1 -1 |
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -67,21 +67,21 @@ │ │ │ │ o6 = {ambient dimension => 3 } │ │ │ │ number of generating polyhedra => 6 │ │ │ │ top dimension of the polyhedra => 2 │ │ │ │ │ │ │ │ o6 : PolyhedralComplex │ │ │ │ i7 : apply(maxPolyhedra PC1,vertices) │ │ │ │ │ │ │ │ -o7 = {| -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 1 -1 1 | | -1 │ │ │ │ - | -1 -1 -1 -1 | | -1 -1 1 1 | | -1 -1 1 1 | | -1 -1 1 1 | | -1 │ │ │ │ +o7 = {| -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 -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 -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 -1 -1 -1 | │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** WWaayyss ttoo uussee sskkeelleettoonn:: ********** │ │ │ │ * skeleton(ZZ,Fan) │ │ │ │ * skeleton(ZZ,PolyhedralComplex) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _s_k_e_l_e_t_o_n is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n. │ │ ├── ./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} │ │ │ ); │ │ │ - -- .336629s elapsed │ │ │ - -- .387447s elapsed │ │ │ - -- .63134s elapsed │ │ │ - -- .282018s elapsed │ │ │ - -- .394787s elapsed │ │ │ - -- .354219s elapsed │ │ │ - -- .546896s elapsed │ │ │ - -- .467107s elapsed │ │ │ - -- .498606s elapsed │ │ │ - -- .40083s elapsed │ │ │ - -- .178026s elapsed │ │ │ + -- .276851s elapsed │ │ │ + -- .324744s elapsed │ │ │ + -- .558347s elapsed │ │ │ + -- .260463s elapsed │ │ │ + -- .291897s elapsed │ │ │ + -- .335862s elapsed │ │ │ + -- .608529s elapsed │ │ │ + -- .503822s elapsed │ │ │ + -- .5519s elapsed │ │ │ + -- .319487s elapsed │ │ │ + -- .195836s 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} │ │ │ ); │ │ │ - -- .446604s elapsed │ │ │ - -- .493194s elapsed │ │ │ - -- .946836s elapsed │ │ │ - -- 1.33882s elapsed │ │ │ - -- .694353s elapsed │ │ │ - -- .984104s elapsed │ │ │ - -- 1.06479s elapsed │ │ │ - -- 1.27644s elapsed │ │ │ - -- .767216s elapsed │ │ │ - -- .774475s elapsed │ │ │ - -- .389898s elapsed │ │ │ - -- .38865s elapsed │ │ │ - -- .530581s elapsed │ │ │ - -- .705463s elapsed │ │ │ - -- .908256s elapsed │ │ │ - -- 1.33533s elapsed │ │ │ - -- 1.04093s elapsed │ │ │ - -- 1.09386s elapsed │ │ │ - -- 1.39789s elapsed │ │ │ - -- 1.08182s elapsed │ │ │ - -- .975381s elapsed │ │ │ - -- 1.05864s elapsed │ │ │ - -- 1.45235s elapsed │ │ │ - -- 1.1937s elapsed │ │ │ - -- .496277s elapsed │ │ │ - -- .660466s elapsed │ │ │ - -- 1.24494s elapsed │ │ │ - -- .706896s elapsed │ │ │ - -- .643648s elapsed │ │ │ - -- .768961s elapsed │ │ │ - -- 1.09997s elapsed │ │ │ - -- .798938s elapsed │ │ │ - -- .659494s elapsed │ │ │ - -- 1.27724s elapsed │ │ │ - -- .836666s elapsed │ │ │ - -- 1.12851s elapsed │ │ │ - -- 1.17895s elapsed │ │ │ - -- 1.42692s elapsed │ │ │ - -- 1.41956s elapsed │ │ │ - -- .801944s elapsed │ │ │ - -- .598258s elapsed │ │ │ - -- 1.04613s elapsed │ │ │ - -- 1.32322s elapsed │ │ │ - -- 1.80661s elapsed │ │ │ - -- 1.15534s elapsed │ │ │ - -- 1.12495s elapsed │ │ │ - -- 1.47107s elapsed │ │ │ - -- 1.1308s elapsed │ │ │ - -- .893136s elapsed │ │ │ - -- .990024s elapsed │ │ │ - -- .955383s elapsed │ │ │ - -- .706861s elapsed │ │ │ - -- .690691s elapsed │ │ │ - -- .942116s elapsed │ │ │ - -- .628566s elapsed │ │ │ - -- 1.07741s elapsed │ │ │ - -- 1.18738s elapsed │ │ │ - -- 1.2581s elapsed │ │ │ - -- .611208s elapsed │ │ │ - -- .459871s elapsed │ │ │ - -- .367176s elapsed │ │ │ + -- .44258s elapsed │ │ │ + -- .513715s elapsed │ │ │ + -- 1.00739s elapsed │ │ │ + -- 1.27023s elapsed │ │ │ + -- .691829s elapsed │ │ │ + -- .867164s elapsed │ │ │ + -- 1.07984s elapsed │ │ │ + -- 1.06087s elapsed │ │ │ + -- .773137s elapsed │ │ │ + -- .724939s elapsed │ │ │ + -- .397615s elapsed │ │ │ + -- .423242s elapsed │ │ │ + -- .555206s elapsed │ │ │ + -- .628289s elapsed │ │ │ + -- .921131s elapsed │ │ │ + -- 1.30675s elapsed │ │ │ + -- 1.01288s elapsed │ │ │ + -- .868172s elapsed │ │ │ + -- 1.22486s elapsed │ │ │ + -- 1.07537s elapsed │ │ │ + -- .845468s elapsed │ │ │ + -- .974881s elapsed │ │ │ + -- 1.32815s elapsed │ │ │ + -- 1.29229s elapsed │ │ │ + -- .506935s elapsed │ │ │ + -- .682259s elapsed │ │ │ + -- 1.31575s elapsed │ │ │ + -- .715746s elapsed │ │ │ + -- .559782s elapsed │ │ │ + -- .80893s elapsed │ │ │ + -- 1.06642s elapsed │ │ │ + -- .859931s elapsed │ │ │ + -- .599381s elapsed │ │ │ + -- 1.09896s elapsed │ │ │ + -- .824233s elapsed │ │ │ + -- 1.1094s elapsed │ │ │ + -- 1.00216s elapsed │ │ │ + -- 1.22857s elapsed │ │ │ + -- 1.25841s elapsed │ │ │ + -- .782931s elapsed │ │ │ + -- .67639s elapsed │ │ │ + -- 1.18208s elapsed │ │ │ + -- 1.47219s elapsed │ │ │ + -- 1.79184s elapsed │ │ │ + -- 1.1565s elapsed │ │ │ + -- 1.16065s elapsed │ │ │ + -- 1.43167s elapsed │ │ │ + -- 1.24789s elapsed │ │ │ + -- 1.01459s elapsed │ │ │ + -- 1.07771s elapsed │ │ │ + -- 1.11083s elapsed │ │ │ + -- .807341s elapsed │ │ │ + -- .815102s elapsed │ │ │ + -- 1.0648s elapsed │ │ │ + -- .654405s elapsed │ │ │ + -- 1.14168s elapsed │ │ │ + -- 1.30557s elapsed │ │ │ + -- 1.35444s elapsed │ │ │ + -- .755466s elapsed │ │ │ + -- .484864s elapsed │ │ │ + -- .364781s 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 │ │ │ - -- .747s elapsed │ │ │ + -- .974s elapsed │ │ │ -- found extra exotic solutions for graph Graph{0 => {1, 4}} -- │ │ │ 1 => {0, 2} │ │ │ 2 => {1, 3} │ │ │ 3 => {2, 4} │ │ │ 4 => {0, 3} │ │ │ +----+-----+-----+----+-----+-----+-----+-----+ │ │ │ |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951| │ │ │ @@ -60,15 +60,15 @@ │ │ │ +---+---+---+---+ │ │ │ |72 |144|216|288| │ │ │ +---+---+---+---+ │ │ │ |288|216|144|72 | │ │ │ +---+---+---+---+ │ │ │ |0 |0 |0 |0 | │ │ │ +---+---+---+---+ │ │ │ - -- .812s elapsed │ │ │ + -- 1.02s elapsed │ │ │ │ │ │ o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809, │ │ │ ------------------------------------------------------------------------ │ │ │ -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}} │ │ │ │ │ │ o6 : List │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ @@ -44,19 +44,19 @@ │ │ │ │ │ │ i5 : printingPrecision = 3 │ │ │ │ │ │ o5 = 3 │ │ │ │ │ │ i6 : for G in Gs list showExoticSolutions G; │ │ │ warning: some solutions are not regular: {37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 88, 89} │ │ │ - -- .828s elapsed │ │ │ + -- .8s elapsed │ │ │ warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88, 89, 90, 91, 97} │ │ │ - -- .63s elapsed │ │ │ - -- .921s elapsed │ │ │ - -- 1.11s elapsed │ │ │ + -- .767s elapsed │ │ │ + -- .938s elapsed │ │ │ + -- 1.21s elapsed │ │ │ -- found extra exotic solutions for graph Graph{0 => {2, 3}} -- │ │ │ 1 => {3, 4} │ │ │ 2 => {0, 4} │ │ │ 3 => {0, 1} │ │ │ 4 => {2, 1} │ │ │ +-----+----+----+-----+-----+-----+-----+-----+ │ │ │ |-.809|.309|.309|-.809|.588 |-.951|.951 |-.588| │ │ │ @@ -69,20 +69,20 @@ │ │ │ +---+---+---+---+ │ │ │ |144|288|72 |216| │ │ │ +---+---+---+---+ │ │ │ |0 |0 |0 |0 | │ │ │ +---+---+---+---+ │ │ │ |216|72 |288|144| │ │ │ +---+---+---+---+ │ │ │ - -- 1.27s elapsed │ │ │ - -- 1.46s elapsed │ │ │ + -- 1.38s elapsed │ │ │ + -- 1.54s elapsed │ │ │ warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47, 53, 54, 56, 57, 59, 60} │ │ │ - -- 1.6s elapsed │ │ │ + -- 1.84s elapsed │ │ │ warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34} │ │ │ - -- 1.28s elapsed │ │ │ - -- 1.34s elapsed │ │ │ + -- 1.59s elapsed │ │ │ + -- 1.6s elapsed │ │ │ warning: some solutions are not regular: {26, 29, 30, 32, 33} │ │ │ - -- 1.57s elapsed │ │ │ + -- 1.73s elapsed │ │ │ warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67, 72, 77, 78} │ │ │ - -- 1.43s elapsed │ │ │ + -- 1.56s elapsed │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out │ │ │ @@ -1,15 +1,15 @@ │ │ │ -- -*- M2-comint -*- hash: 1729328129346969841 │ │ │ │ │ │ i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}}); │ │ │ │ │ │ i2 : getLinearlyStableSolutions(G) │ │ │ -- warning: experimental computation over inexact field begun │ │ │ -- results not reliable (one warning given per session) │ │ │ - -- .327118s elapsed │ │ │ + -- .271401s elapsed │ │ │ warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} │ │ │ │ │ │ o2 = {{1, 1, 1, 0, 0, 0}} │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ 4 => {0, 3} │ │ │ │ │ │ o1 : Graph │ │ │ │ │ │ i2 : showExoticSolutions G │ │ │ -- warning: experimental computation over inexact field begun │ │ │ -- results not reliable (one warning given per session) │ │ │ - -- .890937s elapsed │ │ │ + -- 1.0325s elapsed │ │ │ -- found extra exotic solutions for graph Graph{0 => {1, 4}} -- │ │ │ 1 => {0, 2} │ │ │ 2 => {1, 3} │ │ │ 3 => {2, 4} │ │ │ 4 => {0, 3} │ │ │ +-------+--------+--------+-------+--------+--------+--------+--------+ │ │ │ |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057| │ │ │ @@ -50,14 +50,14 @@ │ │ │ 2 => {1, 3, 4} │ │ │ 3 => {2, 4} │ │ │ 4 => {0, 2, 3} │ │ │ │ │ │ o3 : Graph │ │ │ │ │ │ i4 : showExoticSolutions G │ │ │ - -- 1.30632s elapsed │ │ │ + -- 1.40178s elapsed │ │ │ │ │ │ o4 = {{1, 1, 1, 1, 0, 0, 0, 0}} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ @@ -295,25 +295,25 @@ │ │ │ │ │ │ │ │ │ 
    i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .336629s elapsed
│ │ │ - -- .387447s elapsed
│ │ │ - -- .63134s elapsed
│ │ │ - -- .282018s elapsed
│ │ │ - -- .394787s elapsed
│ │ │ - -- .354219s elapsed
│ │ │ - -- .546896s elapsed
│ │ │ - -- .467107s elapsed
│ │ │ - -- .498606s elapsed
│ │ │ - -- .40083s elapsed
│ │ │ - -- .178026s elapsed
    │ │ │ + -- .276851s elapsed │ │ │ + -- .324744s elapsed │ │ │ + -- .558347s elapsed │ │ │ + -- .260463s elapsed │ │ │ + -- .291897s elapsed │ │ │ + -- .335862s elapsed │ │ │ + -- .608529s elapsed │ │ │ + -- .503822s elapsed │ │ │ + -- .5519s elapsed │ │ │ + -- .319487s elapsed │ │ │ + -- .195836s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │ @@ -380,75 +380,75 @@
│ │ │            
│ │ │              
│ │ │                
    i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .446604s elapsed
│ │ │ - -- .493194s elapsed
│ │ │ - -- .946836s elapsed
│ │ │ - -- 1.33882s elapsed
│ │ │ - -- .694353s elapsed
│ │ │ - -- .984104s elapsed
│ │ │ - -- 1.06479s elapsed
│ │ │ - -- 1.27644s elapsed
│ │ │ - -- .767216s elapsed
│ │ │ - -- .774475s elapsed
│ │ │ - -- .389898s elapsed
│ │ │ - -- .38865s elapsed
│ │ │ - -- .530581s elapsed
│ │ │ - -- .705463s elapsed
│ │ │ - -- .908256s elapsed
│ │ │ - -- 1.33533s elapsed
│ │ │ - -- 1.04093s elapsed
│ │ │ - -- 1.09386s elapsed
│ │ │ - -- 1.39789s elapsed
│ │ │ - -- 1.08182s elapsed
│ │ │ - -- .975381s elapsed
│ │ │ - -- 1.05864s elapsed
│ │ │ - -- 1.45235s elapsed
│ │ │ - -- 1.1937s elapsed
│ │ │ - -- .496277s elapsed
│ │ │ - -- .660466s elapsed
│ │ │ - -- 1.24494s elapsed
│ │ │ - -- .706896s elapsed
│ │ │ - -- .643648s elapsed
│ │ │ - -- .768961s elapsed
│ │ │ - -- 1.09997s elapsed
│ │ │ - -- .798938s elapsed
│ │ │ - -- .659494s elapsed
│ │ │ - -- 1.27724s elapsed
│ │ │ - -- .836666s elapsed
│ │ │ - -- 1.12851s elapsed
│ │ │ - -- 1.17895s elapsed
│ │ │ - -- 1.42692s elapsed
│ │ │ - -- 1.41956s elapsed
│ │ │ - -- .801944s elapsed
│ │ │ - -- .598258s elapsed
│ │ │ - -- 1.04613s elapsed
│ │ │ - -- 1.32322s elapsed
│ │ │ - -- 1.80661s elapsed
│ │ │ - -- 1.15534s elapsed
│ │ │ - -- 1.12495s elapsed
│ │ │ - -- 1.47107s elapsed
│ │ │ - -- 1.1308s elapsed
│ │ │ - -- .893136s elapsed
│ │ │ - -- .990024s elapsed
│ │ │ - -- .955383s elapsed
│ │ │ - -- .706861s elapsed
│ │ │ - -- .690691s elapsed
│ │ │ - -- .942116s elapsed
│ │ │ - -- .628566s elapsed
│ │ │ - -- 1.07741s elapsed
│ │ │ - -- 1.18738s elapsed
│ │ │ - -- 1.2581s elapsed
│ │ │ - -- .611208s elapsed
│ │ │ - -- .459871s elapsed
│ │ │ - -- .367176s elapsed
    
│ │ │ + -- .44258s elapsed
│ │ │ + -- .513715s elapsed
│ │ │ + -- 1.00739s elapsed
│ │ │ + -- 1.27023s elapsed
│ │ │ + -- .691829s elapsed
│ │ │ + -- .867164s elapsed
│ │ │ + -- 1.07984s elapsed
│ │ │ + -- 1.06087s elapsed
│ │ │ + -- .773137s elapsed
│ │ │ + -- .724939s elapsed
│ │ │ + -- .397615s elapsed
│ │ │ + -- .423242s elapsed
│ │ │ + -- .555206s elapsed
│ │ │ + -- .628289s elapsed
│ │ │ + -- .921131s elapsed
│ │ │ + -- 1.30675s elapsed
│ │ │ + -- 1.01288s elapsed
│ │ │ + -- .868172s elapsed
│ │ │ + -- 1.22486s elapsed
│ │ │ + -- 1.07537s elapsed
│ │ │ + -- .845468s elapsed
│ │ │ + -- .974881s elapsed
│ │ │ + -- 1.32815s elapsed
│ │ │ + -- 1.29229s elapsed
│ │ │ + -- .506935s elapsed
│ │ │ + -- .682259s elapsed
│ │ │ + -- 1.31575s elapsed
│ │ │ + -- .715746s elapsed
│ │ │ + -- .559782s elapsed
│ │ │ + -- .80893s elapsed
│ │ │ + -- 1.06642s elapsed
│ │ │ + -- .859931s elapsed
│ │ │ + -- .599381s elapsed
│ │ │ + -- 1.09896s elapsed
│ │ │ + -- .824233s elapsed
│ │ │ + -- 1.1094s elapsed
│ │ │ + -- 1.00216s elapsed
│ │ │ + -- 1.22857s elapsed
│ │ │ + -- 1.25841s elapsed
│ │ │ + -- .782931s elapsed
│ │ │ + -- .67639s elapsed
│ │ │ + -- 1.18208s elapsed
│ │ │ + -- 1.47219s elapsed
│ │ │ + -- 1.79184s elapsed
│ │ │ + -- 1.1565s elapsed
│ │ │ + -- 1.16065s elapsed
│ │ │ + -- 1.43167s elapsed
│ │ │ + -- 1.24789s elapsed
│ │ │ + -- 1.01459s elapsed
│ │ │ + -- 1.07771s elapsed
│ │ │ + -- 1.11083s elapsed
│ │ │ + -- .807341s elapsed
│ │ │ + -- .815102s elapsed
│ │ │ + -- 1.0648s elapsed
│ │ │ + -- .654405s elapsed
│ │ │ + -- 1.14168s elapsed
│ │ │ + -- 1.30557s elapsed
│ │ │ + -- 1.35444s elapsed
│ │ │ + -- .755466s elapsed
│ │ │ + -- .484864s elapsed
│ │ │ + -- .364781s 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}
│ │ │ │            );
│ │ │ │ - -- .336629s elapsed
│ │ │ │ - -- .387447s elapsed
│ │ │ │ - -- .63134s elapsed
│ │ │ │ - -- .282018s elapsed
│ │ │ │ - -- .394787s elapsed
│ │ │ │ - -- .354219s elapsed
│ │ │ │ - -- .546896s elapsed
│ │ │ │ - -- .467107s elapsed
│ │ │ │ - -- .498606s elapsed
│ │ │ │ - -- .40083s elapsed
│ │ │ │ - -- .178026s elapsed
│ │ │ │ + -- .276851s elapsed
│ │ │ │ + -- .324744s elapsed
│ │ │ │ + -- .558347s elapsed
│ │ │ │ + -- .260463s elapsed
│ │ │ │ + -- .291897s elapsed
│ │ │ │ + -- .335862s elapsed
│ │ │ │ + -- .608529s elapsed
│ │ │ │ + -- .503822s elapsed
│ │ │ │ + -- .5519s elapsed
│ │ │ │ + -- .319487s elapsed
│ │ │ │ + -- .195836s 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}
│ │ │ │            );
│ │ │ │ - -- .446604s elapsed
│ │ │ │ - -- .493194s elapsed
│ │ │ │ - -- .946836s elapsed
│ │ │ │ - -- 1.33882s elapsed
│ │ │ │ - -- .694353s elapsed
│ │ │ │ - -- .984104s elapsed
│ │ │ │ - -- 1.06479s elapsed
│ │ │ │ - -- 1.27644s elapsed
│ │ │ │ - -- .767216s elapsed
│ │ │ │ - -- .774475s elapsed
│ │ │ │ - -- .389898s elapsed
│ │ │ │ - -- .38865s elapsed
│ │ │ │ - -- .530581s elapsed
│ │ │ │ - -- .705463s elapsed
│ │ │ │ - -- .908256s elapsed
│ │ │ │ - -- 1.33533s elapsed
│ │ │ │ - -- 1.04093s elapsed
│ │ │ │ - -- 1.09386s elapsed
│ │ │ │ - -- 1.39789s elapsed
│ │ │ │ - -- 1.08182s elapsed
│ │ │ │ - -- .975381s elapsed
│ │ │ │ - -- 1.05864s elapsed
│ │ │ │ - -- 1.45235s elapsed
│ │ │ │ - -- 1.1937s elapsed
│ │ │ │ - -- .496277s elapsed
│ │ │ │ - -- .660466s elapsed
│ │ │ │ - -- 1.24494s elapsed
│ │ │ │ - -- .706896s elapsed
│ │ │ │ - -- .643648s elapsed
│ │ │ │ - -- .768961s elapsed
│ │ │ │ - -- 1.09997s elapsed
│ │ │ │ - -- .798938s elapsed
│ │ │ │ - -- .659494s elapsed
│ │ │ │ - -- 1.27724s elapsed
│ │ │ │ - -- .836666s elapsed
│ │ │ │ - -- 1.12851s elapsed
│ │ │ │ - -- 1.17895s elapsed
│ │ │ │ - -- 1.42692s elapsed
│ │ │ │ - -- 1.41956s elapsed
│ │ │ │ - -- .801944s elapsed
│ │ │ │ - -- .598258s elapsed
│ │ │ │ - -- 1.04613s elapsed
│ │ │ │ - -- 1.32322s elapsed
│ │ │ │ - -- 1.80661s elapsed
│ │ │ │ - -- 1.15534s elapsed
│ │ │ │ - -- 1.12495s elapsed
│ │ │ │ - -- 1.47107s elapsed
│ │ │ │ - -- 1.1308s elapsed
│ │ │ │ - -- .893136s elapsed
│ │ │ │ - -- .990024s elapsed
│ │ │ │ - -- .955383s elapsed
│ │ │ │ - -- .706861s elapsed
│ │ │ │ - -- .690691s elapsed
│ │ │ │ - -- .942116s elapsed
│ │ │ │ - -- .628566s elapsed
│ │ │ │ - -- 1.07741s elapsed
│ │ │ │ - -- 1.18738s elapsed
│ │ │ │ - -- 1.2581s elapsed
│ │ │ │ - -- .611208s elapsed
│ │ │ │ - -- .459871s elapsed
│ │ │ │ - -- .367176s elapsed
│ │ │ │ + -- .44258s elapsed
│ │ │ │ + -- .513715s elapsed
│ │ │ │ + -- 1.00739s elapsed
│ │ │ │ + -- 1.27023s elapsed
│ │ │ │ + -- .691829s elapsed
│ │ │ │ + -- .867164s elapsed
│ │ │ │ + -- 1.07984s elapsed
│ │ │ │ + -- 1.06087s elapsed
│ │ │ │ + -- .773137s elapsed
│ │ │ │ + -- .724939s elapsed
│ │ │ │ + -- .397615s elapsed
│ │ │ │ + -- .423242s elapsed
│ │ │ │ + -- .555206s elapsed
│ │ │ │ + -- .628289s elapsed
│ │ │ │ + -- .921131s elapsed
│ │ │ │ + -- 1.30675s elapsed
│ │ │ │ + -- 1.01288s elapsed
│ │ │ │ + -- .868172s elapsed
│ │ │ │ + -- 1.22486s elapsed
│ │ │ │ + -- 1.07537s elapsed
│ │ │ │ + -- .845468s elapsed
│ │ │ │ + -- .974881s elapsed
│ │ │ │ + -- 1.32815s elapsed
│ │ │ │ + -- 1.29229s elapsed
│ │ │ │ + -- .506935s elapsed
│ │ │ │ + -- .682259s elapsed
│ │ │ │ + -- 1.31575s elapsed
│ │ │ │ + -- .715746s elapsed
│ │ │ │ + -- .559782s elapsed
│ │ │ │ + -- .80893s elapsed
│ │ │ │ + -- 1.06642s elapsed
│ │ │ │ + -- .859931s elapsed
│ │ │ │ + -- .599381s elapsed
│ │ │ │ + -- 1.09896s elapsed
│ │ │ │ + -- .824233s elapsed
│ │ │ │ + -- 1.1094s elapsed
│ │ │ │ + -- 1.00216s elapsed
│ │ │ │ + -- 1.22857s elapsed
│ │ │ │ + -- 1.25841s elapsed
│ │ │ │ + -- .782931s elapsed
│ │ │ │ + -- .67639s elapsed
│ │ │ │ + -- 1.18208s elapsed
│ │ │ │ + -- 1.47219s elapsed
│ │ │ │ + -- 1.79184s elapsed
│ │ │ │ + -- 1.1565s elapsed
│ │ │ │ + -- 1.16065s elapsed
│ │ │ │ + -- 1.43167s elapsed
│ │ │ │ + -- 1.24789s elapsed
│ │ │ │ + -- 1.01459s elapsed
│ │ │ │ + -- 1.07771s elapsed
│ │ │ │ + -- 1.11083s elapsed
│ │ │ │ + -- .807341s elapsed
│ │ │ │ + -- .815102s elapsed
│ │ │ │ + -- 1.0648s elapsed
│ │ │ │ + -- .654405s elapsed
│ │ │ │ + -- 1.14168s elapsed
│ │ │ │ + -- 1.30557s elapsed
│ │ │ │ + -- 1.35444s elapsed
│ │ │ │ + -- .755466s elapsed
│ │ │ │ + -- .484864s elapsed
│ │ │ │ + -- .364781s elapsed
│ │ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │ │  
│ │ │ │        +------------------------+------------------------+
│ │ │ │  o24 = |codimensions            |degrees                 |
│ │ │ │        +------------------------+------------------------+
│ │ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ │ │        +------------------------+------------------------+
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .747s elapsed
│ │ │ + -- .974s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -131,15 +131,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │ - -- .812s elapsed
│ │ │ + -- 1.02s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809,
│ │ │       ------------------------------------------------------------------------
│ │ │       -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -43,15 +43,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ .954}} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : elapsedTime stablesolsPent = showExoticSolutions Pent │ │ │ │ - -- .747s elapsed │ │ │ │ + -- .974s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {1, 4}} -- │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ +----+-----+-----+----+-----+-----+-----+-----+ │ │ │ │ |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951| │ │ │ │ @@ -64,15 +64,15 @@ │ │ │ │ +---+---+---+---+ │ │ │ │ |72 |144|216|288| │ │ │ │ +---+---+---+---+ │ │ │ │ |288|216|144|72 | │ │ │ │ +---+---+---+---+ │ │ │ │ |0 |0 |0 |0 | │ │ │ │ +---+---+---+---+ │ │ │ │ - -- .812s elapsed │ │ │ │ + -- 1.02s elapsed │ │ │ │ │ │ │ │ o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {.309, -.809, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ -.809, .309, -.951, -.588, .588, .951}, {1, 1, 1, 1, 0, 0, 0, 0}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ Computing the (linearly) stable solutions for K5C5 takes a minute or two: │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ @@ -115,19 +115,19 @@ │ │ │ o5 = 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 88, 89}
│ │ │ - -- .828s elapsed
│ │ │ + -- .8s elapsed
│ │ │  warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88, 89, 90, 91, 97}
│ │ │ - -- .63s elapsed
│ │ │ - -- .921s elapsed
│ │ │ - -- 1.11s elapsed
│ │ │ + -- .767s elapsed
│ │ │ + -- .938s elapsed
│ │ │ + -- 1.21s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |-.809|.309|.309|-.809|.588 |-.951|.951 |-.588|
│ │ │ @@ -140,25 +140,25 @@
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │ - -- 1.27s elapsed
│ │ │ - -- 1.46s elapsed
│ │ │ + -- 1.38s elapsed
│ │ │ + -- 1.54s elapsed
│ │ │  warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47, 53, 54, 56, 57, 59, 60}
│ │ │ - -- 1.6s elapsed
│ │ │ + -- 1.84s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34}
│ │ │ - -- 1.28s elapsed
│ │ │ - -- 1.34s elapsed
│ │ │ + -- 1.59s elapsed
│ │ │ + -- 1.6s elapsed
│ │ │  warning: some solutions are not regular: {26, 29, 30, 32, 33}
│ │ │ - -- 1.57s elapsed
│ │ │ + -- 1.73s elapsed
│ │ │  warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67, 72, 77, 78}
│ │ │ - -- 1.43s elapsed
    │ │ │ + -- 1.56s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,21 +48,21 @@ │ │ │ │ i5 : printingPrecision = 3 │ │ │ │ │ │ │ │ o5 = 3 │ │ │ │ i6 : for G in Gs list showExoticSolutions G; │ │ │ │ warning: some solutions are not regular: {37, 38, 40, 41, 42, 43, 44, 45, 46, │ │ │ │ 48, 50, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, │ │ │ │ 79, 81, 83, 85, 86, 87, 88, 89} │ │ │ │ - -- .828s elapsed │ │ │ │ + -- .8s elapsed │ │ │ │ warning: some solutions are not regular: {43, 44, 47, 49, 50, 51, 52, 53, 55, │ │ │ │ 57, 59, 61, 62, 63, 64, 65, 66, 68, 72, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88, │ │ │ │ 89, 90, 91, 97} │ │ │ │ - -- .63s elapsed │ │ │ │ - -- .921s elapsed │ │ │ │ - -- 1.11s elapsed │ │ │ │ + -- .767s elapsed │ │ │ │ + -- .938s elapsed │ │ │ │ + -- 1.21s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {2, 3}} -- │ │ │ │ 1 => {3, 4} │ │ │ │ 2 => {0, 4} │ │ │ │ 3 => {0, 1} │ │ │ │ 4 => {2, 1} │ │ │ │ +-----+----+----+-----+-----+-----+-----+-----+ │ │ │ │ |-.809|.309|.309|-.809|.588 |-.951|.951 |-.588| │ │ │ │ @@ -75,24 +75,24 @@ │ │ │ │ +---+---+---+---+ │ │ │ │ |144|288|72 |216| │ │ │ │ +---+---+---+---+ │ │ │ │ |0 |0 |0 |0 | │ │ │ │ +---+---+---+---+ │ │ │ │ |216|72 |288|144| │ │ │ │ +---+---+---+---+ │ │ │ │ - -- 1.27s elapsed │ │ │ │ - -- 1.46s elapsed │ │ │ │ + -- 1.38s elapsed │ │ │ │ + -- 1.54s elapsed │ │ │ │ warning: some solutions are not regular: {27, 31, 32, 34, 37, 38, 44, 46, 47, │ │ │ │ 53, 54, 56, 57, 59, 60} │ │ │ │ - -- 1.6s elapsed │ │ │ │ + -- 1.84s elapsed │ │ │ │ warning: some solutions are not regular: {16, 17, 18, 19, 20, 21, 22, 23, 24, │ │ │ │ 25, 26, 27, 28, 29, 31, 34} │ │ │ │ - -- 1.28s elapsed │ │ │ │ - -- 1.34s elapsed │ │ │ │ + -- 1.59s elapsed │ │ │ │ + -- 1.6s elapsed │ │ │ │ warning: some solutions are not regular: {26, 29, 30, 32, 33} │ │ │ │ - -- 1.57s elapsed │ │ │ │ + -- 1.73s elapsed │ │ │ │ warning: some solutions are not regular: {38, 40, 42, 53, 54, 55, 62, 63, 67, │ │ │ │ 72, 77, 78} │ │ │ │ - -- 1.43s elapsed │ │ │ │ + -- 1.56s elapsed │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.25.11+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:812:0. │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ @@ -77,15 +77,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : getLinearlyStableSolutions(G)
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .327118s elapsed
│ │ │ + -- .271401s elapsed
│ │ │  warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ of each oscillator is given by the Kuramoto model. The linear stability of a │ │ │ │ solution is determined by the eigenvalues of the Jacobian matrix of the system │ │ │ │ evaluated at the solution. │ │ │ │ i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}}); │ │ │ │ i2 : getLinearlyStableSolutions(G) │ │ │ │ -- warning: experimental computation over inexact field begun │ │ │ │ -- results not reliable (one warning given per session) │ │ │ │ - -- .327118s elapsed │ │ │ │ + -- .271401s elapsed │ │ │ │ warning: some solutions are not regular: {4, 5, 7, 9, 10, 12, 13, 14, 15, 16, │ │ │ │ 17, 18, 19, 20, 21} │ │ │ │ │ │ │ │ o2 = {{1, 1, 1, 0, 0, 0}} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : showExoticSolutions G
│ │ │  -- warning: experimental computation over inexact field begun
│ │ │  --          results not reliable (one warning given per session)
│ │ │ - -- .890937s elapsed
│ │ │ + -- 1.0325s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -147,15 +147,15 @@
│ │ │  
│ │ │  o3 : Graph
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : showExoticSolutions G
│ │ │ - -- 1.30632s elapsed
│ │ │ + -- 1.40178s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ │ │ │ │ o1 : Graph │ │ │ │ i2 : showExoticSolutions G │ │ │ │ -- warning: experimental computation over inexact field begun │ │ │ │ -- results not reliable (one warning given per session) │ │ │ │ - -- .890937s elapsed │ │ │ │ + -- 1.0325s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {1, 4}} -- │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ +-------+--------+--------+-------+--------+--------+--------+--------+ │ │ │ │ |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057| │ │ │ │ @@ -78,15 +78,15 @@ │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3, 4} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 2, 3} │ │ │ │ │ │ │ │ o3 : Graph │ │ │ │ i4 : showExoticSolutions G │ │ │ │ - -- 1.30632s elapsed │ │ │ │ + -- 1.40178s elapsed │ │ │ │ │ │ │ │ o4 = {{1, 1, 1, 1, 0, 0, 0, 0}} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_t_L_i_n_e_a_r_l_y_S_t_a_b_l_e_S_o_l_u_t_i_o_n_s -- Compute linearly stable solutions for the │ │ │ │ Kuramoto oscillator system associated to a graph │ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ @@ -80,20 +80,20 @@ │ │ │ i19 : needsPackage "MultigradedImplicitization"; │ │ │ │ │ │ i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S); │ │ │ warning: computation begun over finite field. resulting polynomials may not lie in the ideal │ │ │ computing total degree: 1 │ │ │ number of monomials = 9 │ │ │ number of distinct multidegrees = 1 │ │ │ - -- .00915863s elapsed │ │ │ + -- .0105515s 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 │ │ │ - -- .544497s elapsed │ │ │ + -- .569209s elapsed │ │ │ │ │ │ o20 : Ideal of S │ │ │ │ │ │ i21 : dim I │ │ │ │ │ │ o21 = 5 │ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ @@ -208,20 +208,20 @@ │ │ │ │ │ │ │ │ │ 
    i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 9
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .00915863s elapsed
│ │ │ + -- .0105515s 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
│ │ │ - -- .544497s elapsed
│ │ │ + -- .569209s 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
│ │ │ │ - -- .00915863s elapsed
│ │ │ │ + -- .0105515s 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
│ │ │ │ - -- .544497s elapsed
│ │ │ │ + -- .569209s elapsed
│ │ │ │  
│ │ │ │  o20 : Ideal of S
│ │ │ │  i21 : dim I
│ │ │ │  
│ │ │ │  o21 = 5
│ │ │ │  i22 : isPrime I
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out
│ │ │ @@ -41,15 +41,15 @@
│ │ │  i3 : g=3
│ │ │  
│ │ │  o3 = 3
│ │ │  
│ │ │  i4 : kk= ZZ/101;
│ │ │  
│ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- 1.1672s elapsed
│ │ │ + -- 1.09938s 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 ();
│ │ │ - -- .368769s elapsed
│ │ │ + -- .413888s 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
│ │ │ - -- 19.7949s elapsed
│ │ │ + -- 15.136s 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);
│ │ │ - -- .733014s elapsed
│ │ │ + -- .653137s 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);
│ │ │ - -- 2.32385s elapsed
│ │ │ + -- 1.89125s elapsed
│ │ │  
│ │ │  i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │  o16 = total: 12 24 12
│ │ │            0: 12 24 12
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html
│ │ │ @@ -128,15 +128,15 @@
│ │ │              
│ │ │                
    i4 : kk= ZZ/101;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- 1.1672s elapsed
    
│ │ │ + -- 1.09938s elapsed
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │ @@ -172,15 +172,15 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .368769s elapsed
    │ │ │ + -- .413888s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │ @@ -193,15 +193,15 @@
│ │ │  
│ │ │  o12 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 19.7949s elapsed
    │ │ │ + -- 15.136s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │ ├── html2text {}
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │              -- will give an Ulrich bundle, with betti table
│ │ │ │              -- 16 32 16
│ │ │ │  i3 : g=3
│ │ │ │  
│ │ │ │  o3 = 3
│ │ │ │  i4 : kk= ZZ/101;
│ │ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ │ - -- 1.1672s elapsed
│ │ │ │ + -- 1.09938s 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 ();
│ │ │ │ - -- .368769s elapsed
│ │ │ │ + -- .413888s 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
│ │ │ │ - -- 19.7949s elapsed
│ │ │ │ + -- 15.136s elapsed
│ │ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │ │  
│ │ │ │                32      32
│ │ │ │  o14 : Matrix S   <-- S
│ │ │ │  i15 : r=2
│ │ │ │  
│ │ │ │  o15 = 2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_search__Ulrich.html
│ │ │ @@ -161,15 +161,15 @@
│ │ │  
│ │ │  o11 : CliffordModule
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- .733014s elapsed
    │ │ │ + -- .653137s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.32385s elapsed
    │ │ │ + -- 1.89125s 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);
│ │ │ │ - -- .733014s elapsed
│ │ │ │ + -- .653137s 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);
│ │ │ │ - -- 2.32385s elapsed
│ │ │ │ + -- 1.89125s elapsed
│ │ │ │  i16 : betti freeResolution Ulr3
│ │ │ │  
│ │ │ │                0  1  2
│ │ │ │  o16 = total: 12 24 12
│ │ │ │            0: 12 24 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out
│ │ │ @@ -66,17 +66,17 @@
│ │ │  i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3}
│ │ │  
│ │ │  o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.69184s elapsed
│ │ │ + -- 2.018s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 5.13187s elapsed
│ │ │ + -- 4.34458s elapsed
│ │ │  
│ │ │  i12 : G==H
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html
│ │ │ @@ -177,21 +177,21 @@
│ │ │  
│ │ │  o9 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.69184s elapsed
    │ │ │ + -- 2.018s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 5.13187s elapsed
    │ │ │ + -- 4.34458s 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); │ │ │ │ - -- 2.69184s elapsed │ │ │ │ + -- 2.018s elapsed │ │ │ │ i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R); │ │ │ │ - -- 5.13187s elapsed │ │ │ │ + -- 4.34458s 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/PolyominoIdeals/example-output/_polyo__Ideal.out │ │ │ @@ -18,31 +18,31 @@ │ │ │ 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 , │ │ │ - 2,4 1,2 2,2 1,4 4,4 3,2 4,2 3,4 2,4 1,3 2,3 1,4 │ │ │ + 2,3 1,1 2,1 1,3 4,3 3,1 4,1 3,3 4,4 1,3 4,3 1,4 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ - 4,4 3,3 4,3 3,4 3,2 1,1 3,1 1,2 4,4 2,3 4,3 2,4 2,2 1,1 │ │ │ + 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 │ │ │ ------------------------------------------------------------------------ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ - 2,1 1,2 4,2 3,1 4,1 3,2 2,3 1,1 2,1 1,3 4,3 3,1 │ │ │ + 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 4,2 3,3 3,4 1,3 │ │ │ ------------------------------------------------------------------------ │ │ │ x x , x x - x x , x x - x x , x x - x x , │ │ │ - 4,1 3,3 4,4 1,3 4,3 1,4 3,4 2,3 3,3 2,4 4,2 2,1 4,1 2,2 │ │ │ + 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 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ - 2,3 1,2 2,2 1,3 2,4 1,1 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 │ │ │ + 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 │ │ │ ------------------------------------------------------------------------ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ - 4,2 3,3 3,4 1,3 3,3 1,4 3,2 2,1 3,1 2,2 4,2 1,1 │ │ │ + 3,1 1,2 4,4 2,3 4,3 2,4 2,2 1,1 2,1 1,2 4,2 3,1 │ │ │ ------------------------------------------------------------------------ │ │ │ x x ) │ │ │ - 4,1 1,2 │ │ │ + 4,1 3,2 │ │ │ │ │ │ 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 │ │ │ @@ -105,33 +105,33 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : I = polyoIdeal Q
│ │ │  
│ │ │  o4 = ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -             2,4 1,2    2,2 1,4   4,4 3,2    4,2 3,4   2,4 1,3    2,3 1,4 
│ │ │ +             2,3 1,1    2,1 1,3   4,3 3,1    4,1 3,3   4,4 1,3    4,3 1,4 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ -      4,4 3,3    4,3 3,4   3,2 1,1    3,1 1,2   4,4 2,3    4,3 2,4   2,2 1,1
│ │ │ +      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
│ │ │       ------------------------------------------------------------------------
│ │ │       - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    -
│ │ │ -        2,1 1,2   4,2 3,1    4,1 3,2   2,3 1,1    2,1 1,3   4,3 3,1  
│ │ │ +        2,1 1,4   4,4 3,1    4,1 3,4   4,3 3,2    4,2 3,3   3,4 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   , x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -      4,1 3,3   4,4 1,3    4,3 1,4   3,4 2,3    3,3 2,4   4,2 2,1    4,1 2,2 
│ │ │ +      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 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ -      2,3 1,2    2,2 1,3   2,4 1,1    2,1 1,4   4,4 3,1    4,1 3,4   4,3 3,2
│ │ │ +      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
│ │ │       ------------------------------------------------------------------------
│ │ │       - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    -
│ │ │ -        4,2 3,3   3,4 1,3    3,3 1,4   3,2 2,1    3,1 2,2   4,2 1,1  
│ │ │ +        3,1 1,2   4,4 2,3    4,3 2,4   2,2 1,1    2,1 1,2   4,2 3,1  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   )
│ │ │ -      4,1 1,2
│ │ │ +      4,1 3,2
│ │ │  
│ │ │  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 {} │ │ │ │ @@ -41,33 +41,33 @@ │ │ │ │ │ │ │ │ │ │ │ │ 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 , │ │ │ │ - 2,4 1,2 2,2 1,4 4,4 3,2 4,2 3,4 2,4 1,3 2,3 1,4 │ │ │ │ + 2,3 1,1 2,1 1,3 4,3 3,1 4,1 3,3 4,4 1,3 4,3 1,4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ - 4,4 3,3 4,3 3,4 3,2 1,1 3,1 1,2 4,4 2,3 4,3 2,4 2,2 1,1 │ │ │ │ + 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 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ │ - 2,1 1,2 4,2 3,1 4,1 3,2 2,3 1,1 2,1 1,3 4,3 3,1 │ │ │ │ + 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 4,2 3,3 3,4 1,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x , x x - x x , x x - x x , x x - x x , │ │ │ │ - 4,1 3,3 4,4 1,3 4,3 1,4 3,4 2,3 3,3 2,4 4,2 2,1 4,1 2,2 │ │ │ │ + 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 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ - 2,3 1,2 2,2 1,3 2,4 1,1 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 │ │ │ │ + 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 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ │ - 4,2 3,3 3,4 1,3 3,3 1,4 3,2 2,1 3,1 2,2 4,2 1,1 │ │ │ │ + 3,1 1,2 4,4 2,3 4,3 2,4 2,2 1,1 2,1 1,2 4,2 3,1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ - 4,1 1,2 │ │ │ │ + 4,1 3,2 │ │ │ │ │ │ │ │ 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/Posets/example-output/___Precompute.out │ │ │ @@ -31,27 +31,27 @@ │ │ │ o5 = CacheTable{name => P} │ │ │ │ │ │ i6 : C == P │ │ │ │ │ │ o6 = true │ │ │ │ │ │ i7 : time isDistributive C │ │ │ - -- used 1.6391e-05s (cpu); 1.2082e-05s (thread); 0s (gc) │ │ │ + -- used 2.1872e-05s (cpu); 9.621e-06s (thread); 0s (gc) │ │ │ │ │ │ o7 = true │ │ │ │ │ │ i8 : time isDistributive P │ │ │ - -- used 6.46167s (cpu); 4.39193s (thread); 0s (gc) │ │ │ + -- used 8.28384s (cpu); 4.83387s (thread); 0s (gc) │ │ │ │ │ │ o8 = true │ │ │ │ │ │ i9 : C' = dual C; │ │ │ │ │ │ i10 : time isDistributive C' │ │ │ - -- used 6.662e-06s (cpu); 6.161e-06s (thread); 0s (gc) │ │ │ + -- used 8.81e-06s (cpu); 5.05e-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 0.390356s (cpu); 0.248423s (thread); 0s (gc) │ │ │ + -- used 0.541816s (cpu); 0.285728s (thread); 0s (gc) │ │ │ │ │ │ o4 = Partition{9, 2} │ │ │ │ │ │ o4 : Partition │ │ │ │ │ │ i5 : time greeneKleitmanPartition(D, Strategy => "chains") │ │ │ - -- used 9.808e-06s (cpu); 9.527e-06s (thread); 0s (gc) │ │ │ + -- used 2.5451e-05s (cpu); 1.8083e-05s (thread); 0s (gc) │ │ │ │ │ │ o5 = Partition{9, 2} │ │ │ │ │ │ o5 : Partition │ │ │ │ │ │ i6 : greeneKleitmanPartition chain 10 │ │ ├── ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ @@ -107,23 +107,23 @@ │ │ │ │ │ │ o6 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isDistributive C
│ │ │ - -- used 1.6391e-05s (cpu); 1.2082e-05s (thread); 0s (gc)
│ │ │ + -- used 2.1872e-05s (cpu); 9.621e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time isDistributive P
│ │ │ - -- used 6.46167s (cpu); 4.39193s (thread); 0s (gc)
│ │ │ + -- used 8.28384s (cpu); 4.83387s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │ @@ -133,15 +133,15 @@ │ │ │ │ │ │ 
    i9 : C' = dual C;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time isDistributive C'
│ │ │ - -- used 6.662e-06s (cpu); 6.161e-06s (thread); 0s (gc)
│ │ │ + -- used 8.81e-06s (cpu); 5.05e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : peek C'.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,26 +41,26 @@
│ │ │ │  i5 : peek P.cache
│ │ │ │  
│ │ │ │  o5 = CacheTable{name => P}
│ │ │ │  i6 : C == P
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : time isDistributive C
│ │ │ │ - -- used 1.6391e-05s (cpu); 1.2082e-05s (thread); 0s (gc)
│ │ │ │ + -- used 2.1872e-05s (cpu); 9.621e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 6.46167s (cpu); 4.39193s (thread); 0s (gc)
│ │ │ │ + -- used 8.28384s (cpu); 4.83387s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  We also know that the dual of a distributive lattice is again a distributive
│ │ │ │  lattice. Other information is copied when possible.
│ │ │ │  i9 : C' = dual C;
│ │ │ │  i10 : time isDistributive C'
│ │ │ │ - -- used 6.662e-06s (cpu); 6.161e-06s (thread); 0s (gc)
│ │ │ │ + -- used 8.81e-06s (cpu); 5.05e-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
│ │ │ @@ -100,25 +100,25 @@
│ │ │              
│ │ │                
    i3 : D = dominanceLattice 6;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ - -- used 0.390356s (cpu); 0.248423s (thread); 0s (gc)
│ │ │ + -- used 0.541816s (cpu); 0.285728s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 9.808e-06s (cpu); 9.527e-06s (thread); 0s (gc)
│ │ │ + -- used 2.5451e-05s (cpu); 1.8083e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,21 +28,21 @@
│ │ │ │  
│ │ │ │  o2 : Partition
│ │ │ │  The conjugate of $l$ has the same property, but with chains replaced by
│ │ │ │  _a_n_t_i_c_h_a_i_n_s. Because of this, it is often better to count via antichains instead
│ │ │ │  of chains. This can be done by passing "antichains" as the Strategy.
│ │ │ │  i3 : D = dominanceLattice 6;
│ │ │ │  i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ │ - -- used 0.390356s (cpu); 0.248423s (thread); 0s (gc)
│ │ │ │ + -- used 0.541816s (cpu); 0.285728s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o4 : Partition
│ │ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ │ - -- used 9.808e-06s (cpu); 9.527e-06s (thread); 0s (gc)
│ │ │ │ + -- used 2.5451e-05s (cpu); 1.8083e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o5 : Partition
│ │ │ │  The Greene-Kleitman partition of the $n$ _c_h_a_i_n is the partition of $n$ with $1$
│ │ │ │  part.
│ │ │ │  i6 : greeneKleitmanPartition chain 10
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated__Primes.out
│ │ │ @@ -125,24 +125,24 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │ +o20 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      b, a}, {e, d, c, b, a}}
│ │ │ +      c, b, a}, {e, d, b, a}}
│ │ │  
│ │ │  o20 : Set
│ │ │  
│ │ │  i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │ +o21 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      b, a}, {e, d, c, b, a}}
│ │ │ +      c, b, a}, {e, d, 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)
│ │ │ - -- .083304s elapsed
│ │ │ + -- .107234s 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)
│ │ │ - -- .0268711s elapsed
│ │ │ + -- .0221419s 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)
│ │ │ - -- .106461s elapsed
│ │ │ + -- .0583743s 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
│ │ │ - -- .0926228s elapsed
│ │ │ + -- .0659616s 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)
│ │ │ - -- .00845388s elapsed
│ │ │ + -- .00894585s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html
│ │ │ @@ -286,28 +286,28 @@
│ │ │  o19 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │ +o20 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      b, a}, {e, d, c, b, a}}
│ │ │ +      c, b, a}, {e, d, b, a}}
│ │ │  
│ │ │  o20 : Set
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │ +o21 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      b, a}, {e, d, c, b, a}}
│ │ │ +      c, b, a}, {e, d, b, a}}
│ │ │  
│ │ │  o21 : Set
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i22 : assert(M1 === M2)
    │ │ │ ├── html2text {} │ │ │ │ @@ -155,24 +155,24 @@ │ │ │ │ o19 = {ideal (a, e), ideal (a, b, c), ideal (a, b, d), ideal (a, b, c, d), │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)} │ │ │ │ │ │ │ │ o19 : List │ │ │ │ i20 : M1 = set apply(L1, I -> sort flatten entries gens I) │ │ │ │ │ │ │ │ -o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d, │ │ │ │ +o20 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - b, a}, {e, d, c, b, a}} │ │ │ │ + c, b, a}, {e, d, b, a}} │ │ │ │ │ │ │ │ o20 : Set │ │ │ │ i21 : M2 = set apply(L2, I -> sort flatten entries gens I) │ │ │ │ │ │ │ │ -o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d, │ │ │ │ +o21 = set {{e, c, b, a}, {e, d, c, b, a}, {d, b, a}, {e, a}, {c, b, a}, {d, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - b, a}, {e, d, c, b, a}} │ │ │ │ + c, b, a}, {e, d, b, a}} │ │ │ │ │ │ │ │ o21 : Set │ │ │ │ i22 : assert(M1 === M2) │ │ │ │ The method using Ext modules comes from Eisenbud-Huneke-Vasconcelos, Invent. │ │ │ │ Math 110 (1992) 207-235. │ │ │ │ Original author (for ideals): _C_._ _Y_a_c_k_e_l. Updated for modules by J. Chen. │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ @@ -107,41 +107,41 @@ │ │ │ 3 │ │ │ o3 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .083304s elapsed
│ │ │ + -- .107234s 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)
│ │ │ - -- .0268711s elapsed
│ │ │ + -- .0221419s 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)
│ │ │ - -- .106461s elapsed
│ │ │ + -- .0583743s 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) │ │ │ │ - -- .083304s elapsed │ │ │ │ + -- .107234s 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) │ │ │ │ - -- .0268711s elapsed │ │ │ │ + -- .0221419s 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) │ │ │ │ - -- .106461s elapsed │ │ │ │ + -- .0583743s elapsed │ │ │ │ │ │ │ │ o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 0 1 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 0 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .0926228s elapsed
│ │ │ + -- .0659616s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │ │ │ │ │ │ │ @@ -148,15 +148,15 @@ │ │ │ │ │ │ o6 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00845388s elapsed
│ │ │ + -- .00894585s 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 │ │ │ │ - -- .0926228s elapsed │ │ │ │ + -- .0659616s 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) │ │ │ │ - -- .00845388s elapsed │ │ │ │ + -- .00894585s elapsed │ │ │ │ │ │ │ │ 2 3 2 2 8 3 2 2 │ │ │ │ o7 = ideal (h*l - l - 4l*s + h*y, h + l s - h x, s + h + l s - h x) │ │ │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_d_i_c_a_l -- the radical of an ideal │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_iterator_lp__Python__Object_rp.out │ │ │ @@ -10,12 +10,12 @@ │ │ │ │ │ │ o2 = range(0, 3) │ │ │ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ i3 : i = iterator x │ │ │ │ │ │ -o3 = │ │ │ +o3 = │ │ │ │ │ │ o3 : PythonObject of class range_iterator │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_next_lp__Python__Object_rp.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ │ │ │ o2 = range(0, 3) │ │ │ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ i3 : i = iterator x │ │ │ │ │ │ -o3 = │ │ │ +o3 = │ │ │ │ │ │ o3 : PythonObject of class range_iterator │ │ │ │ │ │ i4 : next i │ │ │ │ │ │ o4 = 0 │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_python__Run__Script.out │ │ │ @@ -1,22 +1,22 @@ │ │ │ -- -*- M2-comint -*- hash: 447449196062331972 │ │ │ │ │ │ i1 : pyfile = temporaryFileName() | ".py" │ │ │ │ │ │ -o1 = /tmp/M2-47329-0/0.py │ │ │ +o1 = /tmp/M2-73758-0/0.py │ │ │ │ │ │ i2 : pyfile << "import math" << endl │ │ │ │ │ │ -o2 = /tmp/M2-47329-0/0.py │ │ │ +o2 = /tmp/M2-73758-0/0.py │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : pyfile << "x = math.sin(3.4)" << endl << close │ │ │ │ │ │ -o3 = /tmp/M2-47329-0/0.py │ │ │ +o3 = /tmp/M2-73758-0/0.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 0x7f894b69ccc0> │ │ │ +o9 = at 0x7f825ee5ccc0> │ │ │ │ │ │ o9 : PythonObject of class function │ │ │ │ │ │ i10 : x │ │ │ │ │ │ o10 = 5 │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_iterator_lp__Python__Object_rp.html │ │ │ @@ -90,15 +90,15 @@ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f894b6e57d0>
│ │ │ +o3 = <range_iterator object at 0x7f825eea56b0>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,15 +22,15 @@ │ │ │ │ i2 : x = builtins@@range 3 │ │ │ │ │ │ │ │ o2 = range(0, 3) │ │ │ │ │ │ │ │ o2 : PythonObject of class range │ │ │ │ i3 : i = iterator x │ │ │ │ │ │ │ │ -o3 = │ │ │ │ +o3 = │ │ │ │ │ │ │ │ o3 : PythonObject of class range_iterator │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _n_e_x_t_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- retrieve the next item from a python iterator │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _i_t_e_r_a_t_o_r_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- get iterator of iterable python object │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_next_lp__Python__Object_rp.html │ │ │ @@ -86,15 +86,15 @@ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f894b6d9e90>
│ │ │ +o3 = <range_iterator object at 0x7f825ee99d70>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : next i
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : x = builtins@@range 3
│ │ │ │  
│ │ │ │  o2 = range(0, 3)
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range
│ │ │ │  i3 : i = iterator x
│ │ │ │  
│ │ │ │ -o3 = 
│ │ │ │ +o3 = 
│ │ │ │  
│ │ │ │  o3 : PythonObject of class range_iterator
│ │ │ │  i4 : next i
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  
│ │ │ │  o4 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_python__Run__Script.html
│ │ │ @@ -76,31 +76,31 @@
│ │ │            
    The return value is a Python dictionary containing all the variables defined in the global scope.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-47329-0/0.py
    │ │ │ +o1 = /tmp/M2-73758-0/0.py │ │ │
    │ │ │ 
    i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-47329-0/0.py
│ │ │ +o2 = /tmp/M2-73758-0/0.py
│ │ │  
│ │ │  o2 : File
    │ │ │
    │ │ │ 
    i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-47329-0/0.py
│ │ │ +o3 = /tmp/M2-73758-0/0.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/M2-47329-0/0.py
│ │ │ │ +o1 = /tmp/M2-73758-0/0.py
│ │ │ │  i2 : pyfile << "import math" << endl
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-47329-0/0.py
│ │ │ │ +o2 = /tmp/M2-73758-0/0.py
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-47329-0/0.py
│ │ │ │ +o3 = /tmp/M2-73758-0/0.py
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : get pyfile
│ │ │ │  
│ │ │ │  o4 = import math
│ │ │ │       x = math.sin(3.4)
│ │ │ │  i5 : pythonRunScript oo
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_to__Python.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │  o12 : FunctionClosure
    │ │ │
    │ │ │ 
    i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = <built-in method m2sqrt of PyCapsule object at 0x7f894b6bac50>
│ │ │ +o13 = <built-in method m2sqrt of PyCapsule object at 0x7f825ee7ec00>
│ │ │  
│ │ │  o13 : PythonObject of class builtin_function_or_method
    │ │ │
    │ │ │ 
    i14 : pysqrt 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │            sqrt x)
│ │ │ │  
│ │ │ │  o12 = m2sqrt
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : pysqrt = toPython m2sqrt
│ │ │ │  
│ │ │ │ -o13 = 
│ │ │ │ +o13 = 
│ │ │ │  
│ │ │ │  o13 : PythonObject of class builtin_function_or_method
│ │ │ │  i14 : pysqrt 2
│ │ │ │  calling Macaulay2 code from Python!
│ │ │ │  
│ │ │ │  o14 = 1.4142135623730951
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_use_lp__Python__Context_rp.html
│ │ │ @@ -124,15 +124,15 @@
│ │ │                
    i8 : use ctx
    
│ │ │
    │ │ │ 
    i9 : f
│ │ │  
│ │ │ -o9 = <function <lambda> at 0x7f894b69ccc0>
│ │ │ +o9 = <function <lambda> at 0x7f825ee5ccc0>
│ │ │  
│ │ │  o9 : PythonObject of class function
    │ │ │
    │ │ │ 
    i10 : x
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  
│ │ │ │  o7 = y
│ │ │ │  
│ │ │ │  o7 : Symbol
│ │ │ │  i8 : use ctx
│ │ │ │  i9 : f
│ │ │ │  
│ │ │ │ -o9 =  at 0x7f894b69ccc0>
│ │ │ │ +o9 =  at 0x7f825ee5ccc0>
│ │ │ │  
│ │ │ │  o9 : PythonObject of class function
│ │ │ │  i10 : x
│ │ │ │  
│ │ │ │  o10 = 5
│ │ │ │  
│ │ │ │  o10 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out
│ │ │ @@ -180,15 +180,15 @@
│ │ │  i21 : L = trim groebnerStratum F;
│ │ │  
│ │ │  o21 : Ideal of T
│ │ │  
│ │ │  i22 : assert(dim L == 18)
│ │ │  
│ │ │  i23 : elapsedTime isPrime L
│ │ │ - -- 2.73075s elapsed
│ │ │ + -- 2.49002s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : I = pointsIdeal randomPoints(S, 6)
│ │ │  
│ │ │                               2                              2   2          
│ │ │  o24 = ideal (a*c - 7b*c - 49c  + 40a*d - 42b*d + 12c*d + 28d , b  - 36b*c -
│ │ │ @@ -302,15 +302,15 @@
│ │ │  o38 = true
│ │ │  
│ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │  
│ │ │  o39 : Ideal of T
│ │ │  
│ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 1.74846s elapsed
│ │ │ + -- 1.35766s elapsed
│ │ │  
│ │ │  i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
│ │ │  
│ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ │  
│ │ │ @@ -320,37 +320,37 @@
│ │ │  
│ │ │  i43 : compsL441/dim == {16, 14}
│ │ │  
│ │ │  o43 = true
│ │ │  
│ │ │  i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | -49 11 -25 17 44 -16 -27 40 -34 20 -19 29 41 19 -40 42 -13 5 17 39 31
│ │ │ +o44 = | 32 -41 22 15 22 -46 43 42 -27 -27 -13 10 -24 19 -25 48 31 10 41 49 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      10 45 26 43 49 -32 -29 19 -50 15 18 37 -10 34 -28 |
│ │ │ +      -28 -29 -10 -48 5 15 18 45 19 49 37 -32 34 26 -50 |
│ │ │  
│ │ │                 1       36
│ │ │  o44 : Matrix kk  <-- kk
│ │ │  
│ │ │  i45 : Fa = sub(F, (vars S) | pta)
│ │ │  
│ │ │                2              2                              2               
│ │ │ -o45 = ideal (a  - 40b*c + 20c  + 29a*d - 16b*d - 25c*d - 49d , a*b + 43b*c +
│ │ │ +o45 = ideal (a  - 25b*c - 27c  + 10a*d - 46b*d + 22c*d + 32d , a*b - 48b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │           2                              2   2              2                
│ │ │ -      17c  + 31a*d + 41b*d - 34c*d + 11d , b  + 34b*c - 29c  + 37a*d + 10b*d
│ │ │ +      41c  + 39a*d - 24b*d - 27c*d - 41d , b  + 26b*c + 18c  - 32a*d - 28b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2                   2                              2     2
│ │ │ -      + 42c*d - 27d , a*c + 18b*c + 49c  + 19a*d + 39b*d + 19c*d + 44d , b*c 
│ │ │ +                   2                  2                              2     2
│ │ │ +      + 48c*d + 43d , a*c + 37b*c + 5c  + 45a*d + 49b*d + 19c*d + 22d , b*c 
│ │ │        -----------------------------------------------------------------------
│ │ │                       2         2        2        2      3   3            
│ │ │ -      - 50b*c*d + 45c d - 32a*d  - 13b*d  - 19c*d  + 17d , c  - 28b*c*d +
│ │ │ +      + 19b*c*d - 29c d + 15a*d  + 31b*d  - 13c*d  + 15d , c  - 50b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2         2        2       2      3
│ │ │ -      15c d - 10a*d  + 26b*d  + 5c*d  + 40d )
│ │ │ +         2         2        2        2      3
│ │ │ +      49c d + 34a*d  - 10b*d  + 10c*d  + 42d )
│ │ │  
│ │ │  o45 : Ideal of S
│ │ │  
│ │ │  i46 : betti res Fa
│ │ │  
│ │ │               0 1 2 3
│ │ │  o46 = total: 1 6 8 3
│ │ │ @@ -358,81 +358,83 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o46 : BettiTally
│ │ │  
│ │ │  i47 : netList decompose Fa -- this one is 5 points on a plane, and another point
│ │ │  
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -o47 = |ideal (c + 19d, b - 10d, a + 6d)                                                                                                                                   |
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                            2              2                      2   3                2        2        2      3     2                2         2        2      3 |
│ │ │ -      |ideal (a + 18b + 49c - 3d, b  + 34b*c - 29c  - 50b*d + 47c*d - 17d , c  - 28b*c*d + 15c d + 4b*d  - 10c*d  + 10d , b*c  - 50b*c*d + 45c d - 43b*d  + 34c*d  + 22d )|
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +o47 = |ideal (c + 45d, b - 34d, a - 35d)                                                             |
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +      |ideal (c + 18d, b - 12d, a + 47d)                                                             |
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │ +      |                                   2                      2   2      2                      2 |
│ │ │ +      |ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c  - 14b*d + 27c*d - 38d )|
│ │ │ +      +----------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                     2  
│ │ │ -o48 = {ideal (c + 19d, b - 10d, a + 6d), ideal (a + 18b + 49c - 3d, b  +
│ │ │ +                                                                            
│ │ │ +o48 = {ideal (c + 45d, b - 34d, a - 35d), ideal (c + 18d, b - 12d, a + 47d),
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                      2   3                2        2  
│ │ │ -      34b*c - 29c  - 50b*d + 47c*d - 17d , c  - 28b*c*d + 15c d + 4b*d  -
│ │ │ +                                         2                      2   2      2
│ │ │ +      ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2      3     2                2         2        2      3
│ │ │ -      10c*d  + 10d , b*c  - 50b*c*d + 45c d - 43b*d  + 34c*d  + 22d )}
│ │ │ +                           2
│ │ │ +      - 14b*d + 27c*d - 38d )}
│ │ │  
│ │ │  o48 : List
│ │ │  
│ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a + 18b + 49c - 3d
│ │ │ +o49 = c + 18d
│ │ │  
│ │ │  o49 : S
│ │ │  
│ │ │  i50 : CFa/degree
│ │ │  
│ │ │ -o50 = {1, 5}
│ │ │ +o50 = {1, 1, 4}
│ │ │  
│ │ │  o50 : List
│ │ │  
│ │ │  i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │  
│ │ │ -o51 = {false, true}
│ │ │ +o51 = {false, true, false}
│ │ │  
│ │ │  o51 : List
│ │ │  
│ │ │  i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of the 5 points
│ │ │  
│ │ │ -o52 = 5
│ │ │ +o52 = 1
│ │ │  
│ │ │  i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │  
│ │ │ -o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │ +o53 = | -31 -3 -29 -17 -21 5 -32 33 -24 2 26 -26 -45 -4 16 -22 2 -37 16 -23
│ │ │        -----------------------------------------------------------------------
│ │ │ -      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │ +      -42 19 -29 21 7 2 17 9 -15 -9 -47 -13 0 38 47 21 |
│ │ │  
│ │ │                 1       36
│ │ │  o53 : Matrix kk  <-- kk
│ │ │  
│ │ │  i54 : Fb = sub(F, (vars S) | ptb)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │ +              2             2                             2              
│ │ │ +o54 = ideal (a  + 16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , a*b + 7b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2   2              2                  
│ │ │ -      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*d -
│ │ │ +         2                             2   2             2                  
│ │ │ +      16c  - 42a*d - 45b*d - 24c*d - 3d , b  + 47b*c + 9c  + 19b*d - 22c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                   2                              2     2          
│ │ │ -      32d , a*c + 47b*c + 21c  - 20a*d - 23b*d - 13c*d - 19d , b*c  + 21b*c*d
│ │ │ +         2                  2                             2     2           
│ │ │ +      32d , a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b*c  - 9b*c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2         2       2        2     3   3               2        2  
│ │ │ -      - 9c d + 38a*d  - 4b*d  - 13c*d  + 9d , c  + 9b*c*d - 29c d + 2a*d  -
│ │ │ +         2         2       2        2      3   3                2         2  
│ │ │ +      29c d + 17a*d  + 2b*d  + 26c*d  - 17d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2       2      3
│ │ │ -      47b*d  - 4c*d  + 27d )
│ │ │ +           2        2      3
│ │ │ +      21b*d  - 37c*d  + 33d )
│ │ │  
│ │ │  o54 : Ideal of S
│ │ │  
│ │ │  i55 : betti res Fb
│ │ │  
│ │ │               0 1 2 3
│ │ │  o55 = total: 1 6 8 3
│ │ │ @@ -440,80 +442,84 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o55 : BettiTally
│ │ │  
│ │ │  i56 : netList decompose Fb --
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                      2              2                                                                                                                                                               |
│ │ │ -o56 = |ideal (b - 50c - 43d, a + 15c - 46d, c  + 12c*d - 37d )                                                                                                                                                              |
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                             2                                 2                                   2   2                            2                                   2   2                             2 |
│ │ │ -      |ideal (c  + 46a*d - 39b*d + 2c*d - 24d , b*c - 9a*d + 16b*d + 2c*d + 27d , a*c + 43a*d + 44b*d - 48c*d + 24d , b  - 4a*d - 40b*d - 9c*d - 39d , a*b + 16a*d + 26b*d + 37c*d - 24d , a  - 25a*d + 47b*d + 34c*d + 8d )|
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +o56 = |ideal (c - 17d, b - 26d, a - 49d)                              |
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +      |                                     3      2         2      3 |
│ │ │ +      |ideal (b + 39c - 21d, a + 4c - 27d, c  + 43c d + 39c*d  - 15d )|
│ │ │ +      +---------------------------------------------------------------+
│ │ │ +      |                                    2             2            |
│ │ │ +      |ideal (b + 8c + 40d, a + 5c - 33d, c  + 4c*d + 45d )           |
│ │ │ +      +---------------------------------------------------------------+
│ │ │  
│ │ │  i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -o57 = ++
│ │ │ -      ++
│ │ │ +      +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                       2                             2   2             2                      2                  2                             2   2             2                             2   3                2         2        2        2      3     2               2         2       2        2      3 |
│ │ │ +o57 = |ideal (a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b  + 47b*c + 9c  + 19b*d - 22c*d - 32d , a*b + 7b*c + 16c  - 42a*d - 45b*d - 24c*d - 3d , a  + 16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , c  + 21b*c*d - 47c d + 38a*d  + 21b*d  - 37c*d  + 33d , b*c  - 9b*c*d - 29c d + 17a*d  + 2b*d  + 26c*d  - 17d )|
│ │ │ +      +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │  
│ │ │ -o58 = | 44 15 26 -28 -4 33 49 -19 -6 -18 -37 -34 18 -42 -23 23 38 -40 27 26
│ │ │ +o58 = | 49 0 -30 -36 -1 0 -9 17 37 29 34 13 19 8 -10 -47 21 -24 -44 42 9 46
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -39 -24 -21 -13 38 39 -33 15 8 -18 18 -22 -2 -29 -23 46 |
│ │ │ +      15 -29 35 -40 18 -22 -21 -42 39 -2 -33 -23 -13 -18 |
│ │ │  
│ │ │                 1       36
│ │ │  o58 : Matrix kk  <-- kk
│ │ │  
│ │ │  i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │  
│ │ │ -o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │ +o59 = | -5 -40 -6 3 -28 -8 -25 15 15 29 26 -37 11 -14 31 14 1 -50 43 37 5 50
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │ +      10 3 -3 -35 -18 32 -7 -15 33 46 0 21 -49 3 |
│ │ │  
│ │ │                 1       36
│ │ │  o59 : Matrix kk  <-- kk
│ │ │  
│ │ │  i60 : I0 = sub(sub(F, (vars ring F) | sub(pt0, ring F)), S)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o60 = ideal (a  - 23b*c - 18c  - 34a*d + 33b*d + 26c*d + 44d , a*b + 38b*c +
│ │ │ +              2              2                      2                   2  
│ │ │ +o60 = ideal (a  - 10b*c + 29c  + 13a*d - 30c*d + 49d , a*b + 35b*c - 44c  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                 
│ │ │ -      27c  - 39a*d + 18b*d - 6c*d + 15d , b  - 23b*c + 15c  - 2a*d - 24b*d +
│ │ │ +                             2              2                             2 
│ │ │ +      9a*d + 19b*d + 37c*d, b  - 13b*c - 22c  - 33a*d + 46b*d - 47c*d - 9d ,
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                            2     2  
│ │ │ -      23c*d + 49d , a*c - 22b*c + 39c  + 8a*d + 26b*d - 42c*d - 4d , b*c  -
│ │ │ +                      2                           2     2                2   
│ │ │ +      a*c - 2b*c - 40c  - 21a*d + 42b*d + 8c*d - d , b*c  - 42b*c*d + 15c d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      18b*c*d - 21c d - 33a*d  + 38b*d  - 37c*d  - 28d , c  + 46b*c*d + 18c d
│ │ │ +           2        2        2      3   3                2         2        2
│ │ │ +      18a*d  + 21b*d  + 34c*d  - 36d , c  - 18b*c*d + 39c d - 23a*d  - 29b*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 29a*d  - 13b*d  - 40c*d  - 19d )
│ │ │ +             2      3
│ │ │ +      - 24c*d  + 17d )
│ │ │  
│ │ │  o60 : Ideal of S
│ │ │  
│ │ │  i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │  
│ │ │ -              2            2                             2               
│ │ │ -o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │ +              2              2                           2                  2
│ │ │ +o61 = ideal (a  + 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , a*b - 3b*c + 43c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2   2              2                
│ │ │ -      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │ +                                  2   2              2                  
│ │ │ +      + 5a*d + 11b*d + 15c*d - 40d , b  - 49b*c + 32c  + 50b*d + 14c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2                  2                             2     2         
│ │ │ -      - 38d , a*c - 49b*c + 3c  - 9a*d + 37b*d + 46c*d + 50d , b*c  + 3b*c*d
│ │ │ +         2                   2                             2     2          
│ │ │ +      25d , a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b*c  - 15b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2         2       2       2      3   3                2         2
│ │ │ -      - 15c d + 21a*d  - 2b*d  - 2c*d  - 44d , c  + 32b*c*d + 10c d - 35a*d 
│ │ │ +           2         2      2        2     3   3               2         2  
│ │ │ +      + 10c d - 18a*d  + b*d  + 26c*d  + 3d , c  + 3b*c*d + 33c d + 21a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2      3
│ │ │ -      + 33b*d  - 14c*d  + 33d )
│ │ │ +          2        2      3
│ │ │ +      3b*d  - 50c*d  + 15d )
│ │ │  
│ │ │  o61 : Ideal of S
│ │ │  
│ │ │  i62 : betti res I0
│ │ │  
│ │ │               0 1 2 3
│ │ │  o62 = total: 1 6 8 3
│ │ │ @@ -531,34 +537,33 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o63 : BettiTally
│ │ │  
│ │ │  i64 : netList decompose I0
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c + 8d, b + 5d, a - 25d)                                                                                                                             |
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2              2              2   3                2         2        2      3     2                2         2        2     3 |
│ │ │ -      |ideal (a - 22b + 39c + 50d, b  - 23b*c + 15c  + 33b*d + 48d , c  + 46b*c*d + 18c d - 45b*d  - 20c*d  + 17d , b*c  - 18b*c*d - 21c d + 19b*d  + 38c*d  + 6d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +---------------------------------------------------+
│ │ │ +o64 = |ideal (c - 21d, b - 26d, a - 50d)                  |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |ideal (c - 49d, b - 33d, a - 30d)                  |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |ideal (c + 41d, b + 33d, a - 35d)                  |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |ideal (c + d, b + 40d, a - 5d)                     |
│ │ │ +      +---------------------------------------------------+
│ │ │ +      |                                    2            2 |
│ │ │ +      |ideal (b + c - 47d, a - 38c - 17d, c  + 4c*d - 8d )|
│ │ │ +      +---------------------------------------------------+
│ │ │  
│ │ │  i65 : netList decompose I1
│ │ │  
│ │ │ -      +---------------------------------+
│ │ │ -o65 = |ideal (c - 9d, b + 15d, a + 27d) |
│ │ │ -      +---------------------------------+
│ │ │ -      |ideal (c + 48d, b + 11d, a - 37d)|
│ │ │ -      +---------------------------------+
│ │ │ -      |ideal (c + 29d, b + 46d, a + 18d)|
│ │ │ -      +---------------------------------+
│ │ │ -      |ideal (c + 24d, b + 46d, a + 33d)|
│ │ │ -      +---------------------------------+
│ │ │ -      |ideal (c + 22d, b + 38d, a - 50d)|
│ │ │ -      +---------------------------------+
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                        2                             2   2              2                      2                  2                             2   2              2                           2   3               2         2       2        2      3     2                2         2      2        2     3 |
│ │ │ +o65 = |ideal (a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b  - 49b*c + 32c  + 50b*d + 14c*d - 25d , a*b - 3b*c + 43c  + 5a*d + 11b*d + 15c*d - 40d , a  + 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , c  + 3b*c*d + 33c d + 21a*d  + 3b*d  - 50c*d  + 15d , b*c  - 15b*c*d + 10c d - 18a*d  + b*d  + 26c*d  + 3d )|
│ │ │ +      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │  
│ │ │  o66 : Ideal of T
│ │ │  
│ │ │  i67 : C = res(I, FastNonminimal => true)
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html
│ │ │ @@ -344,15 +344,15 @@
│ │ │
    		│ │ │ 
    i22 : assert(dim L == 18)
    │ │ │
    │ │ │ 
    i23 : elapsedTime isPrime L
│ │ │ - -- 2.73075s elapsed
│ │ │ + -- 2.49002s elapsed
│ │ │  
│ │ │  o23 = true
    │ │ │
    │ │ │
    │ │ │

    The Schreyer resolution and minimal Betti numbers

    │ │ │ @@ -556,15 +556,15 @@ │ │ │ │ │ │ o39 : Ideal of T     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 1.74846s elapsed
    │ │ │ + -- 1.35766s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
    │ │ │ @@ -591,40 +591,40 @@ │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │  
│ │ │ -o44 = | -49 11 -25 17 44 -16 -27 40 -34 20 -19 29 41 19 -40 42 -13 5 17 39 31
│ │ │ +o44 = | 32 -41 22 15 22 -46 43 42 -27 -27 -13 10 -24 19 -25 48 31 10 41 49 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      10 45 26 43 49 -32 -29 19 -50 15 18 37 -10 34 -28 |
│ │ │ +      -28 -29 -10 -48 5 15 18 45 19 49 37 -32 34 26 -50 |
│ │ │  
│ │ │                 1       36
│ │ │  o44 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i45 : Fa = sub(F, (vars S) | pta)
│ │ │  
│ │ │                2              2                              2               
│ │ │ -o45 = ideal (a  - 40b*c + 20c  + 29a*d - 16b*d - 25c*d - 49d , a*b + 43b*c +
│ │ │ +o45 = ideal (a  - 25b*c - 27c  + 10a*d - 46b*d + 22c*d + 32d , a*b - 48b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │           2                              2   2              2                
│ │ │ -      17c  + 31a*d + 41b*d - 34c*d + 11d , b  + 34b*c - 29c  + 37a*d + 10b*d
│ │ │ +      41c  + 39a*d - 24b*d - 27c*d - 41d , b  + 26b*c + 18c  - 32a*d - 28b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2                   2                              2     2
│ │ │ -      + 42c*d - 27d , a*c + 18b*c + 49c  + 19a*d + 39b*d + 19c*d + 44d , b*c 
│ │ │ +                   2                  2                              2     2
│ │ │ +      + 48c*d + 43d , a*c + 37b*c + 5c  + 45a*d + 49b*d + 19c*d + 22d , b*c 
│ │ │        -----------------------------------------------------------------------
│ │ │                       2         2        2        2      3   3            
│ │ │ -      - 50b*c*d + 45c d - 32a*d  - 13b*d  - 19c*d  + 17d , c  - 28b*c*d +
│ │ │ +      + 19b*c*d - 29c d + 15a*d  + 31b*d  - 13c*d  + 15d , c  - 50b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2         2        2       2      3
│ │ │ -      15c d - 10a*d  + 26b*d  + 5c*d  + 40d )
│ │ │ +         2         2        2        2      3
│ │ │ +      49c d + 34a*d  - 10b*d  + 10c*d  + 42d )
│ │ │  
│ │ │  o45 : Ideal of S
    │ │ │
    │ │ │ 
    i46 : betti res Fa
│ │ │ @@ -638,104 +638,106 @@
│ │ │  o46 : BettiTally
    │ │ │
    │ │ │ 
    i47 : netList decompose Fa -- this one is 5 points on a plane, and another point
│ │ │  
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -o47 = |ideal (c + 19d, b - 10d, a + 6d)                                                                                                                                   |
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                            2              2                      2   3                2        2        2      3     2                2         2        2      3 |
│ │ │ -      |ideal (a + 18b + 49c - 3d, b  + 34b*c - 29c  - 50b*d + 47c*d - 17d , c  - 28b*c*d + 15c d + 4b*d  - 10c*d  + 10d , b*c  - 50b*c*d + 45c d - 43b*d  + 34c*d  + 22d )|
│ │ │ -      +-------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    │ │ │ + +----------------------------------------------------------------------------------------------+ │ │ │ +o47 = |ideal (c + 45d, b - 34d, a - 35d) | │ │ │ + +----------------------------------------------------------------------------------------------+ │ │ │ + |ideal (c + 18d, b - 12d, a + 47d) | │ │ │ + +----------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a + 37b + 5c - 4d, b*c - 13c + 45b*d - 28c*d - 29d , b - 48c - 14b*d + 27c*d - 38d )| │ │ │ + +----------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ 
    i48 : CFa = minimalPrimes Fa
│ │ │  
│ │ │ -                                                                     2  
│ │ │ -o48 = {ideal (c + 19d, b - 10d, a + 6d), ideal (a + 18b + 49c - 3d, b  +
│ │ │ +                                                                            
│ │ │ +o48 = {ideal (c + 45d, b - 34d, a - 35d), ideal (c + 18d, b - 12d, a + 47d),
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                      2   3                2        2  
│ │ │ -      34b*c - 29c  - 50b*d + 47c*d - 17d , c  - 28b*c*d + 15c d + 4b*d  -
│ │ │ +                                         2                      2   2      2
│ │ │ +      ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2      3     2                2         2        2      3
│ │ │ -      10c*d  + 10d , b*c  - 50b*c*d + 45c d - 43b*d  + 34c*d  + 22d )}
│ │ │ +                           2
│ │ │ +      - 14b*d + 27c*d - 38d )}
│ │ │  
│ │ │  o48 : List
    │ │ │
    │ │ │ 
    i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │  
│ │ │ -o49 = a + 18b + 49c - 3d
│ │ │ +o49 = c + 18d
│ │ │  
│ │ │  o49 : S
    │ │ │
    │ │ │ 
    i50 : CFa/degree
│ │ │  
│ │ │ -o50 = {1, 5}
│ │ │ +o50 = {1, 1, 4}
│ │ │  
│ │ │  o50 : List
    │ │ │
    │ │ │ 
    i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │  
│ │ │ -o51 = {false, true}
│ │ │ +o51 = {false, true, false}
│ │ │  
│ │ │  o51 : List
    │ │ │
    │ │ │ 
    i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of the 5 points
│ │ │  
│ │ │ -o52 = 5
    │ │ │ +o52 = 1 │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │  
│ │ │ -o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │ +o53 = | -31 -3 -29 -17 -21 5 -32 33 -24 2 26 -26 -45 -4 16 -22 2 -37 16 -23
│ │ │        -----------------------------------------------------------------------
│ │ │ -      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │ +      -42 19 -29 21 7 2 17 9 -15 -9 -47 -13 0 38 47 21 |
│ │ │  
│ │ │                 1       36
│ │ │  o53 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i54 : Fb = sub(F, (vars S) | ptb)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │ +              2             2                             2              
│ │ │ +o54 = ideal (a  + 16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , a*b + 7b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2   2              2                  
│ │ │ -      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*d -
│ │ │ +         2                             2   2             2                  
│ │ │ +      16c  - 42a*d - 45b*d - 24c*d - 3d , b  + 47b*c + 9c  + 19b*d - 22c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                   2                              2     2          
│ │ │ -      32d , a*c + 47b*c + 21c  - 20a*d - 23b*d - 13c*d - 19d , b*c  + 21b*c*d
│ │ │ +         2                  2                             2     2           
│ │ │ +      32d , a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b*c  - 9b*c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -          2         2       2        2     3   3               2        2  
│ │ │ -      - 9c d + 38a*d  - 4b*d  - 13c*d  + 9d , c  + 9b*c*d - 29c d + 2a*d  -
│ │ │ +         2         2       2        2      3   3                2         2  
│ │ │ +      29c d + 17a*d  + 2b*d  + 26c*d  - 17d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2       2      3
│ │ │ -      47b*d  - 4c*d  + 27d )
│ │ │ +           2        2      3
│ │ │ +      21b*d  - 37c*d  + 33d )
│ │ │  
│ │ │  o54 : Ideal of S
    │ │ │
    │ │ │ 
    i55 : betti res Fb
│ │ │ @@ -749,102 +751,106 @@
│ │ │  o55 : BettiTally
    │ │ │
    │ │ │ 
    i56 : netList decompose Fb --
│ │ │  
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                                      2              2                                                                                                                                                               |
│ │ │ -o56 = |ideal (b - 50c - 43d, a + 15c - 46d, c  + 12c*d - 37d )                                                                                                                                                              |
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |        2                             2                                 2                                   2   2                            2                                   2   2                             2 |
│ │ │ -      |ideal (c  + 46a*d - 39b*d + 2c*d - 24d , b*c - 9a*d + 16b*d + 2c*d + 27d , a*c + 43a*d + 44b*d - 48c*d + 24d , b  - 4a*d - 40b*d - 9c*d - 39d , a*b + 16a*d + 26b*d + 37c*d - 24d , a  - 25a*d + 47b*d + 34c*d + 8d )|
│ │ │ -      +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    │ │ │ + +---------------------------------------------------------------+ │ │ │ +o56 = |ideal (c - 17d, b - 26d, a - 49d) | │ │ │ + +---------------------------------------------------------------+ │ │ │ + | 3 2 2 3 | │ │ │ + |ideal (b + 39c - 21d, a + 4c - 27d, c + 43c d + 39c*d - 15d )| │ │ │ + +---------------------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b + 8c + 40d, a + 5c - 33d, c + 4c*d + 45d ) | │ │ │ + +---------------------------------------------------------------+ │ │ │
    │ │ │ 
    i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │  
│ │ │ -o57 = ++
│ │ │ -      ++
    │ │ │ + +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 | │ │ │ +o57 = |ideal (a*c - 13b*c + 2c - 15a*d - 23b*d - 4c*d - 21d , b + 47b*c + 9c + 19b*d - 22c*d - 32d , a*b + 7b*c + 16c - 42a*d - 45b*d - 24c*d - 3d , a + 16b*c + 2c - 26a*d + 5b*d - 29c*d - 31d , c + 21b*c*d - 47c d + 38a*d + 21b*d - 37c*d + 33d , b*c - 9b*c*d - 29c d + 17a*d + 2b*d + 26c*d - 17d )| │ │ │ + +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │  
│ │ │ -o58 = | 44 15 26 -28 -4 33 49 -19 -6 -18 -37 -34 18 -42 -23 23 38 -40 27 26
│ │ │ +o58 = | 49 0 -30 -36 -1 0 -9 17 37 29 34 13 19 8 -10 -47 21 -24 -44 42 9 46
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -39 -24 -21 -13 38 39 -33 15 8 -18 18 -22 -2 -29 -23 46 |
│ │ │ +      15 -29 35 -40 18 -22 -21 -42 39 -2 -33 -23 -13 -18 |
│ │ │  
│ │ │                 1       36
│ │ │  o58 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │  
│ │ │ -o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │ +o59 = | -5 -40 -6 3 -28 -8 -25 15 15 29 26 -37 11 -14 31 14 1 -50 43 37 5 50
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │ +      10 3 -3 -35 -18 32 -7 -15 33 46 0 21 -49 3 |
│ │ │  
│ │ │                 1       36
│ │ │  o59 : Matrix kk  <-- kk
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i60 : I0 = sub(sub(F, (vars ring F) | sub(pt0, ring F)), S)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o60 = ideal (a  - 23b*c - 18c  - 34a*d + 33b*d + 26c*d + 44d , a*b + 38b*c +
│ │ │ +              2              2                      2                   2  
│ │ │ +o60 = ideal (a  - 10b*c + 29c  + 13a*d - 30c*d + 49d , a*b + 35b*c - 44c  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                 
│ │ │ -      27c  - 39a*d + 18b*d - 6c*d + 15d , b  - 23b*c + 15c  - 2a*d - 24b*d +
│ │ │ +                             2              2                             2 
│ │ │ +      9a*d + 19b*d + 37c*d, b  - 13b*c - 22c  - 33a*d + 46b*d - 47c*d - 9d ,
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                            2     2  
│ │ │ -      23c*d + 49d , a*c - 22b*c + 39c  + 8a*d + 26b*d - 42c*d - 4d , b*c  -
│ │ │ +                      2                           2     2                2   
│ │ │ +      a*c - 2b*c - 40c  - 21a*d + 42b*d + 8c*d - d , b*c  - 42b*c*d + 15c d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      18b*c*d - 21c d - 33a*d  + 38b*d  - 37c*d  - 28d , c  + 46b*c*d + 18c d
│ │ │ +           2        2        2      3   3                2         2        2
│ │ │ +      18a*d  + 21b*d  + 34c*d  - 36d , c  - 18b*c*d + 39c d - 23a*d  - 29b*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 29a*d  - 13b*d  - 40c*d  - 19d )
│ │ │ +             2      3
│ │ │ +      - 24c*d  + 17d )
│ │ │  
│ │ │  o60 : Ideal of S
    │ │ │
    │ │ │ 
    i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │  
│ │ │ -              2            2                             2               
│ │ │ -o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │ +              2              2                           2                  2
│ │ │ +o61 = ideal (a  + 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , a*b - 3b*c + 43c 
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2   2              2                
│ │ │ -      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │ +                                  2   2              2                  
│ │ │ +      + 5a*d + 11b*d + 15c*d - 40d , b  - 49b*c + 32c  + 50b*d + 14c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2                  2                             2     2         
│ │ │ -      - 38d , a*c - 49b*c + 3c  - 9a*d + 37b*d + 46c*d + 50d , b*c  + 3b*c*d
│ │ │ +         2                   2                             2     2          
│ │ │ +      25d , a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b*c  - 15b*c*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -           2         2       2       2      3   3                2         2
│ │ │ -      - 15c d + 21a*d  - 2b*d  - 2c*d  - 44d , c  + 32b*c*d + 10c d - 35a*d 
│ │ │ +           2         2      2        2     3   3               2         2  
│ │ │ +      + 10c d - 18a*d  + b*d  + 26c*d  + 3d , c  + 3b*c*d + 33c d + 21a*d  +
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2      3
│ │ │ -      + 33b*d  - 14c*d  + 33d )
│ │ │ +          2        2      3
│ │ │ +      3b*d  - 50c*d  + 15d )
│ │ │  
│ │ │  o61 : Ideal of S
    │ │ │
    │ │ │ 
    i62 : betti res I0
│ │ │ @@ -873,37 +879,36 @@
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i64 : netList decompose I0
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -o64 = |ideal (c + 8d, b + 5d, a - 25d)                                                                                                                             |
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                             2              2              2   3                2         2        2      3     2                2         2        2     3 |
│ │ │ -      |ideal (a - 22b + 39c + 50d, b  - 23b*c + 15c  + 33b*d + 48d , c  + 46b*c*d + 18c d - 45b*d  - 20c*d  + 17d , b*c  - 18b*c*d - 21c d + 19b*d  + 38c*d  + 6d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------+
    │ │ │ + +---------------------------------------------------+ │ │ │ +o64 = |ideal (c - 21d, b - 26d, a - 50d) | │ │ │ + +---------------------------------------------------+ │ │ │ + |ideal (c - 49d, b - 33d, a - 30d) | │ │ │ + +---------------------------------------------------+ │ │ │ + |ideal (c + 41d, b + 33d, a - 35d) | │ │ │ + +---------------------------------------------------+ │ │ │ + |ideal (c + d, b + 40d, a - 5d) | │ │ │ + +---------------------------------------------------+ │ │ │ + | 2 2 | │ │ │ + |ideal (b + c - 47d, a - 38c - 17d, c + 4c*d - 8d )| │ │ │ + +---------------------------------------------------+ │ │ │
    │ │ │ 
    i65 : netList decompose I1
│ │ │  
│ │ │ -      +---------------------------------+
│ │ │ -o65 = |ideal (c - 9d, b + 15d, a + 27d) |
│ │ │ -      +---------------------------------+
│ │ │ -      |ideal (c + 48d, b + 11d, a - 37d)|
│ │ │ -      +---------------------------------+
│ │ │ -      |ideal (c + 29d, b + 46d, a + 18d)|
│ │ │ -      +---------------------------------+
│ │ │ -      |ideal (c + 24d, b + 46d, a + 33d)|
│ │ │ -      +---------------------------------+
│ │ │ -      |ideal (c + 22d, b + 38d, a - 50d)|
│ │ │ -      +---------------------------------+
    │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 | │ │ │ +o65 = |ideal (a*c + 46b*c - 35c - 7a*d + 37b*d - 14c*d - 28d , b - 49b*c + 32c + 50b*d + 14c*d - 25d , a*b - 3b*c + 43c + 5a*d + 11b*d + 15c*d - 40d , a + 31b*c + 29c - 37a*d - 8b*d - 6c*d - 5d , c + 3b*c*d + 33c d + 21a*d + 3b*d - 50c*d + 15d , b*c - 15b*c*d + 10c d - 18a*d + b*d + 26c*d + 3d )| │ │ │ + +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │ ├── html2text {}
│ │ │ │ @@ -251,15 +251,15 @@
│ │ │ │        |      31         33        32       34        35        36    |
│ │ │ │        +--------------------------------------------------------------+
│ │ │ │  i21 : L = trim groebnerStratum F;
│ │ │ │  
│ │ │ │  o21 : Ideal of T
│ │ │ │  i22 : assert(dim L == 18)
│ │ │ │  i23 : elapsedTime isPrime L
│ │ │ │ - -- 2.73075s elapsed
│ │ │ │ + -- 2.49002s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  ********** TThhee SScchhrreeyyeerr rreessoolluuttiioonn aanndd mmiinniimmaall BBeettttii nnuummbbeerrss **********
│ │ │ │  Schreyer's construction of a nonminimal free resolution starts with a Groebner
│ │ │ │  basis. First, one constructs the SScchhrreeyyeerr ffrraammee (see La Scala, Stillman). This
│ │ │ │  is determined solely from the initial ideal $J$ and its minimal generators (but
│ │ │ │  depends on some choices of ordering, but otherwise is combinatorial). This
│ │ │ │ @@ -415,15 +415,15 @@
│ │ │ │  We now compute the locus in $V(L)$ where the Betti diagram has no cancellation.
│ │ │ │  This is a closed subscheme of $V(L)$, which is a closed subscheme of the
│ │ │ │  Hilbert scheme. Notice that there are two components.
│ │ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │ │  
│ │ │ │  o39 : Ideal of T
│ │ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ │ - -- 1.74846s elapsed
│ │ │ │ + -- 1.35766s elapsed
│ │ │ │  i41 : #compsL441
│ │ │ │  
│ │ │ │  o41 = 2
│ │ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ │ │  
│ │ │ │  o42 = {16, 14}
│ │ │ │  
│ │ │ │ @@ -431,36 +431,36 @@
│ │ │ │  i43 : compsL441/dim == {16, 14}
│ │ │ │  
│ │ │ │  o43 = true
│ │ │ │  Both components are rational, and here are random points, one on each
│ │ │ │  component:
│ │ │ │  i44 : pta = randomPointOnRationalVariety compsL441_0
│ │ │ │  
│ │ │ │ -o44 = | -49 11 -25 17 44 -16 -27 40 -34 20 -19 29 41 19 -40 42 -13 5 17 39 31
│ │ │ │ +o44 = | 32 -41 22 15 22 -46 43 42 -27 -27 -13 10 -24 19 -25 48 31 10 41 49 39
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      10 45 26 43 49 -32 -29 19 -50 15 18 37 -10 34 -28 |
│ │ │ │ +      -28 -29 -10 -48 5 15 18 45 19 49 37 -32 34 26 -50 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o44 : Matrix kk  <-- kk
│ │ │ │  i45 : Fa = sub(F, (vars S) | pta)
│ │ │ │  
│ │ │ │                2              2                              2
│ │ │ │ -o45 = ideal (a  - 40b*c + 20c  + 29a*d - 16b*d - 25c*d - 49d , a*b + 43b*c +
│ │ │ │ +o45 = ideal (a  - 25b*c - 27c  + 10a*d - 46b*d + 22c*d + 32d , a*b - 48b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │           2                              2   2              2
│ │ │ │ -      17c  + 31a*d + 41b*d - 34c*d + 11d , b  + 34b*c - 29c  + 37a*d + 10b*d
│ │ │ │ +      41c  + 39a*d - 24b*d - 27c*d - 41d , b  + 26b*c + 18c  - 32a*d - 28b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2                   2                              2     2
│ │ │ │ -      + 42c*d - 27d , a*c + 18b*c + 49c  + 19a*d + 39b*d + 19c*d + 44d , b*c
│ │ │ │ +                   2                  2                              2     2
│ │ │ │ +      + 48c*d + 43d , a*c + 37b*c + 5c  + 45a*d + 49b*d + 19c*d + 22d , b*c
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                       2         2        2        2      3   3
│ │ │ │ -      - 50b*c*d + 45c d - 32a*d  - 13b*d  - 19c*d  + 17d , c  - 28b*c*d +
│ │ │ │ +      + 19b*c*d - 29c d + 15a*d  + 31b*d  - 13c*d  + 15d , c  - 50b*c*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2         2        2       2      3
│ │ │ │ -      15c d - 10a*d  + 26b*d  + 5c*d  + 40d )
│ │ │ │ +         2         2        2        2      3
│ │ │ │ +      49c d + 34a*d  - 10b*d  + 10c*d  + 42d )
│ │ │ │  
│ │ │ │  o45 : Ideal of S
│ │ │ │  i46 : betti res Fa
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o46 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │ @@ -468,172 +468,177 @@
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o46 : BettiTally
│ │ │ │  i47 : netList decompose Fa -- this one is 5 points on a plane, and another
│ │ │ │  point
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ -------------+
│ │ │ │ -o47 = |ideal (c + 19d, b - 10d, a + 6d)
│ │ │ │ +----------------------+
│ │ │ │ +o47 = |ideal (c + 45d, b - 34d, a - 35d)
│ │ │ │  |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ -------------+
│ │ │ │ -      |                            2              2                      2   3
│ │ │ │ -2        2        2      3     2                2         2        2      3 |
│ │ │ │ -      |ideal (a + 18b + 49c - 3d, b  + 34b*c - 29c  - 50b*d + 47c*d - 17d , c
│ │ │ │ -- 28b*c*d + 15c d + 4b*d  - 10c*d  + 10d , b*c  - 50b*c*d + 45c d - 43b*d  +
│ │ │ │ -34c*d  + 22d )|
│ │ │ │ +----------------------+
│ │ │ │ +      |ideal (c + 18d, b - 12d, a + 47d)
│ │ │ │ +|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------------------------------------------------------------------------
│ │ │ │ -------------+
│ │ │ │ +----------------------+
│ │ │ │ +      |                                   2                      2   2      2
│ │ │ │ +2 |
│ │ │ │ +      |ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c  -
│ │ │ │ +14b*d + 27c*d - 38d )|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +----------------------+
│ │ │ │  i48 : CFa = minimalPrimes Fa
│ │ │ │  
│ │ │ │ -                                                                     2
│ │ │ │ -o48 = {ideal (c + 19d, b - 10d, a + 6d), ideal (a + 18b + 49c - 3d, b  +
│ │ │ │ +
│ │ │ │ +o48 = {ideal (c + 45d, b - 34d, a - 35d), ideal (c + 18d, b - 12d, a + 47d),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2                      2   3                2        2
│ │ │ │ -      34b*c - 29c  - 50b*d + 47c*d - 17d , c  - 28b*c*d + 15c d + 4b*d  -
│ │ │ │ +                                         2                      2   2      2
│ │ │ │ +      ideal (a + 37b + 5c - 4d, b*c - 13c  + 45b*d - 28c*d - 29d , b  - 48c
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2      3     2                2         2        2      3
│ │ │ │ -      10c*d  + 10d , b*c  - 50b*c*d + 45c d - 43b*d  + 34c*d  + 22d )}
│ │ │ │ +                           2
│ │ │ │ +      - 14b*d + 27c*d - 38d )}
│ │ │ │  
│ │ │ │  o48 : List
│ │ │ │  i49 : lin = CFa_1_0 -- a linear form, defining a plane.
│ │ │ │  
│ │ │ │ -o49 = a + 18b + 49c - 3d
│ │ │ │ +o49 = c + 18d
│ │ │ │  
│ │ │ │  o49 : S
│ │ │ │  i50 : CFa/degree
│ │ │ │  
│ │ │ │ -o50 = {1, 5}
│ │ │ │ +o50 = {1, 1, 4}
│ │ │ │  
│ │ │ │  o50 : List
│ │ │ │  i51 : CFa/(I -> lin % I == 0) -- so 5 points on the plane.
│ │ │ │  
│ │ │ │ -o51 = {false, true}
│ │ │ │ +o51 = {false, true, false}
│ │ │ │  
│ │ │ │  o51 : List
│ │ │ │  i52 : degree(Fa : (Fa : lin))  -- somewhat simpler(?) way to see the ideal of
│ │ │ │  the 5 points
│ │ │ │  
│ │ │ │ -o52 = 5
│ │ │ │ +o52 = 1
│ │ │ │  i53 : ptb = randomPointOnRationalVariety compsL441_1
│ │ │ │  
│ │ │ │ -o53 = | 27 12 -34 9 -19 -43 -32 27 40 45 -13 29 -41 -13 22 -49 -4 -4 9 -23 43
│ │ │ │ +o53 = | -31 -3 -29 -17 -21 5 -32 33 -24 2 26 -26 -45 -4 16 -22 2 -37 16 -23
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      18 -9 -47 43 21 38 17 -20 21 -29 47 0 2 -37 9 |
│ │ │ │ +      -42 19 -29 21 7 2 17 9 -15 -9 -47 -13 0 38 47 21 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o53 : Matrix kk  <-- kk
│ │ │ │  i54 : Fb = sub(F, (vars S) | ptb)
│ │ │ │  
│ │ │ │ -              2              2                              2
│ │ │ │ -o54 = ideal (a  + 22b*c + 45c  + 29a*d - 43b*d - 34c*d + 27d , a*b + 43b*c +
│ │ │ │ +              2             2                             2
│ │ │ │ +o54 = ideal (a  + 16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , a*b + 7b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                              2   2              2
│ │ │ │ -      9c  + 43a*d - 41b*d + 40c*d + 12d , b  - 37b*c + 17c  + 18b*d - 49c*d -
│ │ │ │ +         2                             2   2             2
│ │ │ │ +      16c  - 42a*d - 45b*d - 24c*d - 3d , b  + 47b*c + 9c  + 19b*d - 22c*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                   2                              2     2
│ │ │ │ -      32d , a*c + 47b*c + 21c  - 20a*d - 23b*d - 13c*d - 19d , b*c  + 21b*c*d
│ │ │ │ +         2                  2                             2     2
│ │ │ │ +      32d , a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b*c  - 9b*c*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -          2         2       2        2     3   3               2        2
│ │ │ │ -      - 9c d + 38a*d  - 4b*d  - 13c*d  + 9d , c  + 9b*c*d - 29c d + 2a*d  -
│ │ │ │ +         2         2       2        2      3   3                2         2
│ │ │ │ +      29c d + 17a*d  + 2b*d  + 26c*d  - 17d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2       2      3
│ │ │ │ -      47b*d  - 4c*d  + 27d )
│ │ │ │ +           2        2      3
│ │ │ │ +      21b*d  - 37c*d  + 33d )
│ │ │ │  
│ │ │ │  o54 : Ideal of S
│ │ │ │  i55 : betti res Fb
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o55 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o55 : BettiTally
│ │ │ │  i56 : netList decompose Fb --
│ │ │ │  
│ │ │ │ +      +---------------------------------------------------------------+
│ │ │ │ +o56 = |ideal (c - 17d, b - 26d, a - 49d)                              |
│ │ │ │ +      +---------------------------------------------------------------+
│ │ │ │ +      |                                     3      2         2      3 |
│ │ │ │ +      |ideal (b + 39c - 21d, a + 4c - 27d, c  + 43c d + 39c*d  - 15d )|
│ │ │ │ +      +---------------------------------------------------------------+
│ │ │ │ +      |                                    2             2            |
│ │ │ │ +      |ideal (b + 8c + 40d, a + 5c - 33d, c  + 4c*d + 45d )           |
│ │ │ │ +      +---------------------------------------------------------------+
│ │ │ │ +i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │ │ +
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ---------------------------------------------------------------+
│ │ │ │ -      |                                      2              2
│ │ │ │ -|
│ │ │ │ -o56 = |ideal (b - 50c - 43d, a + 15c - 46d, c  + 12c*d - 37d )
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ---------------------------------------------------------------+
│ │ │ │ -      |        2                             2
│ │ │ │ -2                                   2   2                            2
│ │ │ │ -2   2                             2 |
│ │ │ │ -      |ideal (c  + 46a*d - 39b*d + 2c*d - 24d , b*c - 9a*d + 16b*d + 2c*d + 27d
│ │ │ │ -, a*c + 43a*d + 44b*d - 48c*d + 24d , b  - 4a*d - 40b*d - 9c*d - 39d , a*b +
│ │ │ │ -16a*d + 26b*d + 37c*d - 24d , a  - 25a*d + 47b*d + 34c*d + 8d )|
│ │ │ │ +---------------------------------------------------------------------------+
│ │ │ │ +      |                       2                             2   2             2
│ │ │ │ +2                  2                             2   2             2
│ │ │ │ +2   3                2         2        2        2      3     2               2
│ │ │ │ +2       2        2      3 |
│ │ │ │ +o57 = |ideal (a*c - 13b*c + 2c  - 15a*d - 23b*d - 4c*d - 21d , b  + 47b*c + 9c
│ │ │ │ ++ 19b*d - 22c*d - 32d , a*b + 7b*c + 16c  - 42a*d - 45b*d - 24c*d - 3d , a  +
│ │ │ │ +16b*c + 2c  - 26a*d + 5b*d - 29c*d - 31d , c  + 21b*c*d - 47c d + 38a*d  +
│ │ │ │ +21b*d  - 37c*d  + 33d , b*c  - 9b*c*d - 29c d + 17a*d  + 2b*d  + 26c*d  - 17d
│ │ │ │ +)|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ---------------------------------------------------------------+
│ │ │ │ -i57 : netList for x in subsets(decompose Fb, 3) list intersect(x#0, x#1, x#2)
│ │ │ │ -
│ │ │ │ -o57 = ++
│ │ │ │ -      ++
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +---------------------------------------------------------------------------+
│ │ │ │  i58 : pt0 = randomPointOnRationalVariety(compsL441_0)
│ │ │ │  
│ │ │ │ -o58 = | 44 15 26 -28 -4 33 49 -19 -6 -18 -37 -34 18 -42 -23 23 38 -40 27 26
│ │ │ │ +o58 = | 49 0 -30 -36 -1 0 -9 17 37 29 34 13 19 8 -10 -47 21 -24 -44 42 9 46
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -39 -24 -21 -13 38 39 -33 15 8 -18 18 -22 -2 -29 -23 46 |
│ │ │ │ +      15 -29 35 -40 18 -22 -21 -42 39 -2 -33 -23 -13 -18 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o58 : Matrix kk  <-- kk
│ │ │ │  i59 : pt1 = randomPointOnRationalVariety(compsL441_1)
│ │ │ │  
│ │ │ │ -o59 = | -8 41 28 -44 50 33 -38 33 -23 1 -2 -47 32 46 30 -22 -2 -14 27 37 15
│ │ │ │ +o59 = | -5 -40 -6 3 -28 -8 -25 15 15 29 26 -37 11 -14 31 14 1 -50 43 37 5 50
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -25 -15 33 -23 3 21 -18 -9 3 10 -49 0 -35 -50 32 |
│ │ │ │ +      10 3 -3 -35 -18 32 -7 -15 33 46 0 21 -49 3 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o59 : Matrix kk  <-- kk
│ │ │ │  We compute the ideal of the corresponding zero dimensional scheme with length
│ │ │ │  6, corresponding to the points pt0, pt1 in Hilb.
│ │ │ │  i60 : I0 = sub(sub(F, (vars ring F) | sub(pt0, ring F)), S)
│ │ │ │  
│ │ │ │ -              2              2                              2
│ │ │ │ -o60 = ideal (a  - 23b*c - 18c  - 34a*d + 33b*d + 26c*d + 44d , a*b + 38b*c +
│ │ │ │ +              2              2                      2                   2
│ │ │ │ +o60 = ideal (a  - 10b*c + 29c  + 13a*d - 30c*d + 49d , a*b + 35b*c - 44c  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                             2   2              2
│ │ │ │ -      27c  - 39a*d + 18b*d - 6c*d + 15d , b  - 23b*c + 15c  - 2a*d - 24b*d +
│ │ │ │ +                             2              2                             2
│ │ │ │ +      9a*d + 19b*d + 37c*d, b  - 13b*c - 22c  - 33a*d + 46b*d - 47c*d - 9d ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2                   2                            2     2
│ │ │ │ -      23c*d + 49d , a*c - 22b*c + 39c  + 8a*d + 26b*d - 42c*d - 4d , b*c  -
│ │ │ │ +                      2                           2     2                2
│ │ │ │ +      a*c - 2b*c - 40c  - 21a*d + 42b*d + 8c*d - d , b*c  - 42b*c*d + 15c d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2         2        2        2      3   3                2
│ │ │ │ -      18b*c*d - 21c d - 33a*d  + 38b*d  - 37c*d  - 28d , c  + 46b*c*d + 18c d
│ │ │ │ +           2        2        2      3   3                2         2        2
│ │ │ │ +      18a*d  + 21b*d  + 34c*d  - 36d , c  - 18b*c*d + 39c d - 23a*d  - 29b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2        2      3
│ │ │ │ -      - 29a*d  - 13b*d  - 40c*d  - 19d )
│ │ │ │ +             2      3
│ │ │ │ +      - 24c*d  + 17d )
│ │ │ │  
│ │ │ │  o60 : Ideal of S
│ │ │ │  i61 : I1 = sub(sub(F, (vars ring F) | sub(pt1, ring F)), S)
│ │ │ │  
│ │ │ │ -              2            2                             2
│ │ │ │ -o61 = ideal (a  + 30b*c + c  - 47a*d + 33b*d + 28c*d - 8d , a*b - 23b*c +
│ │ │ │ +              2              2                           2                  2
│ │ │ │ +o61 = ideal (a  + 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , a*b - 3b*c + 43c
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                              2   2              2
│ │ │ │ -      27c  + 15a*d + 32b*d - 23c*d + 41d , b  - 50b*c - 18c  - 25b*d - 22c*d
│ │ │ │ +                                  2   2              2
│ │ │ │ +      + 5a*d + 11b*d + 15c*d - 40d , b  - 49b*c + 32c  + 50b*d + 14c*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2                  2                             2     2
│ │ │ │ -      - 38d , a*c - 49b*c + 3c  - 9a*d + 37b*d + 46c*d + 50d , b*c  + 3b*c*d
│ │ │ │ +         2                   2                             2     2
│ │ │ │ +      25d , a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b*c  - 15b*c*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -           2         2       2       2      3   3                2         2
│ │ │ │ -      - 15c d + 21a*d  - 2b*d  - 2c*d  - 44d , c  + 32b*c*d + 10c d - 35a*d
│ │ │ │ +           2         2      2        2     3   3               2         2
│ │ │ │ +      + 10c d - 18a*d  + b*d  + 26c*d  + 3d , c  + 3b*c*d + 33c d + 21a*d  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2      3
│ │ │ │ -      + 33b*d  - 14c*d  + 33d )
│ │ │ │ +          2        2      3
│ │ │ │ +      3b*d  - 50c*d  + 15d )
│ │ │ │  
│ │ │ │  o61 : Ideal of S
│ │ │ │  i62 : betti res I0
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o62 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │ @@ -648,43 +653,44 @@
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o63 : BettiTally
│ │ │ │  i64 : netList decompose I0
│ │ │ │  
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +o64 = |ideal (c - 21d, b - 26d, a - 50d)                  |
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +      |ideal (c - 49d, b - 33d, a - 30d)                  |
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +      |ideal (c + 41d, b + 33d, a - 35d)                  |
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +      |ideal (c + d, b + 40d, a - 5d)                     |
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +      |                                    2            2 |
│ │ │ │ +      |ideal (b + c - 47d, a - 38c - 17d, c  + 4c*d - 8d )|
│ │ │ │ +      +---------------------------------------------------+
│ │ │ │ +i65 : netList decompose I1
│ │ │ │ +
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ------+
│ │ │ │ -o64 = |ideal (c + 8d, b + 5d, a - 25d)
│ │ │ │ -|
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ------+
│ │ │ │ -      |                             2              2              2   3
│ │ │ │ -2         2        2      3     2                2         2        2     3 |
│ │ │ │ -      |ideal (a - 22b + 39c + 50d, b  - 23b*c + 15c  + 33b*d + 48d , c  +
│ │ │ │ -46b*c*d + 18c d - 45b*d  - 20c*d  + 17d , b*c  - 18b*c*d - 21c d + 19b*d  +
│ │ │ │ -38c*d  + 6d )|
│ │ │ │ +-------------------------------------------------------------------------+
│ │ │ │ +      |                        2                             2   2
│ │ │ │ +2                      2                  2                             2   2
│ │ │ │ +2                           2   3               2         2       2        2
│ │ │ │ +3     2                2         2      2        2     3 |
│ │ │ │ +o65 = |ideal (a*c + 46b*c - 35c  - 7a*d + 37b*d - 14c*d - 28d , b  - 49b*c +
│ │ │ │ +32c  + 50b*d + 14c*d - 25d , a*b - 3b*c + 43c  + 5a*d + 11b*d + 15c*d - 40d , a
│ │ │ │ ++ 31b*c + 29c  - 37a*d - 8b*d - 6c*d - 5d , c  + 3b*c*d + 33c d + 21a*d  + 3b*d
│ │ │ │ +- 50c*d  + 15d , b*c  - 15b*c*d + 10c d - 18a*d  + b*d  + 26c*d  + 3d )|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ------+
│ │ │ │ -i65 : netList decompose I1
│ │ │ │ -
│ │ │ │ -      +---------------------------------+
│ │ │ │ -o65 = |ideal (c - 9d, b + 15d, a + 27d) |
│ │ │ │ -      +---------------------------------+
│ │ │ │ -      |ideal (c + 48d, b + 11d, a - 37d)|
│ │ │ │ -      +---------------------------------+
│ │ │ │ -      |ideal (c + 29d, b + 46d, a + 18d)|
│ │ │ │ -      +---------------------------------+
│ │ │ │ -      |ideal (c + 24d, b + 46d, a + 33d)|
│ │ │ │ -      +---------------------------------+
│ │ │ │ -      |ideal (c + 22d, b + 38d, a - 50d)|
│ │ │ │ -      +---------------------------------+
│ │ │ │ +-------------------------------------------------------------------------------
│ │ │ │ +-------------------------------------------------------------------------+
│ │ │ │  i66 : L430 = (trim minors(2, M1)) + groebnerStratum F;
│ │ │ │  
│ │ │ │  o66 : Ideal of T
│ │ │ │  i67 : C = res(I, FastNonminimal => true)
│ │ │ │  
│ │ │ │         1      4      5      2
│ │ │ │  o67 = S  <-- S  <-- S  <-- S  <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/_canonical__Curve.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  i2 : g=14;
│ │ │  
│ │ │  i3 : FF=ZZ/10007;
│ │ │  
│ │ │  i4 : R=FF[x_0..x_(g-1)];
│ │ │  
│ │ │  i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 8.18095s (cpu); 5.66026s (thread); 0s (gc)
│ │ │ + -- used 8.63426s (cpu); 6.46439s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │
    		│ │ │ 
    i4 : R=FF[x_0..x_(g-1)];
    │ │ │
    │ │ │ 
    i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 8.18095s (cpu); 5.66026s (thread); 0s (gc)
│ │ │ + -- used 8.63426s (cpu); 6.46439s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
    │ │ │ ├── html2text {} │ │ │ │ @@ -17,15 +17,15 @@ │ │ │ │ unirationality of $M_g$ by Severi, Sernesi, Chang-Ran and Verra. │ │ │ │ i1 : setRandomSeed "alpha"; │ │ │ │ -- setting random seed to 10206284518 │ │ │ │ i2 : g=14; │ │ │ │ i3 : FF=ZZ/10007; │ │ │ │ i4 : R=FF[x_0..x_(g-1)]; │ │ │ │ i5 : time betti(I=(random canonicalCurve)(g,R)) │ │ │ │ - -- used 8.18095s (cpu); 5.66026s (thread); 0s (gc) │ │ │ │ + -- used 8.63426s (cpu); 6.46439s (thread); 0s (gc) │ │ │ │ │ │ │ │ 0 1 │ │ │ │ o5 = total: 1 66 │ │ │ │ 0: 1 . │ │ │ │ 1: . 66 │ │ │ │ │ │ │ │ o5 : BettiTally │ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ @@ -7,42 +7,42 @@ │ │ │ │ │ │ o2 = (10, 20) │ │ │ │ │ │ o2 : Sequence │ │ │ │ │ │ i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11) │ │ │ │ │ │ -o3 = ({5, 2.91596e52, 9}, .00214587, .000894177) │ │ │ +o3 = ({5, 2.91596e52, 9}, .00207174, .000941274) │ │ │ │ │ │ o3 : Sequence │ │ │ │ │ │ i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100) │ │ │ │ │ │ -o4 = ({50, 2.30853e454, 98}, .00612829, .0350796) │ │ │ +o4 = ({50, 2.30853e454, 98}, .00585662, .0401955) │ │ │ │ │ │ o4 : Sequence │ │ │ │ │ │ i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2}) │ │ │ │ │ │ -o5 = {{.00550847, .0122005}, {.0050821, .00412591}, {.00509518, .00673598}, │ │ │ +o5 = {{.00577326, .0141836}, {.00512962, .00482506}, {.00546436, .00726162}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00505836, .00986236}, {.00516769, .0132322}, {.0869977, .0125279}, │ │ │ + {.00634212, .0110694}, {.00603324, .0153341}, {.121192, .0191168}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00537013, .00809721}, {.00501329, .00743004}, {.00422633, .00533875}, │ │ │ + {.00484695, .00922064}, {.0068809, .0129731}, {.00434803, .00688545}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {.00672373, .00805844}} │ │ │ + {.0056288, .00949632}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : 1/10*sum(L,t->t_0) │ │ │ │ │ │ -o6 = .01342429950000001 │ │ │ +o6 = .01716389680000008 │ │ │ │ │ │ o6 : RR (of precision 53) │ │ │ │ │ │ i7 : 1/10*sum(L,t->t_1) │ │ │ │ │ │ -o7 = .00876092100000001 │ │ │ +o7 = .01103661020000022 │ │ │ │ │ │ o7 : RR (of precision 53) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ @@ -93,57 +93,57 @@ │ │ │ o2 : Sequence │ │ │
    │ │ │ 
    i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00214587, .000894177)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00207174, .000941274)
│ │ │  
│ │ │  o3 : Sequence
    │ │ │
    │ │ │ 
    i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00612829, .0350796)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00585662, .0401955)
│ │ │  
│ │ │  o4 : Sequence
    │ │ │
    │ │ │ 
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {{.00550847, .0122005}, {.0050821, .00412591}, {.00509518, .00673598},
│ │ │ +o5 = {{.00577326, .0141836}, {.00512962, .00482506}, {.00546436, .00726162},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00505836, .00986236}, {.00516769, .0132322}, {.0869977, .0125279},
│ │ │ +     {.00634212, .0110694}, {.00603324, .0153341}, {.121192, .0191168},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00537013, .00809721}, {.00501329, .00743004}, {.00422633, .00533875},
│ │ │ +     {.00484695, .00922064}, {.0068809, .0129731}, {.00434803, .00688545},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {.00672373, .00805844}}
│ │ │ +     {.0056288, .00949632}}
│ │ │  
│ │ │  o5 : List
    │ │ │
    │ │ │ 
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .01342429950000001
│ │ │ +o6 = .01716389680000008
│ │ │  
│ │ │  o6 : RR (of precision 53)
    │ │ │
    │ │ │ 
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .00876092100000001
│ │ │ +o7 = .01103661020000022
│ │ │  
│ │ │  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}, .00214587, .000894177) │ │ │ │ +o3 = ({5, 2.91596e52, 9}, .00207174, .000941274) │ │ │ │ │ │ │ │ o3 : Sequence │ │ │ │ i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100) │ │ │ │ │ │ │ │ -o4 = ({50, 2.30853e454, 98}, .00612829, .0350796) │ │ │ │ +o4 = ({50, 2.30853e454, 98}, .00585662, .0401955) │ │ │ │ │ │ │ │ o4 : Sequence │ │ │ │ i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2}) │ │ │ │ │ │ │ │ -o5 = {{.00550847, .0122005}, {.0050821, .00412591}, {.00509518, .00673598}, │ │ │ │ +o5 = {{.00577326, .0141836}, {.00512962, .00482506}, {.00546436, .00726162}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00505836, .00986236}, {.00516769, .0132322}, {.0869977, .0125279}, │ │ │ │ + {.00634212, .0110694}, {.00603324, .0153341}, {.121192, .0191168}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00537013, .00809721}, {.00501329, .00743004}, {.00422633, .00533875}, │ │ │ │ + {.00484695, .00922064}, {.0068809, .0129731}, {.00434803, .00688545}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {.00672373, .00805844}} │ │ │ │ + {.0056288, .00949632}} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : 1/10*sum(L,t->t_0) │ │ │ │ │ │ │ │ -o6 = .01342429950000001 │ │ │ │ +o6 = .01716389680000008 │ │ │ │ │ │ │ │ o6 : RR (of precision 53) │ │ │ │ i7 : 1/10*sum(L,t->t_1) │ │ │ │ │ │ │ │ -o7 = .00876092100000001 │ │ │ │ +o7 = .01103661020000022 │ │ │ │ │ │ │ │ o7 : RR (of precision 53) │ │ │ │ ********** WWaayyss ttoo uussee tteessttTTiimmeeFFoorrLLLLLLoonnSSyyzzyyggiieess:: ********** │ │ │ │ * testTimeForLLLonSyzygies(ZZ,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _t_e_s_t_T_i_m_e_F_o_r_L_L_L_o_n_S_y_z_y_g_i_e_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out │ │ │ @@ -1,11 +1,11 @@ │ │ │ -- -*- M2-comint -*- hash: 11549527689790345152 │ │ │ │ │ │ i1 : time ICan = smoothCanonicalCurve(11,5); │ │ │ - -- used 1.35363s (cpu); 1.07786s (thread); 0s (gc) │ │ │ + -- used 1.76494s (cpu); 1.39524s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o1 : Ideal of --[t ..t ] │ │ │ 5 0 10 │ │ │ │ │ │ i2 : (dim ICan, genus ICan, degree ICan) │ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html │ │ │ @@ -82,15 +82,15 @@ │ │ │

    If the option Printing is set to true then printings about the current step in the construction are displayed.

    │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ For g<=10 the curves are constructed via plane models. │ │ │ │ For g<=13 the curves are constructed via space models. │ │ │ │ For g=14 the curves are constructed by Verra's method. │ │ │ │ For g=15 the curves are constructed via matrix factorizations. │ │ │ │ If the option Printing is set to true then printings about the current step in │ │ │ │ the construction are displayed. │ │ │ │ i1 : time ICan = smoothCanonicalCurve(11,5); │ │ │ │ - -- used 1.35363s (cpu); 1.07786s (thread); 0s (gc) │ │ │ │ + -- used 1.76494s (cpu); 1.39524s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o1 : Ideal of --[t ..t ] │ │ │ │ 5 0 10 │ │ │ │ i2 : (dim ICan, genus ICan, degree ICan) │ │ │ │ │ │ │ │ o2 = (2, 11, 20) │ │ ├── ./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ -- setting random seed to 10206284518 │ │ │ │ │ │ i2 : FF=ZZ/10007; │ │ │ │ │ │ i3 : S=FF[x_0..x_6]; │ │ │ │ │ │ i4 : time I=randomCurveGenus14Degree18inP6(S); │ │ │ - -- used 1.66941s (cpu); 1.35693s (thread); 0s (gc) │ │ │ + -- used 2.01878s (cpu); 1.67858s (thread); 0s (gc) │ │ │ │ │ │ o4 : Ideal of S │ │ │ │ │ │ i5 : betti res I │ │ │ │ │ │ 0 1 2 3 4 5 │ │ │ o5 = total: 1 13 45 56 25 2 │ │ ├── ./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ @@ -93,15 +93,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -66,24 +66,24 @@ │ │ │ │ first in rings with more variables. │ │ │ │ i6 : S=ZZ/103[y_0..y_30]; │ │ │ │ i7 : I=minors(2,random(S^3,S^{3:-1})); │ │ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, │ │ │ │ DecompositionStrategy=>MultiplicationTable) │ │ │ │ - -- 3.13643s elapsed │ │ │ │ + -- 3.05751s elapsed │ │ │ │ │ │ │ │ o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, │ │ │ │ DecompositionStrategy=>Decompose) │ │ │ │ - -- 2.23953s elapsed │ │ │ │ + -- 2.10492s elapsed │ │ │ │ │ │ │ │ o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}} │ │ │ │ │ │ │ │ o9 : List │ │ │ │ ********** WWaayyss ttoo uussee rraannddoommPPooiinnttss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out │ │ │ @@ -49,15 +49,15 @@ │ │ │ i12 : Q=QQ[x,y,z,t,u]; │ │ │ │ │ │ i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}}); │ │ │ │ │ │ o13 : RingMap Q <-- Q │ │ │ │ │ │ i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0) │ │ │ - -- used 0.600555s (cpu); 0.418226s (thread); 0s (gc) │ │ │ + -- used 0.877538s (cpu); 0.480262s (thread); 0s (gc) │ │ │ │ │ │ 125 124 120 5 124 100 25 104 20 108 15 2 112 10 3 116 5 4 120 5 124 125 4 120 8 115 2 12 110 3 16 105 4 20 100 5 24 95 6 28 90 7 32 85 8 36 80 9 40 75 10 44 70 11 48 65 12 52 60 13 56 55 14 60 50 15 64 45 16 68 40 17 72 35 18 76 30 19 80 25 20 84 20 21 88 15 22 92 10 23 96 5 24 100 25 24 100 28 95 32 90 2 36 85 3 40 80 4 44 75 5 48 70 6 52 65 7 56 60 8 60 55 9 64 50 10 68 45 11 72 40 12 76 35 13 80 30 14 84 25 15 88 20 16 92 15 17 96 10 18 100 5 19 104 20 48 75 2 52 70 2 56 65 2 2 60 60 3 2 64 55 4 2 68 50 5 2 72 45 6 2 76 40 7 2 80 35 8 2 84 30 9 2 88 25 10 2 92 20 11 2 96 15 12 2 100 10 13 2 104 5 14 2 108 15 2 72 50 3 76 45 3 80 40 2 3 84 35 3 3 88 30 4 3 92 25 5 3 96 20 6 3 100 15 7 3 104 10 8 3 108 5 9 3 112 10 3 96 25 4 100 20 4 104 15 2 4 108 10 3 4 112 5 4 4 116 5 4 120 5 124 │ │ │ o14 = Proj Q - - - > Proj Q {x , x y, - x y + x z, x y - 5x y z + 10x y z - 10x y z + 5x y z - x z + x t, - y + 25x y z - 300x y z + 2300x y z - 12650x y z + 53130x y z - 177100x y z + 480700x y z - 1081575x y z + 2042975x y z - 3268760x y z + 4457400x y z - 5200300x y z + 5200300x y z - 4457400x y z + 3268760x y z - 2042975x y z + 1081575x y z - 480700x y z + 177100x y z - 53130x y z + 12650x y z - 2300x y z + 300x y z - 25x y z + x z - 5x y t + 100x y z*t - 950x y z t + 5700x y z t - 24225x y z t + 77520x y z t - 193800x y z t + 387600x y z t - 629850x y z t + 839800x y z t - 923780x y z t + 839800x y z t - 629850x y z t + 387600x y z t - 193800x y z t + 77520x y z t - 24225x y z t + 5700x y z t - 950x y z t + 100x y z t - 5x z t - 10x y t + 150x y z*t - 1050x y z t + 4550x y z t - 13650x y z t + 30030x y z t - 50050x y z t + 64350x y z t - 64350x y z t + 50050x y z t - 30030x y z t + 13650x y z t - 4550x y z t + 1050x y z t - 150x y z t + 10x z t - 10x y t + 100x y z*t - 450x y z t + 1200x y z t - 2100x y z t + 2520x y z t - 2100x y z t + 1200x y z t - 450x y z t + 100x y z t - 10x z t - 5x y t + 25x y z*t - 50x y z t + 50x y z t - 25x y z t + 5x z t - x t + x u} │ │ │ │ │ │ o14 : RationalMapping │ │ │ │ │ │ i15 : R=QQ[x,y,z,t]/(z-2*t); │ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html │ │ │ @@ -189,15 +189,15 @@ │ │ │ │ │ │ o13 : RingMap Q <-- Q │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ o11 : Ideal of blowUpSubvar │ │ │ │ The next example is a birational map on $\mathbb{P}^4$. │ │ │ │ i12 : Q=QQ[x,y,z,t,u]; │ │ │ │ i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}}); │ │ │ │ │ │ │ │ o13 : RingMap Q <-- Q │ │ │ │ i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0) │ │ │ │ - -- used 0.600555s (cpu); 0.418226s (thread); 0s (gc) │ │ │ │ + -- used 0.877538s (cpu); 0.480262s (thread); 0s (gc) │ │ │ │ │ │ │ │ 125 124 120 5 124 100 25 104 │ │ │ │ 20 108 15 2 112 10 3 116 5 4 120 5 124 125 4 120 │ │ │ │ 8 115 2 12 110 3 16 105 4 20 100 5 24 95 6 │ │ │ │ 28 90 7 32 85 8 36 80 9 40 75 10 44 70 │ │ │ │ 11 48 65 12 52 60 13 56 55 14 60 50 15 │ │ │ │ 64 45 16 68 40 17 72 35 18 76 30 19 80 25 │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out │ │ │ @@ -48,15 +48,15 @@ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ ZZ │ │ │ o13 : Ideal of ---[u ..u ] │ │ │ 101 0 10 │ │ │ │ │ │ i14 : time rationalPoints(I, Amount => true) │ │ │ - -- used 0.00328519s (cpu); 0.00328099s (thread); 0s (gc) │ │ │ + -- used 0.00434048s (cpu); 0.00434084s (thread); 0s (gc) │ │ │ │ │ │ o14 = 110462212541120451001 │ │ │ │ │ │ i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z); │ │ │ │ │ │ o16 : Ideal of QQ[x..z] │ │ │ │ │ │ @@ -142,23 +142,23 @@ │ │ │ │ │ │ i31 : nodes = I + ideal jacobian I; │ │ │ │ │ │ o31 : Ideal of R │ │ │ │ │ │ i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true); │ │ │ -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25) │ │ │ - -- used 0.892058s (cpu); 0.754813s (thread); 0s (gc) │ │ │ + -- used 1.25853s (cpu); 0.961587s (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 0.285089s (cpu); 0.224588s (thread); 0s (gc) │ │ │ + -- used 0.366938s (cpu); 0.256372s (thread); 0s (gc) │ │ │ │ │ │ o35 = 31 │ │ │ │ │ │ i36 : │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html │ │ │ @@ -178,15 +178,15 @@ │ │ │ o13 : Ideal of ---[u ..u ] │ │ │ 101 0 10 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.35363s (cpu); 1.07786s (thread); 0s (gc)
│ │ │ + -- used 1.76494s (cpu); 1.39524s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
    │ │ │
    │ │ │ 
    i3 : S=FF[x_0..x_6];
    │ │ │
    │ │ │ 
    i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.66941s (cpu); 1.35693s (thread); 0s (gc)
│ │ │ + -- used 2.01878s (cpu); 1.67858s (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 1.66941s (cpu); 1.35693s (thread); 0s (gc)
│ │ │ │ + -- used 2.01878s (cpu); 1.67858s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : betti res I
│ │ │ │  
│ │ │ │              0  1  2  3  4 5
│ │ │ │  o5 = total: 1 13 45 56 25 2
│ │ │ │           0: 1  .  .  .  . .
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out
│ │ │ @@ -1,26 +1,24 @@
│ │ │  -- -*- M2-comint -*- hash: 9542801742429495161
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1768844752
│ │ │ + -- setting random seed to 1769302656
│ │ │  
│ │ │ -o1 = 1768844752
│ │ │ +o1 = 1769302656
│ │ │  
│ │ │  i2 : kk=ZZ/101;
│ │ │  
│ │ │  i3 : S=kk[vars(0..5)];
│ │ │  
│ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 3.15015s (cpu); 1.82825s (thread); 0s (gc)
│ │ │ + -- used 4.46986s (cpu); 1.90648s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{3 => 1  }
│ │ │ -           4 => 48
│ │ │ -           5 => 204
│ │ │ -           6 => 167
│ │ │ -           7 => 65
│ │ │ -           8 => 13
│ │ │ +o4 = Tally{4 => 41 }
│ │ │ +           5 => 218
│ │ │ +           6 => 162
│ │ │ +           7 => 64
│ │ │ +           8 => 14
│ │ │             9 => 1
│ │ │ -           10 => 1
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5959465567197821046
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1768844757
│ │ │ + -- setting random seed to 1769302660
│ │ │  
│ │ │ -o1 = 1768844757
│ │ │ +o1 = 1769302660
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -15,13 +15,12 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = a c
│ │ │ +o4 = a*b*c
│ │ │  
│ │ │  o4 : S
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 8876340562021865447
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1768844765
│ │ │ + -- setting random seed to 1769302668
│ │ │  
│ │ │ -o1 = 1768844765
│ │ │ +o1 = 1769302668
│ │ │  
│ │ │  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(b*c*e)
│ │ │ +o5 = ideal(b*d*e)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*e, a*b, b*e, a*d, a*c)
│ │ │ +o6 = ideal (a*b, c*d, c*e, b*c, b*d)
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 10504911213508281315
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1768844761
│ │ │ + -- setting random seed to 1769302664
│ │ │  
│ │ │ -o1 = 1768844761
│ │ │ +o1 = 1769302664
│ │ │  
│ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │ @@ -39,15 +39,15 @@
│ │ │  i7 : J = monomialIdeal 0_S
│ │ │  
│ │ │  o7 = monomialIdeal ()
│ │ │  
│ │ │  o7 : MonomialIdeal of S
│ │ │  
│ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 3.90363s (cpu); 2.77034s (thread); 0s (gc)
│ │ │ + -- used 5.44845s (cpu); 3.38013s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : #T
│ │ │  
│ │ │  o9 = 168
│ │ │  
│ │ │  i10 : T
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html
│ │ │ @@ -71,17 +71,17 @@
│ │ │          
    
│ │ │            
    Chooses a random monomial.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1768844757
│ │ │ + -- setting random seed to 1769302660
│ │ │  
│ │ │ -o1 = 1768844757
    
│ │ │ +o1 = 1769302660
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -98,16 +98,15 @@
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = a c
│ │ │ +o4 = a*b*c
│ │ │  
│ │ │  o4 : S
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,31 +11,30 @@
│ │ │ │            o d, an _i_n_t_e_g_e_r, non-negative
│ │ │ │            o S, a _r_i_n_g, polynomial ring
│ │ │ │      * Outputs:
│ │ │ │            o m, a _r_i_n_g_ _e_l_e_m_e_n_t, monomial of S
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Chooses a random monomial.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1768844757
│ │ │ │ + -- setting random seed to 1769302660
│ │ │ │  
│ │ │ │ -o1 = 1768844757
│ │ │ │ +o1 = 1769302660
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a,b,c]
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : randomMonomial(3,S)
│ │ │ │  
│ │ │ │ -      2
│ │ │ │ -o4 = a c
│ │ │ │ +o4 = a*b*c
│ │ │ │  
│ │ │ │  o4 : S
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_M_o_n_o_m_i_a_l_I_d_e_a_l -- random monomial ideal with given degree generators
│ │ │ │      * _r_a_n_d_o_m_S_q_u_a_r_e_F_r_e_e_M_o_n_o_m_i_a_l_I_d_e_a_l -- random square-free monomial ideal with
│ │ │ │        given degree generators
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommMMoonnoommiiaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html
│ │ │ @@ -71,17 +71,17 @@
│ │ │          
    
│ │ │            
    Choose a random square-free monomial ideal whose generators have the degrees specified by the list or sequence L.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1768844765
│ │ │ + -- setting random seed to 1769302668
│ │ │  
│ │ │ -o1 = 1768844765
    
│ │ │ +o1 = 1769302668
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -108,24 +108,24 @@
│ │ │              
    
│ │ │                
    i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(b*c*e)
│ │ │ +o5 = ideal(b*d*e)
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*e, a*b, b*e, a*d, a*c)
│ │ │ +o6 = ideal (a*b, c*d, c*e, b*c, b*d)
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,17 +13,17 @@
│ │ │ │      * Outputs:
│ │ │ │            o I, an _i_d_e_a_l, square-free monomial ideal with generators of
│ │ │ │              specified degrees
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Choose a random square-free monomial ideal whose generators have the degrees
│ │ │ │  specified by the list or sequence L.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1768844765
│ │ │ │ + -- setting random seed to 1769302668
│ │ │ │  
│ │ │ │ -o1 = 1768844765
│ │ │ │ +o1 = 1769302668
│ │ │ │  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(b*c*e)
│ │ │ │ +o5 = ideal(b*d*e)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (a*e, a*b, b*e, a*d, a*c)
│ │ │ │ +o6 = ideal (a*b, c*d, c*e, b*c, b*d)
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The ideal is constructed degree by degree, starting from the lowest degree
│ │ │ │  specified. If there are not enough monomials of the next specified degree that
│ │ │ │  are not already in the ideal, the function prints a warning and returns an
│ │ │ │  ideal containing all the generators of that degree.
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html
│ │ │ @@ -79,17 +79,17 @@
│ │ │            
    With probability p the routine takes the Alexander dual of I; the default value of p is .05, and it can be set with the option AlexanderProbility.
    
│ │ │            
    Otherwise uses the Metropolis algorithm to produce a random walk on the space of square-free ideals. Note that there are a LOT of square-free ideals; these are the Dedekind numbers, and the sequence (with 1,2,3,4,5,6,7,8 variables) begins 3,6,20,168,7581, 7828354, 2414682040998, 56130437228687557907788. (see the Online Encyclopedia of Integer Sequences for more information). Given I in a polynomial ring S, we make a list ISocgens of the square-free minimal monomial generators of the socle of S/(squares+I) and a list of minimal generators Igens of I. A candidate "next" ideal J is formed as follows: We choose randomly from the union of these lists; if a socle element is chosen, it's added to I; if a minimal generator is chosen, it's replaced by the square-free part of the maximal ideal times it. the chance of making the given move is then 1/(#ISocgens+#Igens), and the chance of making the move back would be the similar quantity for J, so we make the move or not depending on whether random RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1768844761
│ │ │ + -- setting random seed to 1769302664
│ │ │  
│ │ │ -o1 = 1768844761
    
│ │ │ +o1 = 1769302664
│ │ │              
    
│ │ │                
    i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │ @@ -147,15 +147,15 @@
│ │ │  
│ │ │  o7 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 3.90363s (cpu); 2.77034s (thread); 0s (gc)
    
│ │ │ + -- used 5.44845s (cpu); 3.38013s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,17 +35,17 @@
│ │ │ │  choose randomly from the union of these lists; if a socle element is chosen,
│ │ │ │  it's added to I; if a minimal generator is chosen, it's replaced by the square-
│ │ │ │  free part of the maximal ideal times it. the chance of making the given move is
│ │ │ │  then 1/(#ISocgens+#Igens), and the chance of making the move back would be the
│ │ │ │  similar quantity for J, so we make the move or not depending on whether random
│ │ │ │  RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1768844761
│ │ │ │ + -- setting random seed to 1769302664
│ │ │ │  
│ │ │ │ -o1 = 1768844761
│ │ │ │ +o1 = 1769302664
│ │ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : J = monomialIdeal"ab,ad, bcd"
│ │ │ │  
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  i7 : J = monomialIdeal 0_S
│ │ │ │  
│ │ │ │  o7 = monomialIdeal ()
│ │ │ │  
│ │ │ │  o7 : MonomialIdeal of S
│ │ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs
│ │ │ │  (J,AlexanderProbability => .01));
│ │ │ │ - -- used 3.90363s (cpu); 2.77034s (thread); 0s (gc)
│ │ │ │ + -- used 5.44845s (cpu); 3.38013s (thread); 0s (gc)
│ │ │ │  i9 : #T
│ │ │ │  
│ │ │ │  o9 = 168
│ │ │ │  i10 : T
│ │ │ │  
│ │ │ │  o10 = Tally{monomialIdeal () => 45                            }
│ │ │ │              monomialIdeal (a*b*c, a*b*d) => 33
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html
│ │ │ @@ -54,17 +54,17 @@
│ │ │          
    
│ │ │            
    This package can be used to make experiments, trying many ideals, perhaps over small fields. For example...what would you expect the regularities of "typical" monomial ideals with 10 generators of degree 3 in 6 variables to be? Try a bunch of examples -- it's fast. Here we do only 500 -- this takes about a second on a fast machine -- but with a little patience, thousands can be done conveniently.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -72,24 +72,22 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1768844752
│ │ │ + -- setting random seed to 1769302656
│ │ │  
│ │ │ -o1 = 1768844752
    
│ │ │ +o1 = 1769302656
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101;
    
│ │ │              
    
│ │ │                
    i3 : S=kk[vars(0..5)];
    
│ │ │              
    
│ │ │                
    i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 3.15015s (cpu); 1.82825s (thread); 0s (gc)
│ │ │ + -- used 4.46986s (cpu); 1.90648s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{3 => 1  }
│ │ │ -           4 => 48
│ │ │ -           5 => 204
│ │ │ -           6 => 167
│ │ │ -           7 => 65
│ │ │ -           8 => 13
│ │ │ +o4 = Tally{4 => 41 }
│ │ │ +           5 => 218
│ │ │ +           6 => 162
│ │ │ +           7 => 64
│ │ │ +           8 => 14
│ │ │             9 => 1
│ │ │ -           10 => 1
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
    
│ │ │          
    
│ │ │            
    How does this compare with the case of binomial ideals? or pure binomial ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger numbers of examples. Click the link "Finding Extreme Examples" below to see some other, more elaborate ways to search.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,30 +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 1768844752
│ │ │ │ + -- setting random seed to 1769302656
│ │ │ │  
│ │ │ │ -o1 = 1768844752
│ │ │ │ +o1 = 1769302656
│ │ │ │  i2 : kk=ZZ/101;
│ │ │ │  i3 : S=kk[vars(0..5)];
│ │ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ │ - -- used 3.15015s (cpu); 1.82825s (thread); 0s (gc)
│ │ │ │ + -- used 4.46986s (cpu); 1.90648s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -o4 = Tally{3 => 1  }
│ │ │ │ -           4 => 48
│ │ │ │ -           5 => 204
│ │ │ │ -           6 => 167
│ │ │ │ -           7 => 65
│ │ │ │ -           8 => 13
│ │ │ │ +o4 = Tally{4 => 41 }
│ │ │ │ +           5 => 218
│ │ │ │ +           6 => 162
│ │ │ │ +           7 => 64
│ │ │ │ +           8 => 14
│ │ │ │             9 => 1
│ │ │ │ -           10 => 1
│ │ │ │  
│ │ │ │  o4 : Tally
│ │ │ │  How does this compare with the case of binomial ideals? or pure binomial
│ │ │ │  ideals? We invite the reader to experiment, replacing "randomMonomialIdeal"
│ │ │ │  above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger
│ │ │ │  numbers of examples. Click the link "Finding Extreme Examples" below to see
│ │ │ │  some other, more elaborate ways to search.
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_dim__Via__Bezout.out
│ │ │ @@ -5,17 +5,17 @@
│ │ │  i2 : S=kk[y_0..y_8];
│ │ │  
│ │ │  i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2;
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 1.77411s elapsed
│ │ │ + -- 1.39485s elapsed
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : elapsedTime dim I
│ │ │ - -- 3.19442s elapsed
│ │ │ + -- 3.45196s 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)) );
│ │ │ - -- 2.03399s elapsed
│ │ │ + -- 1.70502s elapsed
│ │ │  
│ │ │  i11 : dim J
│ │ │  
│ │ │  o11 = 1
│ │ │  
│ │ │  i12 : i
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out
│ │ │ @@ -27,24 +27,24 @@
│ │ │  i6 : S=ZZ/103[y_0..y_30];
│ │ │  
│ │ │  i7 : I=minors(2,random(S^3,S^{3:-1}));
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 3.13643s elapsed
│ │ │ + -- 3.05751s elapsed
│ │ │  
│ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
│ │ │ - -- 2.23953s elapsed
│ │ │ + -- 2.10492s elapsed
│ │ │  
│ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │  
│ │ │  o9 : List
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html
│ │ │ @@ -95,23 +95,23 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
    
│ │ │                
    i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 1.77411s elapsed
│ │ │ + -- 1.39485s elapsed
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime dim I
│ │ │ - -- 3.19442s elapsed
│ │ │ + -- 3.45196s elapsed
│ │ │  
│ │ │  o5 = 4
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The user may set the MinimumFieldSize to ensure that the field being worked over is big enough.  For instance, there are relatively few linear spaces over a field of characteristic 2, and this can cause incorrect results to be provided.  If no size is provided, the function tries to guess a good size based on ambient ring.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,19 +32,19 @@
│ │ │ │  examples, the built in dim function is much faster.
│ │ │ │  i1 : kk=ZZ/101;
│ │ │ │  i2 : S=kk[y_0..y_8];
│ │ │ │  i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2;
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : elapsedTime dimViaBezout(I)
│ │ │ │ - -- 1.77411s elapsed
│ │ │ │ + -- 1.39485s elapsed
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : elapsedTime dim I
│ │ │ │ - -- 3.19442s elapsed
│ │ │ │ + -- 3.45196s elapsed
│ │ │ │  
│ │ │ │  o5 = 4
│ │ │ │  The user may set the MinimumFieldSize to ensure that the field being worked
│ │ │ │  over is big enough. For instance, there are relatively few linear spaces over a
│ │ │ │  field of characteristic 2, and this can cause incorrect results to be provided.
│ │ │ │  If no size is provided, the function tries to guess a good size based on
│ │ │ │  ambient ring.
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html
│ │ │ @@ -155,15 +155,15 @@
│ │ │  
│ │ │  o9 : Ideal of T
    │ │ │
    │ │ │ 
    i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.03399s elapsed
    │ │ │ + -- 1.70502s 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)) ); │ │ │ │ - -- 2.03399s elapsed │ │ │ │ + -- 1.70502s elapsed │ │ │ │ i11 : dim J │ │ │ │ │ │ │ │ o11 = 1 │ │ │ │ i12 : i │ │ │ │ │ │ │ │ o12 = 4 │ │ │ │ In this particular example, there tend to be about 5 associated primes when │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html │ │ │ @@ -144,27 +144,27 @@ │ │ │ │ │ │ o7 : Ideal of S │ │ │
    │ │ │ 
    i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 3.13643s elapsed
│ │ │ + -- 3.05751s elapsed
│ │ │  
│ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │  
│ │ │  o8 : List
    │ │ │
    │ │ │ 
    i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
│ │ │ - -- 2.23953s elapsed
│ │ │ + -- 2.10492s elapsed
│ │ │  
│ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │  
│ │ │  o9 : List
    │ │ │
    │ │ │ 
    i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.600555s (cpu); 0.418226s (thread); 0s (gc)
│ │ │ + -- used 0.877538s (cpu); 0.480262s (thread); 0s (gc)
│ │ │  
│ │ │                                  125   124      120 5    124    100 25     104 20       108 15 2      112 10 3     116 5 4    120 5    124      125      4 120        8 115 2        12 110 3         16 105 4         20 100 5          24 95 6          28 90 7           32 85 8           36 80 9           40 75 10           44 70 11           48 65 12           52 60 13           56 55 14           60 50 15           64 45 16           68 40 17          72 35 18          76 30 19         80 25 20         84 20 21        88 15 22       92 10 23      96 5 24    100 25     24 100        28 95          32 90 2         36 85 3          40 80 4          44 75 5           48 70 6           52 65 7           56 60 8           60 55 9           64 50 10           68 45 11           72 40 12           76 35 13           80 30 14          84 25 15          88 20 16         92 15 17        96 10 18        100 5 19      104 20       48 75 2       52 70   2        56 65 2 2        60 60 3 2         64 55 4 2         68 50 5 2         72 45 6 2         76 40 7 2         80 35 8 2         84 30 9 2         88 25 10 2         92 20 11 2        96 15 12 2        100 10 13 2       104 5 14 2      108 15 2      72 50 3       76 45   3       80 40 2 3        84 35 3 3        88 30 4 3        92 25 5 3        96 20 6 3        100 15 7 3       104 10 8 3       108 5 9 3      112 10 3     96 25 4      100 20   4      104 15 2 4      108 10 3 4      112 5 4 4     116 5 4    120 5    124
│ │ │  o14 = Proj Q - - - > Proj Q   {x   , x   y, - x   y  + x   z, x   y   - 5x   y  z + 10x   y  z  - 10x   y  z  + 5x   y z  - x   z  + x   t, - y    + 25x y   z - 300x y   z  + 2300x  y   z  - 12650x  y   z  + 53130x  y   z  - 177100x  y  z  + 480700x  y  z  - 1081575x  y  z  + 2042975x  y  z  - 3268760x  y  z   + 4457400x  y  z   - 5200300x  y  z   + 5200300x  y  z   - 4457400x  y  z   + 3268760x  y  z   - 2042975x  y  z   + 1081575x  y  z   - 480700x  y  z   + 177100x  y  z   - 53130x  y  z   + 12650x  y  z   - 2300x  y  z   + 300x  y  z   - 25x  y z   + x   z   - 5x  y   t + 100x  y  z*t - 950x  y  z t + 5700x  y  z t - 24225x  y  z t + 77520x  y  z t - 193800x  y  z t + 387600x  y  z t - 629850x  y  z t + 839800x  y  z t - 923780x  y  z  t + 839800x  y  z  t - 629850x  y  z  t + 387600x  y  z  t - 193800x  y  z  t + 77520x  y  z  t - 24225x  y  z  t + 5700x  y  z  t - 950x  y  z  t + 100x   y z  t - 5x   z  t - 10x  y  t  + 150x  y  z*t  - 1050x  y  z t  + 4550x  y  z t  - 13650x  y  z t  + 30030x  y  z t  - 50050x  y  z t  + 64350x  y  z t  - 64350x  y  z t  + 50050x  y  z t  - 30030x  y  z  t  + 13650x  y  z  t  - 4550x  y  z  t  + 1050x   y  z  t  - 150x   y z  t  + 10x   z  t  - 10x  y  t  + 100x  y  z*t  - 450x  y  z t  + 1200x  y  z t  - 2100x  y  z t  + 2520x  y  z t  - 2100x  y  z t  + 1200x   y  z t  - 450x   y  z t  + 100x   y z t  - 10x   z  t  - 5x  y  t  + 25x   y  z*t  - 50x   y  z t  + 50x   y  z t  - 25x   y z t  + 5x   z t  - x   t  + x   u}
│ │ │  
│ │ │  o14 : RationalMapping
    │ │ │
    │ │ │ 
    i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.00328519s (cpu); 0.00328099s (thread); 0s (gc)
│ │ │ + -- used 0.00434048s (cpu); 0.00434084s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
    │ │ │
    │ │ │
    │ │ │

    Over number fields

    │ │ │ @@ -348,15 +348,15 @@ │ │ │ o31 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 0.892058s (cpu); 0.754813s (thread); 0s (gc)
    │ │ │ + -- used 1.25853s (cpu); 0.961587s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i33 : #oo
│ │ │  
│ │ │  o33 = 31
    │ │ │ @@ -373,15 +373,15 @@ │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.285089s (cpu); 0.224588s (thread); 0s (gc)
│ │ │ + -- used 0.366938s (cpu); 0.256372s (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 0.00328519s (cpu); 0.00328099s (thread); 0s (gc) │ │ │ │ + -- used 0.00434048s (cpu); 0.00434084s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = 110462212541120451001 │ │ │ │ ****** OOvveerr nnuummbbeerr ffiieellddss ****** │ │ │ │ Over a number field one can use the option Bound to specify a maximal │ │ │ │ multiplicative height given by $(x_0:\dots:x_n)\mapsto \prod_{v}\max_i|x_i|_v ^ │ │ │ │ {d_v/d}$ (this is also available as a method _g_l_o_b_a_l_H_e_i_g_h_t). │ │ │ │ i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z); │ │ │ │ @@ -197,24 +197,24 @@ │ │ │ │ │ │ │ │ o30 : Ideal of R │ │ │ │ i31 : nodes = I + ideal jacobian I; │ │ │ │ │ │ │ │ o31 : Ideal of R │ │ │ │ i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true); │ │ │ │ -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25) │ │ │ │ - -- used 0.892058s (cpu); 0.754813s (thread); 0s (gc) │ │ │ │ + -- used 1.25853s (cpu); 0.961587s (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 0.285089s (cpu); 0.224588s (thread); 0s (gc) │ │ │ │ + -- used 0.366938s (cpu); 0.256372s (thread); 0s (gc) │ │ │ │ │ │ │ │ o35 = 31 │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ For a number field other than QQ, the enumeration of elements with bounded │ │ │ │ height depends on an algorithm by Doyle–Krumm, which is currently only │ │ │ │ implemented in Sage. │ │ │ │ ******** MMeennuu ******** │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/___Plane__Curve__Singularities.out │ │ │ @@ -331,15 +331,15 @@ │ │ │ 2 2 2 2 2 2 2 │ │ │ - p w , p y - p , p w y - p p , p w - p ) │ │ │ 2 1 0 1 0 0 1 2 0 0 2 │ │ │ │ │ │ o47 : Ideal of B2 │ │ │ │ │ │ i48 : time sing2 = ideal singularLocus strictTransform2; │ │ │ - -- used 0.835698s (cpu); 0.720059s (thread); 0s (gc) │ │ │ + -- used 1.12629s (cpu); 0.902991s (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 1.06546s (cpu); 0.804885s (thread); 0s (gc) │ │ │ + -- used 1.25038s (cpu); 0.911888s (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 1.51157s (cpu); 1.25413s (thread); 0s (gc) │ │ │ + -- used 1.82255s (cpu); 1.52027s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ 0 3 │ │ │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ - -- used 1.49227s (cpu); 1.22516s (thread); 0s (gc) │ │ │ + -- used 2.07313s (cpu); 1.66681s (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 0.0293844s (cpu); 0.0272978s (thread); 0s (gc) │ │ │ + -- used 0.241136s (cpu); 0.0593634s (thread); 0s (gc) │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ 0 6 │ │ │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ - -- used 0.118499s (cpu); 0.117729s (thread); 0s (gc) │ │ │ + -- used 0.146926s (cpu); 0.134634s (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 0.0201833s (cpu); 0.019083s (thread); 0s (gc) │ │ │ + -- used 0.0806105s (cpu); 0.0263398s (thread); 0s (gc) │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ - -- used 0.00813817s (cpu); 0.00786665s (thread); 0s (gc) │ │ │ + -- used 0.0218241s (cpu); 0.00959479s (thread); 0s (gc) │ │ │ │ │ │ o10 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i11 : transpose gens I1 │ │ │ │ │ │ o11 = {-1, -3} | aw_1-bw_2 | │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/___Plane__Curve__Singularities.html │ │ │ @@ -587,15 +587,15 @@ │ │ │
    │ │ │

    We compute the singular locus once again:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -325,15 +325,15 @@ │ │ │ │ 2 2 2 2 2 2 2 │ │ │ │ - p w , p y - p , p w y - p p , p w - p ) │ │ │ │ 2 1 0 1 0 0 1 2 0 0 2 │ │ │ │ │ │ │ │ o47 : Ideal of B2 │ │ │ │ We compute the singular locus once again: │ │ │ │ i48 : time sing2 = ideal singularLocus strictTransform2; │ │ │ │ - -- used 0.835698s (cpu); 0.720059s (thread); 0s (gc) │ │ │ │ + -- used 1.12629s (cpu); 0.902991s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o48 : Ideal of -----[p ..p , w ..w , x..y] │ │ │ │ 32003 0 2 0 1 │ │ │ │ i49 : saturate(sing2, sub(irrelTot, ring sing2)) │ │ │ │ │ │ │ │ o49 = ideal 1 │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_expected__Rees__Ideal.html │ │ │ @@ -151,15 +151,15 @@ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 0.835698s (cpu); 0.720059s (thread); 0s (gc)
│ │ │ + -- used 1.12629s (cpu); 0.902991s (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 1.06546s (cpu); 0.804885s (thread); 0s (gc)
│ │ │ + -- used 1.25038s (cpu); 0.911888s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
    │ │ │
    │ │ │ @@ -185,24 +185,24 @@ │ │ │ │ │ │ o11 : Ideal of S │ │ │
    │ │ │ 
    i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.51157s (cpu); 1.25413s (thread); 0s (gc)
│ │ │ + -- used 1.82255s (cpu); 1.52027s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
    │ │ │
    │ │ │ 
    i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 1.49227s (cpu); 1.22516s (thread); 0s (gc)
│ │ │ + -- used 2.07313s (cpu); 1.66681s (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 1.06546s (cpu); 0.804885s (thread); 0s (gc) │ │ │ │ + -- used 1.25038s (cpu); 0.911888s (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 1.51157s (cpu); 1.25413s (thread); 0s (gc) │ │ │ │ + -- used 1.82255s (cpu); 1.52027s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ │ - -- used 1.49227s (cpu); 1.22516s (thread); 0s (gc) │ │ │ │ + -- used 2.07313s (cpu); 1.66681s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_y_m_m_e_t_r_i_c_A_l_g_e_b_r_a_I_d_e_a_l -- Ideal of the symmetric algebra of an ideal or │ │ │ │ module │ │ │ │ * _j_a_c_o_b_i_a_n_D_u_a_l -- Computes the 'jacobian dual', part of a method of finding │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal.html │ │ │ @@ -110,24 +110,24 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0293844s (cpu); 0.0272978s (thread); 0s (gc)
│ │ │ + -- used 0.241136s (cpu); 0.0593634s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.118499s (cpu); 0.117729s (thread); 0s (gc)
│ │ │ + -- used 0.146926s (cpu); 0.134634s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ @@ -164,24 +164,24 @@ │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.0201833s (cpu); 0.019083s (thread); 0s (gc)
│ │ │ + -- used 0.0806105s (cpu); 0.0263398s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00813817s (cpu); 0.00786665s (thread); 0s (gc)
│ │ │ + -- used 0.0218241s (cpu); 0.00959479s (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 0.0293844s (cpu); 0.0272978s (thread); 0s (gc) │ │ │ │ + -- used 0.241136s (cpu); 0.0593634s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ │ 0 6 │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ │ - -- used 0.118499s (cpu); 0.117729s (thread); 0s (gc) │ │ │ │ + -- used 0.146926s (cpu); 0.134634s (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 0.0201833s (cpu); 0.019083s (thread); 0s (gc) │ │ │ │ + -- used 0.0806105s (cpu); 0.0263398s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ │ 0 2 │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ │ - -- used 0.00813817s (cpu); 0.00786665s (thread); 0s (gc) │ │ │ │ + -- used 0.0218241s (cpu); 0.00959479s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 : Ideal of S[w ..w ] │ │ │ │ 0 2 │ │ │ │ i11 : transpose gens I1 │ │ │ │ │ │ │ │ o11 = {-1, -3} | aw_1-bw_2 | │ │ │ │ {-1, -3} | aw_0-bw_1 | │ │ ├── ./usr/share/doc/Macaulay2/Regularity/example-output/_m__Regularity.out │ │ │ @@ -71,15 +71,15 @@ │ │ │ x x x , x + x x - x x - x x x , x + x - x x ) │ │ │ 0 1 3 0 0 1 1 2 0 2 5 0 2 0 5 │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ │ │ i8 : benchmark "mRegularity I1" │ │ │ │ │ │ -o8 = .2701566191999999 │ │ │ +o8 = .2991554037 │ │ │ │ │ │ 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 = .07251379724285713 │ │ │ +o11 = .1054410219833333 │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ │ │ i12 : time regularity I2 │ │ │ - -- used 0.0025403s (cpu); 0.00254037s (thread); 0s (gc) │ │ │ + -- used 0.00244193s (cpu); 0.00244612s (thread); 0s (gc) │ │ │ │ │ │ o12 = 4 │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html │ │ │ @@ -176,15 +176,15 @@ │ │ │ o7 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .2701566191999999
│ │ │ +o8 = .2991554037
│ │ │  
│ │ │  o8 : RR (of precision 53)
    │ │ │ │ │ │ │ │ │ │ │ │

    This is an example where regularity is faster than mRegularity.

    │ │ │ │ │ │ @@ -204,23 +204,23 @@ │ │ │ o10 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .07251379724285713
│ │ │ +o11 = .1054410219833333
│ │ │  
│ │ │  o11 : RR (of precision 53)
    │ │ │
    │ │ │ 
    i12 : time regularity I2  
│ │ │ - -- used 0.0025403s (cpu); 0.00254037s (thread); 0s (gc)
│ │ │ + -- used 0.00244193s (cpu); 0.00244612s (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 = .2701566191999999 │ │ │ │ +o8 = .2991554037 │ │ │ │ │ │ │ │ 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 = .07251379724285713 │ │ │ │ +o11 = .1054410219833333 │ │ │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ i12 : time regularity I2 │ │ │ │ - -- used 0.0025403s (cpu); 0.00254037s (thread); 0s (gc) │ │ │ │ + -- used 0.00244193s (cpu); 0.00244612s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 = 4 │ │ │ │ This symbol is provided by the package Regularity. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_g_u_l_a_r_i_t_y -- compute the Castelnuovo-Mumford regularity │ │ │ │ ********** WWaayyss ttoo uussee mmRReegguullaarriittyy:: ********** │ │ │ │ * mRegularity(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out │ │ │ @@ -5,18 +5,18 @@ │ │ │ o2 = ideal(x x - x x ) │ │ │ 0 1 2 3 │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2); │ │ │ - -- used 0.147778s (cpu); 0.0902586s (thread); 0s (gc) │ │ │ + -- used 0.179957s (cpu); 0.0873078s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1) │ │ │ - -- used 0.172349s (cpu); 0.114048s (thread); 0s (gc) │ │ │ + -- used 0.181685s (cpu); 0.0954323s (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 0.211459s (cpu); 0.150871s (thread); 0s (gc) │ │ │ + -- used 0.20818s (cpu); 0.116486s (thread); 0s (gc) │ │ │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ - -- used 0.205754s (cpu); 0.146078s (thread); 0s (gc) │ │ │ + -- used 0.200054s (cpu); 0.124723s (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 0.149342s (cpu); 0.0893238s (thread); 0s (gc) │ │ │ + -- used 0.147666s (cpu); 0.0717063s (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 0.136163s (cpu); 0.0799244s (thread); 0s (gc) │ │ │ + -- used 0.142257s (cpu); 0.0641754s (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 0.150798s (cpu); 0.101925s (thread); 0s (gc) │ │ │ + -- used 0.316488s (cpu); 0.138719s (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 0.101247s (cpu); 0.0695226s (thread); 0s (gc) │ │ │ + -- used 0.189326s (cpu); 0.101494s (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 4.95821s (cpu); 4.5338s (thread); 0s (gc) │ │ │ + -- used 5.66779s (cpu); 5.18221s (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 1.13682s (cpu); 1.01243s (thread); 0s (gc) │ │ │ + -- used 1.3226s (cpu); 1.23947s (thread); 0s (gc) │ │ │ │ │ │ i5 : -- X-resultant of V │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ - -- used 0.258379s (cpu); 0.194024s (thread); 0s (gc) │ │ │ + -- used 0.335012s (cpu); 0.253339s (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 0.302271s (cpu); 0.184875s (thread); 0s (gc) │ │ │ + -- used 0.329183s (cpu); 0.251721s (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 0.00906152s (cpu); 0.00905929s (thread); 0s (gc) │ │ │ + -- used 0.0105882s (cpu); 0.010588s (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 0.0098473s (cpu); 0.00984721s (thread); 0s (gc) │ │ │ + -- used 0.0119901s (cpu); 0.0119895s (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 0.485638s (cpu); 0.44254s (thread); 0s (gc) │ │ │ + -- used 0.542725s (cpu); 0.456703s (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 0.15704s (cpu); 0.116947s (thread); 0s (gc) │ │ │ + -- used 0.24882s (cpu); 0.168374s (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 0.184837s (cpu); 0.143446s (thread); 0s (gc) │ │ │ + -- used 0.245389s (cpu); 0.173599s (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 0.0575167s (cpu); 0.0575175s (thread); 0s (gc) │ │ │ + -- used 0.0715067s (cpu); 0.0715043s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ 3331 0 9 │ │ │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ - -- used 0.607829s (cpu); 0.556866s (thread); 0s (gc) │ │ │ + -- used 0.787643s (cpu); 0.695145s (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 0.116846s (cpu); 0.0580032s (thread); 0s (gc) │ │ │ + -- used 0.153633s (cpu); 0.0735869s (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 0.0428008s (cpu); 0.0428043s (thread); 0s (gc) │ │ │ + -- used 0.0691325s (cpu); 0.0689578s (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 0.0197034s (cpu); 0.019704s (thread); 0s (gc) │ │ │ + -- used 0.0250117s (cpu); 0.0250093s (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 0.0176275s (cpu); 0.0176287s (thread); 0s (gc) │ │ │ + -- used 0.0232976s (cpu); 0.0233023s (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 0.0387325s (cpu); 0.0387333s (thread); 0s (gc) │ │ │ + -- used 0.0457735s (cpu); 0.0455342s (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 0.00734253s (cpu); 0.00734082s (thread); 0s (gc) │ │ │ + -- used 0.0106714s (cpu); 0.0106719s (thread); 0s (gc) │ │ │ │ │ │ o2 = true │ │ │ │ │ │ i3 : -- random quadric in G(1,3) │ │ │ w' = random(2,Grass(1,3)) │ │ │ │ │ │ 2 5 10 2 2 2 3 │ │ │ @@ -56,12 +56,12 @@ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ o3 : -------------------------------------- │ │ │ p p - p p + p p │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ │ │ i4 : time isCoisotropic w' │ │ │ - -- used 0.00676804s (cpu); 0.00676823s (thread); 0s (gc) │ │ │ + -- used 0.00847863s (cpu); 0.00847095s (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 0.0191452s (cpu); 0.0191451s (thread); 0s (gc) │ │ │ + -- used 0.0241718s (cpu); 0.0241721s (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 0.00384623s (cpu); 0.00384331s (thread); 0s (gc) │ │ │ + -- used 0.0041554s (cpu); 0.00415075s (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 0.00243197s (cpu); 0.00243231s (thread); 0s (gc) │ │ │ + -- used 0.00286655s (cpu); 0.00286359s (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 0.00422213s (cpu); 0.00421985s (thread); 0s (gc) │ │ │ + -- used 0.0063695s (cpu); 0.00636948s (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 0.101398s (cpu); 0.0516067s (thread); 0s (gc) │ │ │ + -- used 0.146792s (cpu); 0.0657964s (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 0.0291609s (cpu); 0.0291643s (thread); 0s (gc) │ │ │ + -- used 0.0493136s (cpu); 0.0493166s (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 0.141574s (cpu); 0.141579s (thread); 0s (gc) │ │ │ + -- used 0.176165s (cpu); 0.176174s (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 0.24907s (cpu); 0.172207s (thread); 0s (gc) │ │ │ + -- used 0.395536s (cpu); 0.221719s (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 0.115275s (cpu); 0.0788707s (thread); 0s (gc) │ │ │ + -- used 0.217093s (cpu); 0.132268s (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 0.440608s (cpu); 0.414364s (thread); 0s (gc) │ │ │ + -- used 0.424916s (cpu); 0.340956s (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 0.605546s (cpu); 0.545555s (thread); 0s (gc) │ │ │ + -- used 0.665423s (cpu); 0.578527s (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 0.0233567s (cpu); 0.0233554s (thread); 0s (gc) │ │ │ + -- used 0.02827s (cpu); 0.0282695s (thread); 0s (gc) │ │ │ │ │ │ 12 11 2 10 3 9 4 8 5 7 6 │ │ │ o3 = - 81t u - 1701t u - 15309t u - 76545t u - 229635t u - 413343t u │ │ │ ------------------------------------------------------------------------ │ │ │ 6 7 5 8 11 10 2 9 3 │ │ │ - 413343t u - 177147t u + 567t u + 10206t u + 76545t u + │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -64,15 +64,15 @@ │ │ │ 3 │ │ │ + c x } │ │ │ 9 2 │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : time resultant F │ │ │ - -- used 2.47769s (cpu); 1.89799s (thread); 0s (gc) │ │ │ + -- used 2.6838s (cpu); 2.0709s (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 0.566774s (cpu); 0.42434s (thread); 0s (gc) │ │ │ + -- used 0.510655s (cpu); 0.424536s (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 0.027275s (cpu); 0.0272754s (thread); 0s (gc) │ │ │ + -- used 0.0366946s (cpu); 0.036695s (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 0.0958885s (cpu); 0.0634772s (thread); 0s (gc) │ │ │ + -- used 0.150481s (cpu); 0.0845254s (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 0.032136s (cpu); 0.0321388s (thread); 0s (gc) │ │ │ + -- used 0.0468657s (cpu); 0.0468702s (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 0.0981634s (cpu); 0.0520542s (thread); 0s (gc) │ │ │ + -- used 0.142387s (cpu); 0.0718118s (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 0.149774s (cpu); 0.100749s (thread); 0s (gc) │ │ │ + -- used 0.186643s (cpu); 0.116899s (thread); 0s (gc) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ @@ -86,24 +86,24 @@ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.147778s (cpu); 0.0902586s (thread); 0s (gc)
    │ │ │ + -- used 0.179957s (cpu); 0.0873078s (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 0.172349s (cpu); 0.114048s (thread); 0s (gc)
│ │ │ + -- used 0.181685s (cpu); 0.0954323s (thread); 0s (gc)
│ │ │  
│ │ │                                                                             
│ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3 
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                              
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ @@ -130,26 +130,26 @@
│ │ │
    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,20 +38,20 @@ │ │ │ │ │ │ │ │ o2 = ideal(x x - x x ) │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ │ 0 3 │ │ │ │ i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2); │ │ │ │ - -- used 0.147778s (cpu); 0.0902586s (thread); 0s (gc) │ │ │ │ + -- used 0.179957s (cpu); 0.0873078s (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 0.172349s (cpu); 0.114048s (thread); 0s (gc) │ │ │ │ + -- used 0.181685s (cpu); 0.0954323s (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 0.211459s (cpu); 0.150871s (thread); 0s (gc) │ │ │ │ + -- used 0.20818s (cpu); 0.116486s (thread); 0s (gc) │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ │ - -- used 0.205754s (cpu); 0.146078s (thread); 0s (gc) │ │ │ │ + -- used 0.200054s (cpu); 0.124723s (thread); 0s (gc) │ │ │ │ i8 : assert(oo == (P1xP1xP2,P1xP1xP2')) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_u_a_l_V_a_r_i_e_t_y -- projective dual variety │ │ │ │ ********** WWaayyss ttoo uussee ccaayylleeyyTTrriicckk:: ********** │ │ │ │ * cayleyTrick(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_a_y_l_e_y_T_r_i_c_k is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ @@ -90,15 +90,15 @@ │ │ │ o2 : Ideal of P3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.211459s (cpu); 0.150871s (thread); 0s (gc)
    │ │ │ + -- used 0.20818s (cpu); 0.116486s (thread); 0s (gc) │ │ │
    │ │ │ 
    i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
    │ │ │
    │ │ │ 
    i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.205754s (cpu); 0.146078s (thread); 0s (gc)
    │ │ │ + -- used 0.200054s (cpu); 0.124723s (thread); 0s (gc) │ │ │
    │ │ │ 
    i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
    │ │ │
    │ │ │ 
    i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.149342s (cpu); 0.0893238s (thread); 0s (gc)
│ │ │ + -- used 0.147666s (cpu); 0.0717063s (thread); 0s (gc)
│ │ │  
│ │ │               2 2    2 2    2 2    4                2      2 2   2      
│ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x 
│ │ │ @@ -162,15 +162,15 @@
│ │ │  o5 : Ideal of P3
    │ │ │
    │ │ │ 
    i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.136163s (cpu); 0.0799244s (thread); 0s (gc)
│ │ │ + -- used 0.142257s (cpu); 0.0641754s (thread); 0s (gc)
│ │ │  
│ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2 
│ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3 
│ │ │       ------------------------------------------------------------------------
│ │ │            2      3 2   3        3 2   4         2         2 2       3    
│ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ @@ -222,30 +222,30 @@
│ │ │  o9 : Ideal of P3
    │ │ │
    │ │ │ 
    i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.150798s (cpu); 0.101925s (thread); 0s (gc)
│ │ │ + -- used 0.316488s (cpu); 0.138719s (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 0.101247s (cpu); 0.0695226s (thread); 0s (gc)
│ │ │ + -- used 0.189326s (cpu); 0.101494s (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 0.149342s (cpu); 0.0893238s (thread); 0s (gc) │ │ │ │ + -- used 0.147666s (cpu); 0.0717063s (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 0.136163s (cpu); 0.0799244s (thread); 0s (gc) │ │ │ │ + -- used 0.142257s (cpu); 0.0641754s (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 0.150798s (cpu); 0.101925s (thread); 0s (gc) │ │ │ │ + -- used 0.316488s (cpu); 0.138719s (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 0.101247s (cpu); 0.0695226s (thread); 0s (gc) │ │ │ │ + -- used 0.189326s (cpu); 0.101494s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = true │ │ │ │ Note that chowEquations(W,0) is not the same as chowEquations W. │ │ │ │ ********** WWaayyss ttoo uussee cchhoowwEEqquuaattiioonnss:: ********** │ │ │ │ * chowEquations(RingElement) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_h_o_w_E_q_u_a_t_i_o_n_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ 3331 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian)
│ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ - -- used 4.95821s (cpu); 4.5338s (thread); 0s (gc)
│ │ │ + -- used 5.66779s (cpu); 5.18221s (thread); 0s (gc)
│ │ │  
│ │ │        4               2              2     2               2            
│ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │        1,2,4     0,2,4 1,2,4 2,3,4    0,2,4 2,3,4     1,2,3 1,2,4 1,2,5  
│ │ │       ------------------------------------------------------------------------
│ │ │        2     2              2                                       
│ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ @@ -227,22 +227,22 @@
│ │ │         2,3,5 1,4,5    1,3,5 2,4,5    1,2,5 3,4,5   2,3,4 1,4,5    1,3,4 2,4,5    1,2,4 3,4,5   2,3,5 0,4,5    0,3,5 2,4,5    0,2,5 3,4,5   1,3,5 0,4,5    0,3,5 1,4,5    0,1,5 3,4,5   1,2,5 0,4,5    0,2,5 1,4,5    0,1,5 2,4,5   2,3,4 0,4,5    0,3,4 2,4,5    0,2,4 3,4,5   1,3,4 0,4,5    0,3,4 1,4,5    0,1,4 3,4,5   1,2,4 0,4,5    0,2,4 1,4,5    0,1,4 2,4,5   1,2,3 0,4,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 1,3,5    1,3,4 2,3,5    1,2,3 3,4,5   1,2,5 0,3,5    0,2,5 1,3,5    0,1,5 2,3,5   2,3,4 0,3,5    0,3,4 2,3,5    0,2,3 3,4,5   1,3,4 0,3,5    0,3,4 1,3,5    0,1,3 3,4,5   1,2,4 0,3,5    0,2,4 1,3,5    0,1,4 2,3,5    0,1,2 3,4,5   1,2,3 0,3,5    0,2,3 1,3,5    0,1,3 2,3,5   2,3,4 1,2,5    1,2,4 2,3,5    1,2,3 2,4,5   1,3,4 1,2,5    1,2,4 1,3,5    1,2,3 1,4,5   0,3,4 1,2,5    0,2,4 1,3,5    0,1,4 2,3,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 0,2,5    0,2,4 2,3,5    0,2,3 2,4,5   1,3,4 0,2,5    0,2,4 1,3,5    0,2,3 1,4,5    0,1,2 3,4,5   0,3,4 0,2,5    0,2,4 0,3,5    0,2,3 0,4,5   1,2,4 0,2,5    0,2,4 1,2,5    0,1,2 2,4,5   1,2,3 0,2,5    0,2,3 1,2,5    0,1,2 2,3,5   2,3,4 0,1,5    0,1,4 2,3,5    0,1,3 2,4,5    0,1,2 3,4,5   1,3,4 0,1,5    0,1,4 1,3,5    0,1,3 1,4,5   0,3,4 0,1,5    0,1,4 0,3,5    0,1,3 0,4,5   1,2,4 0,1,5    0,1,4 1,2,5    0,1,2 1,4,5   0,2,4 0,1,5    0,1,4 0,2,5    0,1,2 0,4,5   1,2,3 0,1,5    0,1,3 1,2,5    0,1,2 1,3,5   0,2,3 0,1,5    0,1,3 0,2,5    0,1,2 0,3,5   1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : -- equivalently (but faster)...
│ │ │       time assert(ChowV === chowForm f)
│ │ │ - -- used 1.13682s (cpu); 1.01243s (thread); 0s (gc)
    │ │ │ + -- used 1.3226s (cpu); 1.23947s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.258379s (cpu); 0.194024s (thread); 0s (gc)
    │ │ │ + -- used 0.335012s (cpu); 0.253339s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │ @@ -257,15 +257,15 @@
│ │ │  o6 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.302271s (cpu); 0.184875s (thread); 0s (gc)
    │ │ │ + -- used 0.329183s (cpu); 0.251721s (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 4.95821s (cpu); 4.5338s (thread); 0s (gc) │ │ │ │ + -- used 5.66779s (cpu); 5.18221s (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 1.13682s (cpu); 1.01243s (thread); 0s (gc) │ │ │ │ + -- used 1.3226s (cpu); 1.23947s (thread); 0s (gc) │ │ │ │ i5 : -- X-resultant of V │ │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ │ - -- used 0.258379s (cpu); 0.194024s (thread); 0s (gc) │ │ │ │ + -- used 0.335012s (cpu); 0.253339s (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 0.302271s (cpu); 0.184875s (thread); 0s (gc) │ │ │ │ + -- used 0.329183s (cpu); 0.251721s (thread); 0s (gc) │ │ │ │ i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring │ │ │ │ Xres)) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety │ │ │ │ * _h_u_r_w_i_t_z_F_o_r_m -- Hurwitz form of a projective variety │ │ │ │ ********** WWaayyss ttoo uussee cchhoowwFFoorrmm:: ********** │ │ │ │ * chowForm(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ @@ -83,15 +83,15 @@ │ │ │ │ │ │ o2 : ZZ[a..c][x..y] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time discriminant F
│ │ │ - -- used 0.00906152s (cpu); 0.00905929s (thread); 0s (gc)
│ │ │ + -- used 0.0105882s (cpu); 0.010588s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
    │ │ │ │ │ │ │ │ │ @@ -104,15 +104,15 @@ │ │ │ │ │ │ o5 : ZZ[a..d][x..y] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time discriminant F
│ │ │ - -- used 0.0098473s (cpu); 0.00984721s (thread); 0s (gc)
│ │ │ + -- used 0.0119901s (cpu); 0.0119895s (thread); 0s (gc)
│ │ │  
│ │ │          2 2       3     3                   2 2
│ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │  
│ │ │  o6 : ZZ[a..d]
    │ │ │ │ │ │ │ │ │ @@ -165,15 +165,15 @@ │ │ │ │ │ │ o12 : R' │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : time D=discriminant pencil
│ │ │ - -- used 0.485638s (cpu); 0.44254s (thread); 0s (gc)
│ │ │ + -- used 0.542725s (cpu); 0.456703s (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 0.00906152s (cpu); 0.00905929s (thread); 0s (gc)
│ │ │ │ + -- used 0.0105882s (cpu); 0.010588s (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 0.0098473s (cpu); 0.00984721s (thread); 0s (gc)
│ │ │ │ + -- used 0.0119901s (cpu); 0.0119895s (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 0.485638s (cpu); 0.44254s (thread); 0s (gc)
│ │ │ │ + -- used 0.542725s (cpu); 0.456703s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             108      106 2       102 6      100 8       98 10       96 12
│ │ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │ │             0        0   1       0   1      0   1       0  1        0  1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            94 14       92 16      90 18      88 20      86 22       84 24
│ │ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html
│ │ │ @@ -90,28 +90,28 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time V' = dualVariety V
│ │ │ - -- used 0.15704s (cpu); 0.116947s (thread); 0s (gc)
│ │ │ + -- used 0.24882s (cpu); 0.168374s (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 0.184837s (cpu); 0.143446s (thread); 0s (gc)
│ │ │ + -- used 0.245389s (cpu); 0.173599s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │

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

    │ │ │ │ │ │ @@ -131,25 +131,25 @@ │ │ │ o4 : ----[a ..a ][x ..x ] │ │ │ 3331 0 9 0 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -31,24 +31,24 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ 0 3 │ │ │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ │ 0 5 │ │ │ │ i2 : time V' = dualVariety V │ │ │ │ - -- used 0.15704s (cpu); 0.116947s (thread); 0s (gc) │ │ │ │ + -- used 0.24882s (cpu); 0.168374s (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 0.184837s (cpu); 0.143446s (thread); 0s (gc) │ │ │ │ + -- used 0.245389s (cpu); 0.173599s (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 0.0575167s (cpu); 0.0575175s (thread); 0s (gc) │ │ │ │ + -- used 0.0715067s (cpu); 0.0715043s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ │ 3331 0 9 │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ │ - -- used 0.607829s (cpu); 0.556866s (thread); 0s (gc) │ │ │ │ + -- used 0.787643s (cpu); 0.695145s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o6 : Ideal of ----[x ..x ] │ │ │ │ 3331 0 9 │ │ │ │ i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z) │ │ │ │ │ │ │ │ o7 = true │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html │ │ │ @@ -85,15 +85,15 @@ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.0575167s (cpu); 0.0575175s (thread); 0s (gc)
│ │ │ + -- used 0.0715067s (cpu); 0.0715043s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
    │ │ │
    │ │ │ 
    i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.607829s (cpu); 0.556866s (thread); 0s (gc)
│ │ │ + -- used 0.787643s (cpu); 0.695145s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
    │ │ │
    │ │ │ 
    i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.116846s (cpu); 0.0580032s (thread); 0s (gc)
│ │ │ + -- used 0.153633s (cpu); 0.0735869s (thread); 0s (gc)
│ │ │  
│ │ │          3   3          2   2              2       2          2   3    
│ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │          0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        2       2               2   2                                   
│ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ @@ -138,15 +138,15 @@
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
    │ │ │
    │ │ │ 
    i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.0428008s (cpu); 0.0428043s (thread); 0s (gc)
│ │ │ + -- used 0.0691325s (cpu); 0.0689578s (thread); 0s (gc)
│ │ │  
│ │ │              3          2                         2                        
│ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │          0,3 1,2    0,2 1,2 1,3    0,2 0,3 1,2    0,2 1,3     0,3 1,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │             2      3      2
│ │ │       2x   x    + x    + x
│ │ │ @@ -179,15 +179,15 @@
│ │ │
    		│ │ │ 
    i5 : w = chowForm C;
    │ │ │
    │ │ │ 
    i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.0197034s (cpu); 0.019704s (thread); 0s (gc)
│ │ │ + -- used 0.0250117s (cpu); 0.0250093s (thread); 0s (gc)
│ │ │  
│ │ │                     3          2          3                       2        
│ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2      2            2   3               2        
│ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ @@ -227,15 +227,15 @@
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.0176275s (cpu); 0.0176287s (thread); 0s (gc)
│ │ │ + -- used 0.0232976s (cpu); 0.0233023s (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 0.116846s (cpu); 0.0580032s (thread); 0s (gc) │ │ │ │ + -- used 0.153633s (cpu); 0.0735869s (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 0.0428008s (cpu); 0.0428043s (thread); 0s (gc) │ │ │ │ + -- used 0.0691325s (cpu); 0.0689578s (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 0.0197034s (cpu); 0.019704s (thread); 0s (gc) │ │ │ │ + -- used 0.0250117s (cpu); 0.0250093s (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 0.0176275s (cpu); 0.0176287s (thread); 0s (gc) │ │ │ │ + -- used 0.0232976s (cpu); 0.0233023s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {2, 2, 2, 2, 2, 2} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** WWaayyss ttoo uussee ffrroommPPlluucckkeerrTTooSSttiieeffeell:: ********** │ │ │ │ * fromPluckerToStiefel(Ideal) │ │ │ │ * fromPluckerToStiefel(Matrix) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_hurwitz__Form.html │ │ │ @@ -92,15 +92,15 @@ │ │ │ o1 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time hurwitzForm Q
│ │ │ - -- used 0.0387325s (cpu); 0.0387333s (thread); 0s (gc)
│ │ │ + -- used 0.0457735s (cpu); 0.0455342s (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 0.0387325s (cpu); 0.0387333s (thread); 0s (gc)
│ │ │ │ + -- used 0.0457735s (cpu); 0.0455342s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2                                 2
│ │ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │ │                0,1            0,1 0,2            0,2           0,1 1,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                                 2
│ │ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time isCoisotropic w
│ │ │ - -- used 0.00734253s (cpu); 0.00734082s (thread); 0s (gc)
│ │ │ + -- used 0.0106714s (cpu); 0.0106719s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- random quadric in G(1,3)
│ │ │ @@ -140,15 +140,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time isCoisotropic w'
│ │ │ - -- used 0.00676804s (cpu); 0.00676823s (thread); 0s (gc)
│ │ │ + -- used 0.00847863s (cpu); 0.00847095s (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 0.00734253s (cpu); 0.00734082s (thread); 0s (gc) │ │ │ │ + -- used 0.0106714s (cpu); 0.0106719s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = true │ │ │ │ i3 : -- random quadric in G(1,3) │ │ │ │ w' = random(2,Grass(1,3)) │ │ │ │ │ │ │ │ 2 5 10 2 2 2 3 │ │ │ │ o3 = 6p + -p p + --p + -p p + 10p p + 5p + --p p │ │ │ │ @@ -72,15 +72,15 @@ │ │ │ │ │ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ │ o3 : -------------------------------------- │ │ │ │ p p - p p + p p │ │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ i4 : time isCoisotropic w' │ │ │ │ - -- used 0.00676804s (cpu); 0.00676823s (thread); 0s (gc) │ │ │ │ + -- used 0.00847863s (cpu); 0.00847095s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = false │ │ │ │ ********** WWaayyss ttoo uussee iissCCooiissoottrrooppiicc:: ********** │ │ │ │ * isCoisotropic(RingElement) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _i_s_C_o_i_s_o_t_r_o_p_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html │ │ │ @@ -117,15 +117,15 @@ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ 33331 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0191452s (cpu); 0.0191451s (thread); 0s (gc)
│ │ │ + -- used 0.0241718s (cpu); 0.0241721s (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 0.0191452s (cpu); 0.0191451s (thread); 0s (gc) │ │ │ │ + -- used 0.0241718s (cpu); 0.0241721s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety │ │ │ │ * _p_l_u_c_k_e_r -- get the Plücker coordinates of a linear subspace │ │ │ │ ********** WWaayyss ttoo uussee iissIInnCCooiissoottrrooppiicc:: ********** │ │ │ │ * isInCoisotropic(Ideal,Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ @@ -87,15 +87,15 @@ │ │ │ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00384623s (cpu); 0.00384331s (thread); 0s (gc)
│ │ │ + -- used 0.0041554s (cpu); 0.00415075s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0  
│ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0  
│ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0  
│ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0  
│ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0  
│ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0  
│ │ │ @@ -158,15 +158,15 @@
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00243197s (cpu); 0.00243231s (thread); 0s (gc)
│ │ │ + -- used 0.00286655s (cpu); 0.00286359s (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 0.00384623s (cpu); 0.00384331s (thread); 0s (gc)
│ │ │ │ + -- used 0.0041554s (cpu); 0.00415075s (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 0.00243197s (cpu); 0.00243231s (thread); 0s (gc)
│ │ │ │ + -- used 0.00286655s (cpu); 0.00286359s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0
│ │ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0
│ │ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0
│ │ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0
│ │ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0
│ │ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o3 : Ideal of P4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time p = plucker L
│ │ │ - -- used 0.00422213s (cpu); 0.00421985s (thread); 0s (gc)
│ │ │ + -- used 0.0063695s (cpu); 0.00636948s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o4 : Ideal of G'1'4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time L' = plucker p
│ │ │ - -- used 0.101398s (cpu); 0.0516067s (thread); 0s (gc)
│ │ │ + -- used 0.146792s (cpu); 0.0657964s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -138,15 +138,15 @@
│ │ │  
│ │ │  o7 : Ideal of G'1'4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.0291609s (cpu); 0.0291643s (thread); 0s (gc)
│ │ │ + -- used 0.0493136s (cpu); 0.0493166s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : (codim W,degree W)
│ │ │ @@ -158,15 +158,15 @@
│ │ │            
│ │ │          
│ │ │          
    In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb{P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ - -- used 0.141574s (cpu); 0.141579s (thread); 0s (gc)
│ │ │ + -- used 0.176165s (cpu); 0.176174s (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 0.00422213s (cpu); 0.00421985s (thread); 0s (gc)
│ │ │ │ + -- used 0.0063695s (cpu); 0.00636948s (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 0.101398s (cpu); 0.0516067s (thread); 0s (gc)
│ │ │ │ + -- used 0.146792s (cpu); 0.0657964s (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 0.0291609s (cpu); 0.0291643s (thread); 0s (gc)
│ │ │ │ + -- used 0.0493136s (cpu); 0.0493166s (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 0.141574s (cpu); 0.141579s (thread); 0s (gc)
│ │ │ │ + -- used 0.176165s (cpu); 0.176174s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of G'1'4
│ │ │ │  i11 : assert(Y' == Y)
│ │ │ │  WWaarrnniinngg: Notice that, by default, the computation is done on a randomly chosen
│ │ │ │  affine chart on the Grassmannian. To change this behavior, you can use the
│ │ │ │  _A_f_f_i_n_e_C_h_a_r_t_G_r_a_s_s option.
│ │ │ │  ********** WWaayyss ttoo uussee pplluucckkeerr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o2 : List
│ │ │              
    
│ │ │                
    i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.24907s (cpu); 0.172207s (thread); 0s (gc)
│ │ │ + -- used 0.395536s (cpu); 0.221719s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o3 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.115275s (cpu); 0.0788707s (thread); 0s (gc)
│ │ │ + -- used 0.217093s (cpu); 0.132268s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -156,15 +156,15 @@
│ │ │  
│ │ │  o4 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.440608s (cpu); 0.414364s (thread); 0s (gc)
│ │ │ + -- used 0.424916s (cpu); 0.340956s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -180,15 +180,15 @@
│ │ │  
│ │ │  o5 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.605546s (cpu); 0.545555s (thread); 0s (gc)
│ │ │ + -- used 0.665423s (cpu); 0.578527s (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 0.24907s (cpu); 0.172207s (thread); 0s (gc)
│ │ │ │ + -- used 0.395536s (cpu); 0.221719s (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 0.115275s (cpu); 0.0788707s (thread); 0s (gc)
│ │ │ │ + -- used 0.217093s (cpu); 0.132268s (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 0.440608s (cpu); 0.414364s (thread); 0s (gc)
│ │ │ │ + -- used 0.424916s (cpu); 0.340956s (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 0.605546s (cpu); 0.545555s (thread); 0s (gc)
│ │ │ │ + -- used 0.665423s (cpu); 0.578527s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time resultant F
│ │ │ - -- used 0.0233567s (cpu); 0.0233554s (thread); 0s (gc)
│ │ │ + -- used 0.02827s (cpu); 0.0282695s (thread); 0s (gc)
│ │ │  
│ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u 
│ │ │       ------------------------------------------------------------------------
│ │ │                6 7          5 8       11          10 2         9 3  
│ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -155,15 +155,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time resultant F
│ │ │ - -- used 2.47769s (cpu); 1.89799s (thread); 0s (gc)
│ │ │ + -- used 2.6838s (cpu); 2.0709s (thread); 0s (gc)
│ │ │  
│ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2  
│ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0  
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2  
│ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ @@ -1790,15 +1790,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time # terms resultant F
│ │ │ - -- used 0.566774s (cpu); 0.42434s (thread); 0s (gc)
│ │ │ + -- used 0.510655s (cpu); 0.424536s (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 0.0233567s (cpu); 0.0233554s (thread); 0s (gc) │ │ │ │ + -- used 0.02827s (cpu); 0.0282695s (thread); 0s (gc) │ │ │ │ │ │ │ │ 12 11 2 10 3 9 4 8 5 7 6 │ │ │ │ o3 = - 81t u - 1701t u - 15309t u - 76545t u - 229635t u - 413343t u │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 6 7 5 8 11 10 2 9 3 │ │ │ │ - 413343t u - 177147t u + 567t u + 10206t u + 76545t u + │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ @@ -86,15 +86,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 │ │ │ │ + c x } │ │ │ │ 9 2 │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : time resultant F │ │ │ │ - -- used 2.47769s (cpu); 1.89799s (thread); 0s (gc) │ │ │ │ + -- used 2.6838s (cpu); 2.0709s (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 0.566774s (cpu); 0.42434s (thread); 0s (gc) │ │ │ │ + -- used 0.510655s (cpu); 0.424536s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 21894 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_h_o_w_F_o_r_m -- Chow form of a projective variety │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_R_i_n_g_E_l_e_m_e_n_t_) │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * resultant(List) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ 0 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.027275s (cpu); 0.0272754s (thread); 0s (gc)
│ │ │ + -- used 0.0366946s (cpu); 0.036695s (thread); 0s (gc)
│ │ │  
│ │ │        2                                                       2        
│ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │        1,3 2,3    1,2 1,3 2,4    0,3 1,3 2,4    0,2 1,4 2,4    1,2 3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │        2
│ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ @@ -118,15 +118,15 @@
│ │ │         2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │       time tangentialChowForm(S,1)
│ │ │ - -- used 0.0958885s (cpu); 0.0634772s (thread); 0s (gc)
│ │ │ + -- used 0.150481s (cpu); 0.0845254s (thread); 0s (gc)
│ │ │  
│ │ │        2     2        2     2               3        2     2      
│ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │               3         3               3            
│ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ @@ -163,43 +163,43 @@
│ │ │         1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S)
│ │ │       time tangentialChowForm(S,2)
│ │ │ - -- used 0.032136s (cpu); 0.0321388s (thread); 0s (gc)
│ │ │ + -- used 0.0468657s (cpu); 0.0468702s (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 0.0981634s (cpu); 0.0520542s (thread); 0s (gc)
│ │ │ + -- used 0.142387s (cpu); 0.0718118s (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 0.149774s (cpu); 0.100749s (thread); 0s (gc)
    │ │ │ + -- used 0.186643s (cpu); 0.116899s (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 0.027275s (cpu); 0.0272754s (thread); 0s (gc) │ │ │ │ + -- used 0.0366946s (cpu); 0.036695s (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 0.0958885s (cpu); 0.0634772s (thread); 0s (gc) │ │ │ │ + -- used 0.150481s (cpu); 0.0845254s (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 0.032136s (cpu); 0.0321388s (thread); 0s (gc) │ │ │ │ + -- used 0.0468657s (cpu); 0.0468702s (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 0.0981634s (cpu); 0.0520542s (thread); 0s (gc) │ │ │ │ + -- used 0.142387s (cpu); 0.0718118s (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 0.149774s (cpu); 0.100749s (thread); 0s (gc) │ │ │ │ + -- used 0.186643s (cpu); 0.116899s (thread); 0s (gc) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_C_o_i_s_o_t_r_o_p_i_c -- whether a hypersurface of a Grassmannian is a tangential │ │ │ │ Chow form │ │ │ │ * _c_h_o_w_F_o_r_m -- Chow form of a projective variety │ │ │ │ ********** WWaayyss ttoo uussee ttaannggeennttiiaallCChhoowwFFoorrmm:: ********** │ │ │ │ * tangentialChowForm(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ time(seconds) 700 │ │ │ file(blocks) unlimited │ │ │ data(kbytes) unlimited │ │ │ stack(kbytes) 8192 │ │ │ coredump(blocks) unlimited │ │ │ memory(kbytes) 850000 │ │ │ locked memory(kbytes) 8192 │ │ │ -process 63817 │ │ │ +process 63520 │ │ │ nofiles 512 │ │ │ vmemory(kbytes) unlimited │ │ │ locks unlimited │ │ │ rtprio 0 │ │ │ │ │ │ o1 = 0 │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ @@ -1,137 +1,137 @@ │ │ │ -- -*- M2-comint -*- hash: 2927978066455787395 │ │ │ │ │ │ i1 : fn=temporaryFileName()|".m2" │ │ │ │ │ │ -o1 = /tmp/M2-29951-0/0.m2 │ │ │ +o1 = /tmp/M2-43121-0/0.m2 │ │ │ │ │ │ i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-29951-0/1.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-43121-0/1.m2" >"/tmp/M2-43121-0/1.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" => 3 │ │ │ + "time used" => 2 │ │ │ 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/M2-29951-0/2.m2" >"/tmp/M2-29951-0/2.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-43121-0/2.m2" >"/tmp/M2-43121-0/2.out" 2>&1 )) │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ i11 : h │ │ │ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29951-0/2.ans} │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43121-0/2.ans} │ │ │ "exit code" => 27 │ │ │ - "output file" => /tmp/M2-29951-0/2.out │ │ │ + "output file" => /tmp/M2-43121-0/2.out │ │ │ "return code" => 6912 │ │ │ "statistics" => null │ │ │ - "time used" => 2 │ │ │ + "time used" => 1 │ │ │ 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/M2-29951-0/3.m2" >"/tmp/M2-29951-0/3.out" 2>&1 )) │ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-43121-0/3.m2" >"/tmp/M2-43121-0/3.out" 2>&1 )) │ │ │ Killed │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ i15 : h │ │ │ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29951-0/3.ans} │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43121-0/3.ans} │ │ │ "exit code" => 0 │ │ │ - "output file" => /tmp/M2-29951-0/3.out │ │ │ + "output file" => /tmp/M2-43121-0/3.out │ │ │ "return code" => 9 │ │ │ "statistics" => null │ │ │ "time used" => 2 │ │ │ value => null │ │ │ │ │ │ o15 : HashTable │ │ │ │ │ │ i16 : if h#"output file" =!= null and fileExists(h#"output file") then get(h#"output file") │ │ │ │ │ │ o16 = │ │ │ - i1 : -- Script /tmp/M2-29951-0/3.m2 automatically generated by RunExternalM2 │ │ │ + i1 : -- Script /tmp/M2-43121-0/3.m2 automatically generated by RunExternalM2 │ │ │ needsPackage("RunExternalM2",Configuration=>{"isChild"=>true}); │ │ │ │ │ │ - i2 : load "/tmp/M2-29951-0/0.m2"; │ │ │ + i2 : load "/tmp/M2-43121-0/0.m2"; │ │ │ │ │ │ - i3 : runExternalM2ReturnAnswer("/tmp/M2-29951-0/3.ans",spin (10)); │ │ │ + i3 : runExternalM2ReturnAnswer("/tmp/M2-43121-0/3.ans",spin (10)); │ │ │ Spinning!! │ │ │ │ │ │ │ │ │ i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file") │ │ │ │ │ │ i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true); │ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-29951-0/4.m2" >"/tmp/M2-29951-0/4.out" 2>&1') >"/tmp/M2-29951-0/4.stat" 2>&1 )) │ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-43121-0/4.m2" >"/tmp/M2-43121-0/4.out" 2>&1') >"/tmp/M2-43121-0/4.stat" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i19 : h#"statistics" │ │ │ │ │ │ -o19 = Command being timed: "sh -c /usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-29951-0/4.m2" >"/tmp/M2-29951-0/4.out" 2>&1" │ │ │ - User time (seconds): 4.69 │ │ │ - System time (seconds): 0.13 │ │ │ - Percent of CPU this job got: 93% │ │ │ - Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.17 │ │ │ +o19 = Command being timed: "sh -c /usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-43121-0/4.m2" >"/tmp/M2-43121-0/4.out" 2>&1" │ │ │ + User time (seconds): 5.07 │ │ │ + System time (seconds): 0.29 │ │ │ + Percent of CPU this job got: 126% │ │ │ + Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.25 │ │ │ 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): 255884 │ │ │ + Maximum resident set size (kbytes): 341940 │ │ │ Average resident set size (kbytes): 0 │ │ │ Major (requiring I/O) page faults: 0 │ │ │ - Minor (reclaiming a frame) page faults: 9336 │ │ │ - Voluntary context switches: 1978 │ │ │ - Involuntary context switches: 1733 │ │ │ + Minor (reclaiming a frame) page faults: 10910 │ │ │ + Voluntary context switches: 7363 │ │ │ + Involuntary context switches: 1429 │ │ │ Swaps: 0 │ │ │ File system inputs: 0 │ │ │ - File system outputs: 0 │ │ │ + File system outputs: 24 │ │ │ Socket messages sent: 0 │ │ │ Socket messages received: 0 │ │ │ Signals delivered: 0 │ │ │ Page size (bytes): 4096 │ │ │ Exit status: 0 │ │ │ │ │ │ │ │ │ i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///; │ │ │ │ │ │ i21 : (runExternalM2(fn,identity,v))#value===v │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-29951-0/6.m2" >"/tmp/M2-29951-0/6.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-43121-0/6.m2" >"/tmp/M2-43121-0/6.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ o21 = true │ │ │ │ │ │ i22 : R=QQ[x,y]; │ │ │ │ │ │ i23 : v=coker random(R^2,R^{3:-1}) │ │ │ @@ -139,54 +139,54 @@ │ │ │ o23 = cokernel | 9/2x+9/4y 7/9x+7/10y 7x+3/7y | │ │ │ | 3/4x+7/4y 7/10x+7/3y 6/7x+6y | │ │ │ │ │ │ 2 │ │ │ o23 : R-module, quotient of R │ │ │ │ │ │ i24 : h=runExternalM2(fn,identity,v) │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-29951-0/7.m2" >"/tmp/M2-29951-0/7.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-43121-0/7.m2" >"/tmp/M2-43121-0/7.out" 2>&1 )) │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29951-0/7.ans} │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43121-0/7.ans} │ │ │ "exit code" => 1 │ │ │ - "output file" => /tmp/M2-29951-0/7.out │ │ │ + "output file" => /tmp/M2-43121-0/7.out │ │ │ "return code" => 256 │ │ │ "statistics" => null │ │ │ - "time used" => 1 │ │ │ + "time used" => 2 │ │ │ value => null │ │ │ │ │ │ o24 : HashTable │ │ │ │ │ │ i25 : get(h#"output file") │ │ │ │ │ │ o25 = │ │ │ - i1 : -- Script /tmp/M2-29951-0/7.m2 automatically generated by RunExternalM2 │ │ │ + i1 : -- Script /tmp/M2-43121-0/7.m2 automatically generated by RunExternalM2 │ │ │ needsPackage("RunExternalM2",Configuration=>{"isChild"=>true}); │ │ │ │ │ │ - i2 : load "/tmp/M2-29951-0/0.m2"; │ │ │ + i2 : load "/tmp/M2-43121-0/0.m2"; │ │ │ │ │ │ - i3 : runExternalM2ReturnAnswer("/tmp/M2-29951-0/7.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/M2-43121-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}})))); │ │ │ stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to objects: │ │ │ R (of class Symbol) │ │ │ ^ 2 (of class ZZ) │ │ │ │ │ │ │ │ │ i26 : fn<"/tmp/M2-29951-0/8.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-43121-0/8.m2" >"/tmp/M2-43121-0/8.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/M2-29951-0/9.m2" >"/tmp/M2-29951-0/9.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-43121-0/9.m2" >"/tmp/M2-43121-0/9.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i30 : h#value │ │ │ │ │ │ o30 = QQ[x..y] │ │ │ │ │ │ o30 : PolynomialRing │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ @@ -75,15 +75,15 @@ │ │ │ time(seconds) 700 │ │ │ file(blocks) unlimited │ │ │ data(kbytes) unlimited │ │ │ stack(kbytes) 8192 │ │ │ coredump(blocks) unlimited │ │ │ memory(kbytes) 850000 │ │ │ locked memory(kbytes) 8192 │ │ │ -process 63817 │ │ │ +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 63817 │ │ │ │ +process 63520 │ │ │ │ nofiles 512 │ │ │ │ vmemory(kbytes) unlimited │ │ │ │ locks unlimited │ │ │ │ rtprio 0 │ │ │ │ │ │ │ │ o1 = 0 │ │ │ │ This starts a new shell and executes the command given, which in this case │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ @@ -84,15 +84,15 @@ │ │ │

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

      │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -115,28 +115,28 @@ │ │ │
    │ │ │

    and then call them:

    │ │ │
    │ │ │
    │ │ │ 
    i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-29951-0/0.m2
    │ │ │ +o1 = /tmp/M2-43121-0/0.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/M2-29951-0/1.m2" >"/tmp/M2-29951-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43121-0/1.m2" >"/tmp/M2-43121-0/1.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" => 3
│ │ │ +               "time used" => 2
│ │ │                 value => 16
│ │ │  
│ │ │  o7 : HashTable
    │ │ │
    │ │ │ @@ -157,29 +157,29 @@ │ │ │

    │ │ │

    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/M2-29951-0/2.m2" >"/tmp/M2-29951-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43121-0/2.m2" >"/tmp/M2-43121-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    │ │ │
    │ │ │ 
    i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29951-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43121-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-29951-0/2.out
│ │ │ +                "output file" => /tmp/M2-43121-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 2
│ │ │ +                "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
    │ │ │
    │ │ │ @@ -199,46 +199,46 @@ │ │ │
    │ │ │

    Here, we use resource limits to limit the routine to 2 seconds of computational time, while the system is asked to use 10 seconds of computational time:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -248,40 +248,40 @@ │ │ │

    │ │ │

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

    │ │ │ │ │ │
    │ │ │ 
    i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29951-0/3.m2" >"/tmp/M2-29951-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43121-0/3.m2" >"/tmp/M2-43121-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    │ │ │
    │ │ │ 
    i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29951-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43121-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-29951-0/3.out
│ │ │ +                "output file" => /tmp/M2-43121-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │                  "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o15 : HashTable
    │ │ │
    │ │ │ 
    i16 : if h#"output file" =!= null and fileExists(h#"output file") then get(h#"output file")
│ │ │  
│ │ │  o16 = 
│ │ │ -      i1 : -- Script /tmp/M2-29951-0/3.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43121-0/3.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29951-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43121-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29951-0/3.ans",spin (10));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43121-0/3.ans",spin (10));
│ │ │        Spinning!!
    │ │ │
    │ │ │ 
    i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file")
    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -295,15 +295,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29951-0/4.m2" >"/tmp/M2-29951-0/4.out" 2>&1') >"/tmp/M2-29951-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43121-0/4.m2" >"/tmp/M2-43121-0/4.out" 2>&1') >"/tmp/M2-43121-0/4.stat" 2>&1 ))
│ │ │  Finished running.
    │ │ │
    │ │ │ 
    i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29951-0/4.m2" >"/tmp/M2-29951-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 4.69
│ │ │ -              System time (seconds): 0.13
│ │ │ -              Percent of CPU this job got: 93%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.17
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43121-0/4.m2" >"/tmp/M2-43121-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 5.07
│ │ │ +              System time (seconds): 0.29
│ │ │ +              Percent of CPU this job got: 126%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.25
│ │ │                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): 255884
│ │ │ +              Maximum resident set size (kbytes): 341940
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 9336
│ │ │ -              Voluntary context switches: 1978
│ │ │ -              Involuntary context switches: 1733
│ │ │ +              Minor (reclaiming a frame) page faults: 10910
│ │ │ +              Voluntary context switches: 7363
│ │ │ +              Involuntary context switches: 1429
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 24
│ │ │                Socket messages sent: 0
│ │ │                Socket messages received: 0
│ │ │                Signals delivered: 0
│ │ │                Page size (bytes): 4096
│ │ │                Exit status: 0
    │ │ │
    │ │ │ 
    i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
    │ │ │
    │ │ │ 
    i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29951-0/6.m2" >"/tmp/M2-29951-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43121-0/6.m2" >"/tmp/M2-43121-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
    │ │ │
    │ │ │
    │ │ │ @@ -325,24 +325,24 @@ │ │ │ 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/M2-29951-0/7.m2" >"/tmp/M2-29951-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43121-0/7.m2" >"/tmp/M2-43121-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29951-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43121-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-29951-0/7.out
│ │ │ +                "output file" => /tmp/M2-43121-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 1
│ │ │ +                "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
    │ │ │
    │ │ │
    │ │ │ @@ -350,20 +350,20 @@ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-29951-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-43121-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-29951-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-43121-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29951-0/7.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/M2-43121-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │        stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
    │ │ │
    │ │ │
    │ │ │ @@ -374,15 +374,15 @@ │ │ │
    		│ │ │ 
    i26 : fn<<///R=QQ[x,y];///<<endl<<flush;
    │ │ │
    │ │ │ 
    i27 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29951-0/8.m2" >"/tmp/M2-29951-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43121-0/8.m2" >"/tmp/M2-43121-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
    │ │ │
    │ │ │
    │ │ │ @@ -394,15 +394,15 @@ │ │ │ │ │ │ 
    i28 : v=R;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-29951-0/9.m2" >"/tmp/M2-29951-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-43121-0/9.m2" >"/tmp/M2-43121-0/9.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i30 : h#value
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,201 +45,201 @@
│ │ │ │  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/M2-29951-0/0.m2
│ │ │ │ +o1 = /tmp/M2-43121-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-29951-0/1.out" 2>&1 ))
│ │ │ │ +M2-43121-0/1.m2" >"/tmp/M2-43121-0/1.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" => 3
│ │ │ │ +               "time used" => 2
│ │ │ │                 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/
│ │ │ │ -M2-29951-0/2.m2" >"/tmp/M2-29951-0/2.out" 2>&1 ))
│ │ │ │ +M2-43121-0/2.m2" >"/tmp/M2-43121-0/2.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i11 : h
│ │ │ │  
│ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-29951-0/2.ans}
│ │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-43121-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-29951-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-43121-0/2.out
│ │ │ │                  "return code" => 6912
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 2
│ │ │ │ +                "time used" => 1
│ │ │ │                  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/M2-29951-0/3.m2" >"/tmp/M2-29951-0/3.out" 2>&1 ))
│ │ │ │ +q  <"/tmp/M2-43121-0/3.m2" >"/tmp/M2-43121-0/3.out" 2>&1 ))
│ │ │ │  Killed
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i15 : h
│ │ │ │  
│ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-29951-0/3.ans}
│ │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-43121-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-29951-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-43121-0/3.out
│ │ │ │                  "return code" => 9
│ │ │ │                  "statistics" => null
│ │ │ │                  "time used" => 2
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o15 : HashTable
│ │ │ │  i16 : if h#"output file" =!= null and fileExists(h#"output file") then get
│ │ │ │  (h#"output file")
│ │ │ │  
│ │ │ │  o16 =
│ │ │ │ -      i1 : -- Script /tmp/M2-29951-0/3.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-43121-0/3.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-29951-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-43121-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29951-0/3.ans",spin (10));
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43121-0/3.ans",spin (10));
│ │ │ │        Spinning!!
│ │ │ │  i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get
│ │ │ │  (h#"answer file")
│ │ │ │  We can get quite a lot of detail on the resources used with the _K_e_e_p_S_t_a_t_i_s_t_i_c_s
│ │ │ │  command:
│ │ │ │  i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ │  Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop -
│ │ │ │ --no-debug --silent  -q  <"/tmp/M2-29951-0/4.m2" >"/tmp/M2-29951-0/4.out" 2>&1')
│ │ │ │ ->"/tmp/M2-29951-0/4.stat" 2>&1 ))
│ │ │ │ +-no-debug --silent  -q  <"/tmp/M2-43121-0/4.m2" >"/tmp/M2-43121-0/4.out" 2>&1')
│ │ │ │ +>"/tmp/M2-43121-0/4.stat" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i19 : h#"statistics"
│ │ │ │  
│ │ │ │  o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug
│ │ │ │ ---silent  -q  <"/tmp/M2-29951-0/4.m2" >"/tmp/M2-29951-0/4.out" 2>&1"
│ │ │ │ -              User time (seconds): 4.69
│ │ │ │ -              System time (seconds): 0.13
│ │ │ │ -              Percent of CPU this job got: 93%
│ │ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.17
│ │ │ │ +--silent  -q  <"/tmp/M2-43121-0/4.m2" >"/tmp/M2-43121-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 5.07
│ │ │ │ +              System time (seconds): 0.29
│ │ │ │ +              Percent of CPU this job got: 126%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:04.25
│ │ │ │                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): 255884
│ │ │ │ +              Maximum resident set size (kbytes): 341940
│ │ │ │                Average resident set size (kbytes): 0
│ │ │ │                Major (requiring I/O) page faults: 0
│ │ │ │ -              Minor (reclaiming a frame) page faults: 9336
│ │ │ │ -              Voluntary context switches: 1978
│ │ │ │ -              Involuntary context switches: 1733
│ │ │ │ +              Minor (reclaiming a frame) page faults: 10910
│ │ │ │ +              Voluntary context switches: 7363
│ │ │ │ +              Involuntary context switches: 1429
│ │ │ │                Swaps: 0
│ │ │ │                File system inputs: 0
│ │ │ │ -              File system outputs: 0
│ │ │ │ +              File system outputs: 24
│ │ │ │                Socket messages sent: 0
│ │ │ │                Socket messages received: 0
│ │ │ │                Signals delivered: 0
│ │ │ │                Page size (bytes): 4096
│ │ │ │                Exit status: 0
│ │ │ │  We can handle most kinds of objects as return values, although _M_u_t_a_b_l_e_M_a_t_r_i_x
│ │ │ │  does not work. Here, we use the built-in _i_d_e_n_t_i_t_y function:
│ │ │ │  i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
│ │ │ │  i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-29951-0/6.m2" >"/tmp/M2-29951-0/6.out" 2>&1 ))
│ │ │ │ +M2-43121-0/6.m2" >"/tmp/M2-43121-0/6.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/
│ │ │ │ -M2-29951-0/7.m2" >"/tmp/M2-29951-0/7.out" 2>&1 ))
│ │ │ │ +M2-43121-0/7.m2" >"/tmp/M2-43121-0/7.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  
│ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-29951-0/7.ans}
│ │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-43121-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-29951-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-43121-0/7.out
│ │ │ │                  "return code" => 256
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 1
│ │ │ │ +                "time used" => 2
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o24 : HashTable
│ │ │ │  To view the error message:
│ │ │ │  i25 : get(h#"output file")
│ │ │ │  
│ │ │ │  o25 =
│ │ │ │ -      i1 : -- Script /tmp/M2-29951-0/7.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-43121-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-29951-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-43121-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-29951-0/7.ans",identity (cokernel
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-43121-0/7.ans",identity (cokernel
│ │ │ │  (map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/
│ │ │ │  4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ │        stdio:4:74:(3):[1]: error: no method for binary operator ^ applied to
│ │ │ │  objects:
│ │ │ │                    R (of class Symbol)
│ │ │ │              ^     2 (of class ZZ)
│ │ │ │  Keep in mind that the object you are passing must make sense in the context of
│ │ │ │  the file containing your function! For instance, here we need to define the
│ │ │ │  ring:
│ │ │ │  i26 : fn<"/tmp/M2-29951-0/8.out" 2>&1 ))
│ │ │ │ +M2-43121-0/8.m2" >"/tmp/M2-43121-0/8.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/
│ │ │ │ -M2-29951-0/9.m2" >"/tmp/M2-29951-0/9.out" 2>&1 ))
│ │ │ │ +M2-43121-0/9.m2" >"/tmp/M2-43121-0/9.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i30 : h#value
│ │ │ │  
│ │ │ │  o30 = QQ[x..y]
│ │ │ │  
│ │ │ │  o30 : PolynomialRing
│ │ │ │  i31 : v===h#value
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out
│ │ │ @@ -30,23 +30,23 @@
│ │ │                                              )
│ │ │  
│ │ │                            "variable positions" => {-1}
│ │ │  
│ │ │  o5 : InterpretedSLProgram
│ │ │  
│ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000318247s (cpu); 0.000230232s (thread); 0s (gc)
│ │ │ + -- used 0.00274013s (cpu); 0.000265969s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i7 : ZZ[y];
│ │ │  
│ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 4.68246s (cpu); 3.4549s (thread); 0s (gc)
│ │ │ + -- used 4.74556s (cpu); 3.45185s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html
│ │ │ @@ -104,29 +104,29 @@
│ │ │  
│ │ │  o5 : InterpretedSLProgram
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000318247s (cpu); 0.000230232s (thread); 0s (gc)
│ │ │ + -- used 0.00274013s (cpu); 0.000265969s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : ZZ[y];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 4.68246s (cpu); 3.4549s (thread); 0s (gc)
│ │ │ + -- used 4.74556s (cpu); 3.45185s (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 0.000318247s (cpu); 0.000230232s (thread); 0s (gc)
│ │ │ │ + -- used 0.00274013s (cpu); 0.000265969s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1       1
│ │ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │ │  i7 : ZZ[y];
│ │ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ │ - -- used 4.68246s (cpu); 3.4549s (thread); 0s (gc)
│ │ │ │ + -- used 4.74556s (cpu); 3.45185s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i3 : r={5,11,3,2}
│ │ │  
│ │ │  o3 = {5, 11, 3, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .00615994s elapsed
│ │ │ + -- .00829506s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : ChainComplex
│ │ │ @@ -51,15 +51,15 @@
│ │ │         53        53         53         53        53
│ │ │                                                  
│ │ │       -1        0          1          2         3
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │  
│ │ │  i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00220763s elapsed
│ │ │ + -- .00325644s elapsed
│ │ │  
│ │ │  i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │                 0 => 3
│ │ │                 1 => 5
│ │ │                 2 => 2
│ │ │ @@ -95,15 +95,15 @@
│ │ │  i12 : maximalEntry chainComplex errors
│ │ │  
│ │ │  o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .0049523s elapsed
│ │ │ + -- .00647094s 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)
│ │ │ - -- .00281902s elapsed
│ │ │ + -- .00336362s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : ChainComplex
│ │ │ @@ -47,25 +47,25 @@
│ │ │         53        53         53         53
│ │ │                                        
│ │ │       0         1          2          3
│ │ │  
│ │ │  o6 : ChainComplex
│ │ │  
│ │ │  i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000611005s elapsed
│ │ │ + -- .000741753s 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)
│ │ │ - -- .00144724s elapsed
│ │ │ + -- .00161551s 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)
│ │ │ - -- .00359791s elapsed
│ │ │ + -- .00426711s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,3}
│ │ │  
│ │ │  o4 = {4, 3, 3}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00291014s elapsed
│ │ │ + -- .00364468s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,3}
│ │ │  
│ │ │  o4 = {4, 3, 3}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00240427s elapsed
│ │ │ + -- .00322441s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html
│ │ │ @@ -105,15 +105,15 @@
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .00615994s elapsed
│ │ │ + -- .00829506s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : ChainComplex
    │ │ │ @@ -150,15 +150,15 @@ │ │ │ │ │ │ o6 : ChainComplex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00220763s elapsed
    │ │ │ + -- .00325644s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │ @@ -212,15 +212,15 @@
│ │ │  
│ │ │  o12 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .0049523s elapsed
    │ │ │ + -- .00647094s 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) │ │ │ │ - -- .00615994s elapsed │ │ │ │ + -- .00829506s elapsed │ │ │ │ │ │ │ │ 6 19 19 7 3 │ │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o4 : ChainComplex │ │ │ │ @@ -70,15 +70,15 @@ │ │ │ │ o6 = RR <-- RR <-- RR <-- RR <-- RR │ │ │ │ 53 53 53 53 53 │ │ │ │ │ │ │ │ -1 0 1 2 3 │ │ │ │ │ │ │ │ o6 : ChainComplex │ │ │ │ i7 : elapsedTime (h,U)=SVDComplex CR; │ │ │ │ - -- .00220763s elapsed │ │ │ │ + -- .00325644s elapsed │ │ │ │ i8 : h │ │ │ │ │ │ │ │ o8 = HashTable{-1 => 1} │ │ │ │ 0 => 3 │ │ │ │ 1 => 5 │ │ │ │ 2 => 2 │ │ │ │ 3 => 1 │ │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ 1)*Sigma.dd_ell*transpose U_ell); │ │ │ │ i12 : maximalEntry chainComplex errors │ │ │ │ │ │ │ │ o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian); │ │ │ │ - -- .0049523s elapsed │ │ │ │ + -- .00647094s elapsed │ │ │ │ i14 : hL === h │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : SigmaL =source U; │ │ │ │ i16 : for i from min CR+1 to max CR list maximalEntry(SigmaL.dd_i -Sigma.dd_i) │ │ │ │ │ │ │ │ o16 = {1.77636e-14, 6.39488e-14, 8.52651e-14, 3.55271e-15} │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ @@ -107,15 +107,15 @@ │ │ │ │ │ │ o3 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00281902s elapsed
│ │ │ + -- .00336362s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : ChainComplex
    │ │ │ @@ -148,28 +148,28 @@ │ │ │ │ │ │ o6 : ChainComplex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000611005s elapsed
│ │ │ + -- .000741753s 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)
│ │ │ - -- .00144724s elapsed
│ │ │ + -- .00161551s 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) │ │ │ │ - -- .00281902s elapsed │ │ │ │ + -- .00336362s elapsed │ │ │ │ │ │ │ │ 5 10 11 5 │ │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o4 : ChainComplex │ │ │ │ @@ -69,24 +69,24 @@ │ │ │ │ o6 = RR <-- RR <-- RR <-- RR │ │ │ │ 53 53 53 53 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o6 : ChainComplex │ │ │ │ i7 : elapsedTime (h,h1)=SVDHomology CR │ │ │ │ - -- .000611005s elapsed │ │ │ │ + -- .000741753s 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) │ │ │ │ - -- .00144724s elapsed │ │ │ │ + -- .00161551s elapsed │ │ │ │ │ │ │ │ o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14) }) │ │ │ │ 1 => 3 1 => (1.71747, 922.381, 2.51496e-13) │ │ │ │ 2 => 5 2 => (922.381, 73.5901, 1.81323e-13) │ │ │ │ 3 => 2 3 => (73.5901, , 2.82914e-13) │ │ │ │ │ │ │ │ o8 : Sequence │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ @@ -110,15 +110,15 @@ │ │ │ │ │ │ o4 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .00359791s elapsed
│ │ │ + -- .00426711s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    │ │ │ ├── html2text {} │ │ │ │ @@ -34,15 +34,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,5} │ │ │ │ │ │ │ │ o4 = {4, 3, 5} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true) │ │ │ │ - -- .00359791s elapsed │ │ │ │ + -- .00426711s elapsed │ │ │ │ │ │ │ │ 6 10 13 8 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ @@ -104,15 +104,15 @@ │ │ │ │ │ │ o4 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00291014s elapsed
│ │ │ + -- .00364468s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .00291014s elapsed │ │ │ │ + -- .00364468s elapsed │ │ │ │ │ │ │ │ 6 10 11 5 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : ChainComplex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html │ │ │ @@ -104,15 +104,15 @@ │ │ │ │ │ │ o4 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00240427s elapsed
│ │ │ + -- .00322441s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : ChainComplex
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .00240427s elapsed │ │ │ │ + -- .00322441s elapsed │ │ │ │ │ │ │ │ 6 10 11 5 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : ChainComplex │ │ ├── ./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 0.377312s (cpu); 0.307705s (thread); 0s (gc) │ │ │ + -- used 0.378379s (cpu); 0.378107s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ - -- used 0.469704s (cpu); 0.469683s (thread); 0s (gc) │ │ │ + -- used 0.770224s (cpu); 0.673985s (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 0.0255412s (cpu); 0.0255412s (thread); 0s (gc) │ │ │ + -- used 0.0288117s (cpu); 0.028821s (thread); 0s (gc) │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ - -- used 0.00728276s (cpu); 0.00728315s (thread); 0s (gc) │ │ │ + -- used 0.00949256s (cpu); 0.00949951s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ @@ -125,23 +125,23 @@ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.377312s (cpu); 0.307705s (thread); 0s (gc)
│ │ │ + -- used 0.378379s (cpu); 0.378107s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.469704s (cpu); 0.469683s (thread); 0s (gc)
│ │ │ + -- used 0.770224s (cpu); 0.673985s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Strategy => Quotient is faster in other cases:

    │ │ │ @@ -158,23 +158,23 @@ │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.0255412s (cpu); 0.0255412s (thread); 0s (gc)
│ │ │ + -- used 0.0288117s (cpu); 0.028821s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00728276s (cpu); 0.00728315s (thread); 0s (gc)
│ │ │ + -- used 0.00949256s (cpu); 0.00949951s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -56,32 +56,32 @@ │ │ │ │ i5 : I = monomialCurveIdeal(S, 1..n-1); │ │ │ │ │ │ │ │ o5 : Ideal of S │ │ │ │ i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5)); │ │ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ i7 : time quotient(I^3, J^2, Strategy => Iterate); │ │ │ │ - -- used 0.377312s (cpu); 0.307705s (thread); 0s (gc) │ │ │ │ + -- used 0.378379s (cpu); 0.378107s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ │ - -- used 0.469704s (cpu); 0.469683s (thread); 0s (gc) │ │ │ │ + -- used 0.770224s (cpu); 0.673985s (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 0.0255412s (cpu); 0.0255412s (thread); 0s (gc) │ │ │ │ + -- used 0.0288117s (cpu); 0.028821s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ │ - -- used 0.00728276s (cpu); 0.00728315s (thread); 0s (gc) │ │ │ │ + -- used 0.00949256s (cpu); 0.00949951s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ ********** RReeffeerreenncceess ********** │ │ │ │ For further information see for example Exercise 15.41 in Eisenbud's │ │ │ │ Commutative Algebra with a View Towards Algebraic Geometry. │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd SSttrraatteeggyy:: ********** │ │ │ │ * addHook(...,Strategy=>...) -- see _a_d_d_H_o_o_k -- add a hook function to an │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ @@ -40,23 +40,23 @@ │ │ │ ) │ │ │ │ │ │ o6 = f │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ │ │ i7 : for n from 2 to 10 list time f n │ │ │ - -- used 0.00531309s (cpu); 0.00530845s (thread); 0s (gc) │ │ │ - -- used 0.00670285s (cpu); 0.00670399s (thread); 0s (gc) │ │ │ - -- used 0.0102064s (cpu); 0.0102079s (thread); 0s (gc) │ │ │ - -- used 0.017188s (cpu); 0.0171899s (thread); 0s (gc) │ │ │ - -- used 0.0321696s (cpu); 0.0321728s (thread); 0s (gc) │ │ │ - -- used 0.0560131s (cpu); 0.0560195s (thread); 0s (gc) │ │ │ - -- used 0.0940609s (cpu); 0.094067s (thread); 0s (gc) │ │ │ - -- used 0.28927s (cpu); 0.195433s (thread); 0s (gc) │ │ │ - -- used 0.229151s (cpu); 0.229159s (thread); 0s (gc) │ │ │ + -- used 0.00725131s (cpu); 0.00726126s (thread); 0s (gc) │ │ │ + -- used 0.0113143s (cpu); 0.0113218s (thread); 0s (gc) │ │ │ + -- used 0.0125746s (cpu); 0.0125849s (thread); 0s (gc) │ │ │ + -- used 0.0204641s (cpu); 0.0204769s (thread); 0s (gc) │ │ │ + -- used 0.0392133s (cpu); 0.0392254s (thread); 0s (gc) │ │ │ + -- used 0.0648603s (cpu); 0.0648754s (thread); 0s (gc) │ │ │ + -- used 0.111559s (cpu); 0.111571s (thread); 0s (gc) │ │ │ + -- used 0.179174s (cpu); 0.179189s (thread); 0s (gc) │ │ │ + -- used 0.480648s (cpu); 0.326504s (thread); 0s (gc) │ │ │ │ │ │ o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, │ │ │ ------------------------------------------------------------------------ │ │ │ 289139638632755625, 520764738758073845321} │ │ │ │ │ │ o7 : List │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ @@ -126,23 +126,23 @@ │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00531309s (cpu); 0.00530845s (thread); 0s (gc)
│ │ │ - -- used 0.00670285s (cpu); 0.00670399s (thread); 0s (gc)
│ │ │ - -- used 0.0102064s (cpu); 0.0102079s (thread); 0s (gc)
│ │ │ - -- used 0.017188s (cpu); 0.0171899s (thread); 0s (gc)
│ │ │ - -- used 0.0321696s (cpu); 0.0321728s (thread); 0s (gc)
│ │ │ - -- used 0.0560131s (cpu); 0.0560195s (thread); 0s (gc)
│ │ │ - -- used 0.0940609s (cpu); 0.094067s (thread); 0s (gc)
│ │ │ - -- used 0.28927s (cpu); 0.195433s (thread); 0s (gc)
│ │ │ - -- used 0.229151s (cpu); 0.229159s (thread); 0s (gc)
│ │ │ + -- used 0.00725131s (cpu); 0.00726126s (thread); 0s (gc)
│ │ │ + -- used 0.0113143s (cpu); 0.0113218s (thread); 0s (gc)
│ │ │ + -- used 0.0125746s (cpu); 0.0125849s (thread); 0s (gc)
│ │ │ + -- used 0.0204641s (cpu); 0.0204769s (thread); 0s (gc)
│ │ │ + -- used 0.0392133s (cpu); 0.0392254s (thread); 0s (gc)
│ │ │ + -- used 0.0648603s (cpu); 0.0648754s (thread); 0s (gc)
│ │ │ + -- used 0.111559s (cpu); 0.111571s (thread); 0s (gc)
│ │ │ + -- used 0.179174s (cpu); 0.179189s (thread); 0s (gc)
│ │ │ + -- used 0.480648s (cpu); 0.326504s (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 0.00531309s (cpu); 0.00530845s (thread); 0s (gc) │ │ │ │ - -- used 0.00670285s (cpu); 0.00670399s (thread); 0s (gc) │ │ │ │ - -- used 0.0102064s (cpu); 0.0102079s (thread); 0s (gc) │ │ │ │ - -- used 0.017188s (cpu); 0.0171899s (thread); 0s (gc) │ │ │ │ - -- used 0.0321696s (cpu); 0.0321728s (thread); 0s (gc) │ │ │ │ - -- used 0.0560131s (cpu); 0.0560195s (thread); 0s (gc) │ │ │ │ - -- used 0.0940609s (cpu); 0.094067s (thread); 0s (gc) │ │ │ │ - -- used 0.28927s (cpu); 0.195433s (thread); 0s (gc) │ │ │ │ - -- used 0.229151s (cpu); 0.229159s (thread); 0s (gc) │ │ │ │ + -- used 0.00725131s (cpu); 0.00726126s (thread); 0s (gc) │ │ │ │ + -- used 0.0113143s (cpu); 0.0113218s (thread); 0s (gc) │ │ │ │ + -- used 0.0125746s (cpu); 0.0125849s (thread); 0s (gc) │ │ │ │ + -- used 0.0204641s (cpu); 0.0204769s (thread); 0s (gc) │ │ │ │ + -- used 0.0392133s (cpu); 0.0392254s (thread); 0s (gc) │ │ │ │ + -- used 0.0648603s (cpu); 0.0648754s (thread); 0s (gc) │ │ │ │ + -- used 0.111559s (cpu); 0.111571s (thread); 0s (gc) │ │ │ │ + -- used 0.179174s (cpu); 0.179189s (thread); 0s (gc) │ │ │ │ + -- used 0.480648s (cpu); 0.326504s (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/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 4.03546s (cpu); 3.39136s (thread); 0s (gc) │ │ │ + -- used 8.89254s (cpu); 4.35037s (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 51.7018s (cpu); 48.4003s (thread); 0s (gc) │ │ │ + -- used 65.3984s (cpu); 59.2419s (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 0.233473s (cpu); 0.153541s (thread); 0s (gc) │ │ │ + -- used 0.498402s (cpu); 0.195409s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o6 = 2H + 4H H + 2H │ │ │ 1 1 2 2 │ │ │ │ │ │ o6 : A │ │ │ │ │ │ i7 : time segre(X,Y,A) │ │ │ - -- used 0.134763s (cpu); 0.0957936s (thread); 0s (gc) │ │ │ + -- used 0.320574s (cpu); 0.149131s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ o7 = 12H H - 6H H - 6H H + 2H + 4H H + 2H │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ o7 : A │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ @@ -162,15 +162,15 @@ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time isComponentContained(X,Y)
│ │ │ - -- used 4.03546s (cpu); 3.39136s (thread); 0s (gc)
│ │ │ + -- used 8.89254s (cpu); 4.35037s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : print "we could confirm this with the computation:"
│ │ │ @@ -189,15 +189,15 @@
│ │ │  
│ │ │  o13 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 51.7018s (cpu); 48.4003s (thread); 0s (gc)
│ │ │ + -- used 65.3984s (cpu); 59.2419s (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 4.03546s (cpu); 3.39136s (thread); 0s (gc) │ │ │ │ + -- used 8.89254s (cpu); 4.35037s (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 51.7018s (cpu); 48.4003s (thread); 0s (gc) │ │ │ │ + -- used 65.3984s (cpu); 59.2419s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = true │ │ │ │ ********** WWaayyss ttoo uussee iissCCoommppoonneennttCCoonnttaaiinneedd:: ********** │ │ │ │ * isComponentContained(Ideal,Ideal) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _i_s_C_o_m_p_o_n_e_n_t_C_o_n_t_a_i_n_e_d is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ @@ -118,27 +118,27 @@ │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.233473s (cpu); 0.153541s (thread); 0s (gc)
│ │ │ + -- used 0.498402s (cpu); 0.195409s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time segre(X,Y,A)
│ │ │ - -- used 0.134763s (cpu); 0.0957936s (thread); 0s (gc)
│ │ │ + -- used 0.320574s (cpu); 0.149131s (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 0.233473s (cpu); 0.153541s (thread); 0s (gc) │ │ │ │ + -- used 0.498402s (cpu); 0.195409s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o6 = 2H + 4H H + 2H │ │ │ │ 1 1 2 2 │ │ │ │ │ │ │ │ o6 : A │ │ │ │ i7 : time segre(X,Y,A) │ │ │ │ - -- used 0.134763s (cpu); 0.0957936s (thread); 0s (gc) │ │ │ │ + -- used 0.320574s (cpu); 0.149131s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o7 = 12H H - 6H H - 6H H + 2H + 4H H + 2H │ │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ │ │ o7 : A │ │ │ │ ********** WWaayyss ttoo uussee sseeggrreeDDiimmXX:: ********** │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_test__Example.out │ │ │ @@ -1,6 +1,6 @@ │ │ │ -- -*- M2-comint -*- hash: 1331702921222 │ │ │ │ │ │ i1 : check SimpleDoc │ │ │ - -- capturing check(0, "SimpleDoc") -- .370859s elapsed │ │ │ + -- capturing check(0, "SimpleDoc") -- .170458s elapsed │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html │ │ │ @@ -74,15 +74,15 @@ │ │ │
    │ │ │

    The check method executes all package tests defined this way.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .370859s elapsed
    │ │ │ + -- capturing check(0, "SimpleDoc") -- .170458s elapsed │ │ │
    │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -10,15 +10,15 @@ │ │ │ │ The variable testExample is a _S_t_r_i_n_g containing an example of the use of _T_E_S_T │ │ │ │ to write a test case. │ │ │ │ TEST /// -* test for simpleDocFrob *- │ │ │ │ assert(simpleDocFrob(2,matrix{{1,2}}) == matrix{{1,2,0,0},{0,0,1,2}}) │ │ │ │ /// │ │ │ │ The _c_h_e_c_k method executes all package tests defined this way. │ │ │ │ i1 : check SimpleDoc │ │ │ │ - -- capturing check(0, "SimpleDoc") -- .370859s elapsed │ │ │ │ + -- capturing check(0, "SimpleDoc") -- .170458s 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/SparseResultants/example-output/_degree__Determinant.out │ │ │ @@ -3,15 +3,15 @@ │ │ │ i1 : n = {2,3,2} │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : time degreeDeterminant n │ │ │ - -- used 8.504e-05s (cpu); 7.9479e-05s (thread); 0s (gc) │ │ │ + -- used 0.000120194s (cpu); 0.000108766s (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 0.0318322s (cpu); 0.030991s (thread); 0s (gc) │ │ │ + -- used 0.125815s (cpu); 0.0460633s (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 0.419831s (cpu); 0.24997s (thread); 0s (gc) │ │ │ + -- used 0.585659s (cpu); 0.290508s (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 0.14522s (cpu); 0.1035s (thread); 0s (gc) │ │ │ + -- used 0.197194s (cpu); 0.118749s (thread); 0s (gc) │ │ │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula │ │ │ - -- used 0.291032s (cpu); 0.247505s (thread); 0s (gc) │ │ │ + -- used 0.376308s (cpu); 0.296521s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ - -- used 0.364826s (cpu); 0.310427s (thread); 0s (gc) │ │ │ + -- used 0.459117s (cpu); 0.4176s (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 0.081294s (cpu); 0.0797257s (thread); 0s (gc) │ │ │ + -- used 0.153735s (cpu); 0.108241s (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 0.522415s (cpu); 0.443252s (thread); 0s (gc) │ │ │ + -- used 0.496742s (cpu); 0.496743s (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 2.60899s (cpu); 2.21012s (thread); 0s (gc) │ │ │ + -- used 3.18436s (cpu); 2.7555s (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 0.493177s (cpu); 0.45013s (thread); 0s (gc) │ │ │ + -- used 0.473328s (cpu); 0.450516s (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 0.00960219s (cpu); 0.00960397s (thread); 0s (gc) │ │ │ + -- used 0.0112789s (cpu); 0.0112782s (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 0.0329425s (cpu); 0.0318668s (thread); 0s (gc) │ │ │ + -- used 0.148581s (cpu); 0.0544028s (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 0.00329367s (cpu); 0.00329382s (thread); 0s (gc) │ │ │ + -- used 0.00451553s (cpu); 0.00451479s (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 0.271788s (cpu); 0.22445s (thread); 0s (gc) │ │ │ + -- used 0.254706s (cpu); 0.195466s (thread); 0s (gc) │ │ │ │ │ │ i13 : quotientRemainder(UnmixedRes,MixedRes) │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ o13 = (b c - b b c c + b b c + b c c - 2b b c c - b b c c + b c , 0) │ │ │ 5 2 4 5 2 4 2 5 4 4 2 5 2 5 2 5 2 4 4 5 2 5 │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ @@ -76,15 +76,15 @@ │ │ │ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : time degreeDeterminant n
│ │ │ - -- used 8.504e-05s (cpu); 7.9479e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000120194s (cpu); 0.000108766s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : M = genericMultidimensionalMatrix n;
│ │ │ @@ -98,15 +98,15 @@
│ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │                                                        0,0,0   1,2,1
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i4 : time degree determinant M
│ │ │ - -- used 0.0318322s (cpu); 0.030991s (thread); 0s (gc)
│ │ │ + -- used 0.125815s (cpu); 0.0460633s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = {6}
│ │ │  
│ │ │  o4 : List
      │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -15,15 +15,15 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : n = {2,3,2} │ │ │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : time degreeDeterminant n │ │ │ │ - -- used 8.504e-05s (cpu); 7.9479e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.000120194s (cpu); 0.000108766s (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 0.0318322s (cpu); 0.030991s (thread); 0s (gc) │ │ │ │ + -- used 0.125815s (cpu); 0.0460633s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = {6} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_t_e_r_m_i_n_a_n_t_(_M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x_) -- hyperdeterminant of a │ │ │ │ multidimensional matrix │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ @@ -80,15 +80,15 @@ │ │ │ │ │ │ 
      i1 : (d,n) := (2,3);
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.419831s (cpu); 0.24997s (thread); 0s (gc)
│ │ │ + -- used 0.585659s (cpu); 0.290508s (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 0.419831s (cpu); 0.24997s (thread); 0s (gc) │ │ │ │ + -- used 0.585659s (cpu); 0.290508s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = Disc │ │ │ │ │ │ │ │ o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 │ │ │ │ 2 |) │ │ │ │ | 0 0 0 1 1 2 0 0 1 │ │ │ │ 0 | │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ @@ -90,27 +90,27 @@ │ │ │ │ │ │ o1 : Sequence │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.14522s (cpu); 0.1035s (thread); 0s (gc)
      │ │ │ + -- used 0.197194s (cpu); 0.118749s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.291032s (cpu); 0.247505s (thread); 0s (gc)
      │ │ │ + -- used 0.376308s (cpu); 0.296521s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.364826s (cpu); 0.310427s (thread); 0s (gc)
      │ │ │ + -- used 0.459117s (cpu); 0.4176s (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 0.14522s (cpu); 0.1035s (thread); 0s (gc) │ │ │ │ + -- used 0.197194s (cpu); 0.118749s (thread); 0s (gc) │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay │ │ │ │ formula │ │ │ │ - -- used 0.291032s (cpu); 0.247505s (thread); 0s (gc) │ │ │ │ + -- used 0.376308s (cpu); 0.296521s (thread); 0s (gc) │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ │ - -- used 0.364826s (cpu); 0.310427s (thread); 0s (gc) │ │ │ │ + -- used 0.459117s (cpu); 0.4176s (thread); 0s (gc) │ │ │ │ i5 : assert(o2 == o3 and o3 == o4) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_p_a_r_s_e_R_e_s_u_l_t_a_n_t -- sparse resultant (A-resultant) │ │ │ │ * _a_f_f_i_n_e_R_e_s_u_l_t_a_n_t -- affine resultant │ │ │ │ * _d_e_n_s_e_D_i_s_c_r_i_m_i_n_a_n_t -- dense discriminant (classical discriminant) │ │ │ │ * _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x -- exponents in one or more polynomials │ │ │ │ * _g_e_n_e_r_i_c_L_a_u_r_e_n_t_P_o_l_y_n_o_m_i_a_l_s -- generic (Laurent) polynomials │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ @@ -84,15 +84,15 @@ │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : time det M
│ │ │ - -- used 0.081294s (cpu); 0.0797257s (thread); 0s (gc)
│ │ │ + -- used 0.153735s (cpu); 0.108241s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i4 : time det M
│ │ │ - -- used 0.522415s (cpu); 0.443252s (thread); 0s (gc)
│ │ │ + -- used 0.496742s (cpu); 0.496743s (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 0.081294s (cpu); 0.0797257s (thread); 0s (gc) │ │ │ │ + -- used 0.153735s (cpu); 0.108241s (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 0.522415s (cpu); 0.443252s (thread); 0s (gc) │ │ │ │ + -- used 0.496742s (cpu); 0.496743s (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 │ │ │ @@ -90,15 +90,15 @@ │ │ │ o1 : ZZ[a ..a ][x ..x , y ..y , z ..z ] │ │ │ 0,0,0 1,2,1 0 1 0 2 0 1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time sparseDiscriminant f
│ │ │ - -- used 2.60899s (cpu); 2.21012s (thread); 0s (gc)
│ │ │ + -- used 3.18436s (cpu); 2.7555s (thread); 0s (gc)
│ │ │  
│ │ │                                                     2                        
│ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │              2     2                                2            
│ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a     x y z  + a     x y z  + a     x y z
│ │ │ │        1,1,1 1 1 1    1,2,0 1 2 0    1,2,1 1 2 1
│ │ │ │  
│ │ │ │  o1 : ZZ[a     ..a     ][x ..x , y ..y , z ..z ]
│ │ │ │           0,0,0   1,2,1   0   1   0   2   0   1
│ │ │ │  i2 : time sparseDiscriminant f
│ │ │ │ - -- used 2.60899s (cpu); 2.21012s (thread); 0s (gc)
│ │ │ │ + -- used 3.18436s (cpu); 2.7555s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                     2
│ │ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2     2                                2
│ │ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    Description
    
│ │ │          
    Alternatively, one can apply the method directly to the list of Laurent polynomials $f_0,\ldots,f_n$. In this case, the matrices $A_0,\ldots,A_n$ are automatically determined by exponentsMatrix. If you want require that $A_0=\cdots=A_n$, then use the option Unmixed=>true (this could be faster). Below we consider some examples.
    
│ │ │          
    In the first example, we calculate the sparse (mixed) resultant associated to the three sets of monomials $(1,x y,x^2 y,x),(y,x^2 y^2,x^2 y,x),(1,y,x y,x)$. Then we evaluate it at the three polynomials $f = c_{(1,1)}+c_{(1,2)} x y+c_{(1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{(2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o3 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ - -- used 0.493177s (cpu); 0.45013s (thread); 0s (gc)
│ │ │ + -- used 0.473328s (cpu); 0.450516s (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 0.00960219s (cpu); 0.00960397s (thread); 0s (gc)
│ │ │ + -- used 0.0112789s (cpu); 0.0112782s (thread); 0s (gc)
│ │ │  
│ │ │          2                       4      2   2               4    
│ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        3       2       3               2                   3        
│ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ @@ -825,15 +825,15 @@
│ │ │            
    
│ │ │          
    In the second example, we calculate the sparse unmixed resultant associated to the set of monomials $(1,x,y,xy)$. Then we evaluate it at the three polynomials $f = a_0 + a_1 x + a_2 y + a_3 x y, g = b_0 + b_1 x + b_2 y + b_3 x y, h = c_0 + c_1 x + c_2 y + c_3 x y$. Moreover, we perform all the computation over $\mathbb{Z}/3331$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,15 +68,15 @@
│ │ │ │  
│ │ │ │  o4 = | 0 1 |
│ │ │ │       | 2 3 |
│ │ │ │       | 4 |
│ │ │ │  
│ │ │ │  o4 : YoungTableau
│ │ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ │ - -- used 0.00153476s (cpu); 0.0015318s (thread); 0s (gc)
│ │ │ │ + -- used 0.00172447s (cpu); 0.00172097s (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 0.0012244s (cpu); 0.00122431s (thread); 0s (gc)
│ │ │ │ + -- used 0.00172632s (cpu); 0.0017267s (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 0.0019364s (cpu); 0.00193684s (thread); 0s (gc)
│ │ │ │ + -- used 0.00243406s (cpu); 0.00243378s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )
│ │ │ │  (x )(x ))
│ │ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4
│ │ │ │  2   0
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_representation__Multiplicity.html
│ │ │ @@ -126,15 +126,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -111,15 +111,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -121,15 +121,15 @@
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0329425s (cpu); 0.0318668s (thread); 0s (gc)
│ │ │ + -- used 0.148581s (cpu); 0.0544028s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
    
│ │ │              
    
│ │ │ @@ -849,15 +849,15 @@
│ │ │  
│ │ │  o8 : Sequence
│ │ │              
    
│ │ │                
    i9 : time Res(f,g,h)
│ │ │ - -- used 0.00329367s (cpu); 0.00329382s (thread); 0s (gc)
│ │ │ + -- used 0.00451553s (cpu); 0.00451479s (thread); 0s (gc)
│ │ │  
│ │ │        2     2            2            2        2 2    2          
│ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2                       2                         
│ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ @@ -938,15 +938,15 @@
│ │ │  
│ │ │  o11 : Sequence
    
│ │ │              
    
│ │ │                
    i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.271788s (cpu); 0.22445s (thread); 0s (gc)
    
│ │ │ + -- used 0.254706s (cpu); 0.195466s (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 0.493177s (cpu); 0.45013s (thread); 0s (gc)
│ │ │ │ + -- used 0.473328s (cpu); 0.450516s (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 0.00960219s (cpu); 0.00960397s (thread); 0s (gc)
│ │ │ │ + -- used 0.0112789s (cpu); 0.0112782s (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 0.0329425s (cpu); 0.0318668s (thread); 0s (gc)
│ │ │ │ + -- used 0.148581s (cpu); 0.0544028s (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 0.00329367s (cpu); 0.00329382s (thread); 0s (gc)
│ │ │ │ + -- used 0.00451553s (cpu); 0.00451479s (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 0.271788s (cpu); 0.22445s (thread); 0s (gc)
│ │ │ │ + -- used 0.254706s (cpu); 0.195466s (thread); 0s (gc)
│ │ │ │  i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │ │  
│ │ │ │          2 2                   2    2                               2 2
│ │ │ │  o13 = (b c  - b b c c  + b b c  + b c c  - 2b b c c  - b b c c  + b c , 0)
│ │ │ │          5 2    4 5 2 4    2 5 4    4 2 5     2 5 2 5    2 4 4 5    2 5
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o4 = | 0 1 |
│ │ │       | 2 3 |
│ │ │       | 4 |
│ │ │  
│ │ │  o4 : YoungTableau
│ │ │  
│ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00153476s (cpu); 0.0015318s (thread); 0s (gc)
│ │ │ + -- used 0.00172447s (cpu); 0.00172097s (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 0.0012244s (cpu); 0.00122431s (thread); 0s (gc)
│ │ │ + -- used 0.00172632s (cpu); 0.0017267s (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 0.0019364s (cpu); 0.00193684s (thread); 0s (gc)
│ │ │ + -- used 0.00243406s (cpu); 0.00243378s (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 0.333052s (cpu); 0.286542s (thread); 0s (gc)
│ │ │ + -- used 0.389839s (cpu); 0.314024s (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 0.667033s (cpu); 0.513099s (thread); 0s (gc)
│ │ │ + -- used 0.914798s (cpu); 0.598814s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │  o4 = HashTable{{0, 1, 2, 3, 4, 5} => HashTable{0 => - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        }
│ │ │                                                        3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6
│ │ │                                                      2 3 2 2     4 2 3 2     2 2 2 3     4 3 2   2   2 2 3   2   2 3   2 2   2   3 2 2   2 2   3 2   4   2 3 2   2 2 2   3   4 2   2 3   2   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     1 3 2 2     1 2 3 2     2 3 2 2     1 3 2 2     2 2 3 2     1 2 3 2     1 3 2 2     2 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 3 2     2 2 2 3     4 2 2 3     2 2 2 3     2 2 2 3     4 2 2 3     2 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 3 2   2   1 2 3   2   2 3 2   2   1 3 2   2   2 2 3   2   1 2 3   2   1 3 2   2   2 3 2   2   1 3 2   2   1 2 3   2   2 2 3   2   1 2 3   2   2 3   2 2   2   3 2 2   4 3   2 2   2 3   2 2   4   3 2 2   2   3 2 2   2 3   2 2   4 3   2 2   2 3   2 2   2   3 2 2   4   3 2 2   2   3 2 2   1 2   3 2   1   2 3 2   2 2   3 2   1 2   3 2   2   2 3 2   1   2 3 2   1 2   3 2   2 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   2 2 2   3   4 2 2   3   2 2 2   3   2 2 2   3   4 2 2   3   2 2 2   3   1 2   2 3   1   2 2 3   2 2   2 3   1 2   2 3   2   2 2 3   1   2 2 3   1 2   2 3   2 2   2 3   1 2   2 3   1   2 2 3   2   2 2 3   1   2 2 3
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html
│ │ │ @@ -125,15 +125,15 @@
│ │ │  
│ │ │  o4 : YoungTableau
    
│ │ │              
    
│ │ │                
    i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00153476s (cpu); 0.0015318s (thread); 0s (gc)
│ │ │ + -- used 0.00172447s (cpu); 0.00172097s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -149,15 +149,15 @@
│ │ │  
│ │ │  o5 : R
    
│ │ │              
    
│ │ │                
    i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.0012244s (cpu); 0.00122431s (thread); 0s (gc)
│ │ │ + -- used 0.00172632s (cpu); 0.0017267s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -173,15 +173,15 @@
│ │ │  
│ │ │  o6 : R
    
│ │ │              
    
│ │ │                
    i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.0019364s (cpu); 0.00193684s (thread); 0s (gc)
│ │ │ + -- used 0.00243406s (cpu); 0.00243378s (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 0.333052s (cpu); 0.286542s (thread); 0s (gc)
│ │ │ + -- used 0.389839s (cpu); 0.314024s (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 0.333052s (cpu); 0.286542s (thread); 0s (gc)
│ │ │ │ + -- used 0.389839s (cpu); 0.314024s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │ │                 Partition{2, 2, 2} => 1
│ │ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.667033s (cpu); 0.513099s (thread); 0s (gc)
    
│ │ │ + -- used 0.914798s (cpu); 0.598814s (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 0.667033s (cpu); 0.513099s (thread); 0s (gc)
│ │ │ │ + -- used 0.914798s (cpu); 0.598814s (thread); 0s (gc)
│ │ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │ │  
│ │ │ │                                                        2 2 2       4 2   2     2
│ │ │ │  2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2
│ │ │ │  1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2
│ │ │ │  1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2
│ │ │ │  2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │       of discriminant 31 = det| 8 1 |
│ │ │                               | 1 4 |
│ │ │       containing a surface of degree 1 and sectional genus 0
│ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │       (This is a classical example of rational fourfold)
│ │ │  
│ │ │  i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ - -- used 1.75614s (cpu); 1.01935s (thread); 0s (gc)
│ │ │ + -- used 2.9359s (cpu); 1.23963s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 14
│ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │  
│ │ │  i3 : time U' = associatedK3surface X;
│ │ │ - -- used 1.73102s (cpu); 1.08457s (thread); 0s (gc)
│ │ │ + -- used 3.29086s (cpu); 1.33721s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time U' = associatedK3surface X;
│ │ │ - -- used 6.08153s (cpu); 4.27606s (thread); 0s (gc)
│ │ │ + -- used 8.87417s (cpu); 5.02022s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : (mu,U,C,f) = building U';
│ │ │  
│ │ │  i5 : ? mu
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -8,28 +8,28 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 3.49765s (cpu); 2.02889s (thread); 0s (gc)
│ │ │ + -- used 3.72818s (cpu); 2.24547s (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 0.378374s (cpu); 0.281225s (thread); 0s (gc)
│ │ │ + -- used 0.350363s (cpu); 0.30771s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │       containing a surface in PP^8 of degree 9 and sectional genus 2
│ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 13.0413s (cpu); 7.16457s (thread); 0s (gc)
│ │ │ + -- used 23.2717s (cpu); 8.88585s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │  
│ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ @@ -29,15 +29,15 @@
│ │ │  i5 : p := point Y -- random point on Y
│ │ │  
│ │ │  o5 = point of coordinates [7214, -1460, 7057, -2440, 15907, -14345, -5937, 13402, 1]
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │  
│ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.353588s (cpu); 0.240421s (thread); 0s (gc)
│ │ │ + -- used 0.534987s (cpu); 0.305916s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │  
│ │ │  i7 : S = surface X;
│ │ │  
│ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1729890813579561111
│ │ │  
│ │ │  i1 : X = specialCubicFourfold "quintic del Pezzo surface";
│ │ │  
│ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1
│ │ │  
│ │ │  i2 : time discriminant X
│ │ │ - -- used 0.442738s (cpu); 0.162062s (thread); 0s (gc)
│ │ │ + -- used 0.424688s (cpu); 0.134221s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 14
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1730220932418738713
│ │ │  
│ │ │  i1 : X = specialGushelMukaiFourfold "tau-quadric";
│ │ │  
│ │ │  o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i2 : time discriminant X
│ │ │ - -- used 0.844046s (cpu); 0.470475s (thread); 0s (gc)
│ │ │ + -- used 1.49935s (cpu); 0.569683s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.285681s (cpu); 0.221584s (thread); 0s (gc)
│ │ │ + -- used 0.611083s (cpu); 0.326388s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Cubic__Fourfold_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = specialCubicFourfold V;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.637213s (cpu); 0.372682s (thread); 0s (gc)
│ │ │ + -- used 0.832281s (cpu); 0.447299s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │  
│ │ │  i3 : X = specialGushelMukaiFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 3.66421s (cpu); 2.33465s (thread); 0s (gc)
│ │ │ + -- used 4.79427s (cpu); 3.26519s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i3 : ? X
│ │ │  
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
│ │ │  
│ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 1.57671s (cpu); 0.755758s (thread); 0s (gc)
│ │ │ + -- used 2.35195s (cpu); 0.969033s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out
│ │ │ @@ -7,22 +7,22 @@
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i4 : X = projectiveVariety ideal(x_1^2*x_3+x_0*x_2*x_3-6*x_1*x_2*x_3-x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+2*x_0^2*x_4-10*x_0*x_1*x_4+13*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
│ │ │  
│ │ │  i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ - -- used 0.0120034s (cpu); 0.00902917s (thread); 0s (gc)
│ │ │ + -- used 0.010489s (cpu); 0.010737s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and sectional genus 0
│ │ │  
│ │ │  i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 0.309932s (cpu); 0.17223s (thread); 0s (gc)
│ │ │ + -- used 1.01011s (cpu); 0.258699s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i7 : assert(F == X)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold.out
│ │ │ @@ -7,22 +7,22 @@
│ │ │  o3 : ProjectiveVariety, surface in PP^8
│ │ │  
│ │ │  i4 : X = projectiveVariety ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8);
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8
│ │ │  
│ │ │  i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ - -- used 2.46891s (cpu); 1.74756s (thread); 0s (gc)
│ │ │ + -- used 2.33172s (cpu); 1.87624s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 5.04095s (cpu); 3.3771s (thread); 0s (gc)
│ │ │ + -- used 6.96927s (cpu); 3.71085s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : X = specialGushelMukaiFourfold(ideal(x_6-x_7, x_5, x_3-x_4, x_1, x_0-x_4, x_2*x_7-x_4*x_8), ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 3.62218s (cpu); 2.4293s (thread); 0s (gc)
│ │ │ + -- used 6.17324s (cpu); 3.24141s (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 4.86801s (cpu); 3.17605s (thread); 0s (gc)
│ │ │ + -- used 6.16626s (cpu); 3.28797s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = specialCubicFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time f = unirationalParametrization X;
│ │ │ - -- used 1.17109s (cpu); 0.597804s (thread); 0s (gc)
│ │ │ + -- used 1.74488s (cpu); 0.826696s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │  
│ │ │  i5 : degreeSequence f
│ │ │  
│ │ │  o5 = {[10]}
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │       (This is a classical example of rational fourfold)
    
│ │ │              
    
│ │ │                
    i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ - -- used 1.75614s (cpu); 1.01935s (thread); 0s (gc)
│ │ │ + -- used 2.9359s (cpu); 1.23963s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  o2 = Complete intersection of 3 quadrics in PP^7
│ │ │ │       of discriminant 31 = det| 8 1 |
│ │ │ │                               | 1 4 |
│ │ │ │       containing a surface of degree 1 and sectional genus 0
│ │ │ │       cut out by 5 hypersurfaces of degree 1
│ │ │ │       (This is a classical example of rational fourfold)
│ │ │ │  i3 : time U' = associatedCastelnuovoSurface X;
│ │ │ │ - -- used 1.75614s (cpu); 1.01935s (thread); 0s (gc)
│ │ │ │ + -- used 2.9359s (cpu); 1.23963s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^7 cut out by 2
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │              
    
│ │ │                
    i3 : time U' = associatedK3surface X;
│ │ │ - -- used 1.73102s (cpu); 1.08457s (thread); 0s (gc)
│ │ │ + -- used 3.29086s (cpu); 1.33721s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 14
│ │ │ │       containing a (smooth) surface of degree 4 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degree 2
│ │ │ │  i3 : time U' = associatedK3surface X;
│ │ │ │ - -- used 1.73102s (cpu); 1.08457s (thread); 0s (gc)
│ │ │ │ + -- used 3.29086s (cpu); 1.33721s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -107,15 +107,15 @@
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │              
    
│ │ │                
    i3 : time U' = associatedK3surface X;
│ │ │ - -- used 6.08153s (cpu); 4.27606s (thread); 0s (gc)
│ │ │ + -- used 8.87417s (cpu); 5.02022s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
    
│ │ │              
    
│ │ │                
    i4 : (mu,U,C,f) = building U';
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 1 of Table 1 in arXiv:2002.07026)
│ │ │ │  i3 : time U' = associatedK3surface X;
│ │ │ │ - -- used 6.08153s (cpu); 4.27606s (thread); 0s (gc)
│ │ │ │ + -- used 8.87417s (cpu); 5.02022s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : (mu,U,C,f) = building U';
│ │ │ │  i5 : ? mu
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │              
    
│ │ │                
    i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 3.49765s (cpu); 2.02889s (thread); 0s (gc)
│ │ │ + -- used 3.72818s (cpu); 2.24547s (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 0.378374s (cpu); 0.281225s (thread); 0s (gc)
│ │ │ + -- used 0.350363s (cpu); 0.30771s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
    
│ │ │              
    
│ │ │                
    i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,28 +30,28 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 3.49765s (cpu); 2.02889s (thread); 0s (gc)
│ │ │ │ + -- used 3.72818s (cpu); 2.24547s (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 0.378374s (cpu); 0.281225s (thread); 0s (gc)
│ │ │ │ + -- used 0.350363s (cpu); 0.30771s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree
│ │ │ │  (C * surface X) == 5 and isSubset(p, C))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_t_e_c_t_C_o_n_g_r_u_e_n_c_e_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_,_Z_Z_) -- detect and return a
│ │ │ │        congruence of (2e-1)-secant curves of degree e inside a del Pezzo
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │              
    
│ │ │                
    i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 13.0413s (cpu); 7.16457s (thread); 0s (gc)
│ │ │ + -- used 23.2717s (cpu); 8.88585s (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 0.353588s (cpu); 0.240421s (thread); 0s (gc)
│ │ │ + -- used 0.534987s (cpu); 0.305916s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
    
│ │ │              
    
│ │ │                
    i7 : S = surface X;
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 20
│ │ │ │       containing a surface in PP^8 of degree 9 and sectional genus 2
│ │ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 13.0413s (cpu); 7.16457s (thread); 0s (gc)
│ │ │ │ + -- used 23.2717s (cpu); 8.88585s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the quadratic map and passing through a
│ │ │ │  general point: 7
│ │ │ │  number 1-secant lines = 6
│ │ │ │  number 3-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  i5 : p := point Y -- random point on Y
│ │ │ │  
│ │ │ │  o5 = point of coordinates [7214, -1460, 7057, -2440, 15907, -14345, -5937,
│ │ │ │  13402, 1]
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ │ - -- used 0.353588s (cpu); 0.240421s (thread); 0s (gc)
│ │ │ │ + -- used 0.534987s (cpu); 0.305916s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │ │  i7 : S = surface X;
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ │ │  i8 : assert(dim C == 1 and degree C == 2 and dim(C*S) == 0 and degree(C*S) == 3
│ │ │ │  and isSubset(p,C) and isSubset(C,Y))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  
│ │ │  o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1
    
│ │ │              
    
│ │ │                
    i2 : time discriminant X
│ │ │ - -- used 0.442738s (cpu); 0.162062s (thread); 0s (gc)
│ │ │ + -- used 0.424688s (cpu); 0.134221s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 14
    
│ │ │              
    
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ thanks to the functions _E_u_l_e_r_C_h_a_r_a_c_t_e_r_i_s_t_i_c and _E_u_l_e_r (the option Algorithm │ │ │ │ allows you to select the method). │ │ │ │ i1 : X = specialCubicFourfold "quintic del Pezzo surface"; │ │ │ │ │ │ │ │ o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and │ │ │ │ sectional genus 1 │ │ │ │ i2 : time discriminant X │ │ │ │ - -- used 0.442738s (cpu); 0.162062s (thread); 0s (gc) │ │ │ │ + -- used 0.424688s (cpu); 0.134221s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 14 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_) -- discriminant of a special │ │ │ │ Gushel-Mukai fourfold │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * discriminant(HodgeSpecialFourfold) │ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.html │ │ │ @@ -80,15 +80,15 @@ │ │ │ │ │ │ o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time discriminant X
│ │ │ - -- used 0.844046s (cpu); 0.470475s (thread); 0s (gc)
│ │ │ + -- used 1.49935s (cpu); 0.569683s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ the functions _c_y_c_l_e_C_l_a_s_s, _E_u_l_e_r_C_h_a_r_a_c_t_e_r_i_s_t_i_c and _E_u_l_e_r (the option Algorithm │ │ │ │ allows you to select the method). │ │ │ │ i1 : X = specialGushelMukaiFourfold "tau-quadric"; │ │ │ │ │ │ │ │ o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and │ │ │ │ sectional genus 0 │ │ │ │ i2 : time discriminant X │ │ │ │ - -- used 0.844046s (cpu); 0.470475s (thread); 0s (gc) │ │ │ │ + -- used 1.49935s (cpu); 0.569683s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 10 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_C_u_b_i_c_F_o_u_r_f_o_l_d_) -- discriminant of a special cubic │ │ │ │ fourfold │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_S_p_e_c_i_a_l_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_) -- discriminant of a special │ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ @@ -88,15 +88,15 @@ │ │ │ │ │ │ o3 : ProjectiveVariety, surface in PP^5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.285681s (cpu); 0.221584s (thread); 0s (gc)
│ │ │ + -- used 0.611083s (cpu); 0.326388s (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 0.285681s (cpu); 0.221584s (thread); 0s (gc)
│ │ │ │ + -- used 0.611083s (cpu); 0.326388s (thread); 0s (gc)
│ │ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3
│ │ │ │  hypersurfaces of degree 2
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 32
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Cubic__Fourfold_rp.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.637213s (cpu); 0.372682s (thread); 0s (gc)
│ │ │ + -- used 0.832281s (cpu); 0.447299s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = specialCubicFourfold V;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 0.637213s (cpu); 0.372682s (thread); 0s (gc)
│ │ │ │ + -- used 0.832281s (cpu); 0.447299s (thread); 0s (gc)
│ │ │ │  S: Veronese surface in PP^5
│ │ │ │  X: smooth cubic hypersurface in PP^5
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 27
│ │ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 3.66421s (cpu); 2.33465s (thread); 0s (gc)
│ │ │ + -- used 4.79427s (cpu); 3.26519s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │ │  i3 : X = specialGushelMukaiFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 3.66421s (cpu); 2.33465s (thread); 0s (gc)
│ │ │ │ + -- used 4.79427s (cpu); 3.26519s (thread); 0s (gc)
│ │ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │ │  X: GM fourfold containing S
│ │ │ │  Y: del Pezzo fivefold containing X
│ │ │ │  h^1(N_{S,Y}) = 0
│ │ │ │  h^0(N_{S,Y}) = 11
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 1.57671s (cpu); 0.755758s (thread); 0s (gc)
│ │ │ + -- used 2.35195s (cpu); 0.969033s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^9
│ │ │ │  i3 : ? X
│ │ │ │  
│ │ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │ │       1^2 2^5
│ │ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ │ - -- used 1.57671s (cpu); 0.755758s (thread); 0s (gc)
│ │ │ │ + -- used 2.35195s (cpu); 0.969033s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7
│ │ │ │  hypersurfaces of degrees 1^2 2^5
│ │ │ │       dominance: true
│ │ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Cubic__Fourfold.html
│ │ │ @@ -95,25 +95,25 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^5
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time F = specialCubicFourfold(S,X,NumNodes=>3);
│ │ │ - -- used 0.0120034s (cpu); 0.00902917s (thread); 0s (gc)
│ │ │ + -- used 0.010489s (cpu); 0.010737s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and sectional genus 0
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 0.309932s (cpu); 0.17223s (thread); 0s (gc)
│ │ │ + -- used 1.01011s (cpu); 0.258699s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -115,24 +115,24 @@ │ │ │ │ 3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2- │ │ │ │ 7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3- │ │ │ │ x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5- │ │ │ │ x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2); │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^5 │ │ │ │ i5 : time F = specialCubicFourfold(S,X,NumNodes=>3); │ │ │ │ - -- used 0.0120034s (cpu); 0.00902917s (thread); 0s (gc) │ │ │ │ + -- used 0.010489s (cpu); 0.010737s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and │ │ │ │ sectional genus 0 │ │ │ │ i6 : time describe F │ │ │ │ warning: clearing value of symbol x to allow access to subscripted variables │ │ │ │ based on it │ │ │ │ : debug with expression debug 9868 or with command line option -- │ │ │ │ debug 9868 │ │ │ │ - -- used 0.309932s (cpu); 0.17223s (thread); 0s (gc) │ │ │ │ + -- used 1.01011s (cpu); 0.258699s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = Special cubic fourfold of discriminant 26 │ │ │ │ containing a 3-nodal surface of degree 7 and sectional genus 0 │ │ │ │ cut out by 13 hypersurfaces of degree 3 │ │ │ │ i7 : assert(F == X) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_p_e_c_i_a_l_C_u_b_i_c_F_o_u_r_f_o_l_d_(_E_m_b_e_d_d_e_d_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- random special cubic │ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Gushel__Mukai__Fourfold.html │ │ │ @@ -93,25 +93,25 @@ │ │ │ │ │ │ o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time F = specialGushelMukaiFourfold(S,X);
│ │ │ - -- used 2.46891s (cpu); 1.74756s (thread); 0s (gc)
│ │ │ + -- used 2.33172s (cpu); 1.87624s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time describe F
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 5.04095s (cpu); 3.3771s (thread); 0s (gc)
│ │ │ + -- used 6.96927s (cpu); 3.71085s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface in PP^8 of degree 2 and sectional genus 0
│ │ │       cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
│ │ │       and with class in G(1,4) given by s_(3,1)+s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 1 of Table 1 in arXiv:2002.07026)
    │ │ │ ├── html2text {} │ │ │ │ @@ -33,24 +33,24 @@ │ │ │ │ x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2- │ │ │ │ x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4- │ │ │ │ 2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2- │ │ │ │ 3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8); │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8 │ │ │ │ i5 : time F = specialGushelMukaiFourfold(S,X); │ │ │ │ - -- used 2.46891s (cpu); 1.74756s (thread); 0s (gc) │ │ │ │ + -- used 2.33172s (cpu); 1.87624s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and │ │ │ │ sectional genus 0 │ │ │ │ i6 : time describe F │ │ │ │ warning: clearing value of symbol x to allow access to subscripted variables │ │ │ │ based on it │ │ │ │ : debug with expression debug 9868 or with command line option -- │ │ │ │ debug 9868 │ │ │ │ - -- used 5.04095s (cpu); 3.3771s (thread); 0s (gc) │ │ │ │ + -- used 6.96927s (cpu); 3.71085s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = Special Gushel-Mukai fourfold of discriminant 10(') │ │ │ │ containing a surface in PP^8 of degree 2 and sectional genus 0 │ │ │ │ cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2) │ │ │ │ and with class in G(1,4) given by s_(3,1)+s_(2,2) │ │ │ │ Type: ordinary │ │ │ │ (case 1 of Table 1 in arXiv:2002.07026) │ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 3.62218s (cpu); 2.4293s (thread); 0s (gc)
│ │ │ + -- used 6.17324s (cpu); 3.24141s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -26,15 +26,15 @@ │ │ │ │ o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and │ │ │ │ sectional genus 0 │ │ │ │ i3 : time toGrass X │ │ │ │ warning: clearing value of symbol x to allow access to subscripted variables │ │ │ │ based on it │ │ │ │ : debug with expression debug 9868 or with command line option -- │ │ │ │ debug 9868 │ │ │ │ - -- used 3.62218s (cpu); 2.4293s (thread); 0s (gc) │ │ │ │ + -- used 6.17324s (cpu); 3.24141s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = multi-rational map consisting of one single rational map │ │ │ │ source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces │ │ │ │ of degree 2 │ │ │ │ target variety: GG(1,4) ⊂ PP^9 │ │ │ │ │ │ │ │ o3 : MultirationalMap (rational map from X to GG(1,4)) │ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html │ │ │ @@ -82,15 +82,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.86801s (cpu); 3.17605s (thread); 0s (gc)
│ │ │ + -- used 6.16626s (cpu); 3.28797s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -25,15 +25,15 @@ │ │ │ │ │ │ │ │ o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8 │ │ │ │ i3 : time toGrass X │ │ │ │ warning: clearing value of symbol x to allow access to subscripted variables │ │ │ │ based on it │ │ │ │ : debug with expression debug 9868 or with command line option -- │ │ │ │ debug 9868 │ │ │ │ - -- used 4.86801s (cpu); 3.17605s (thread); 0s (gc) │ │ │ │ + -- used 6.16626s (cpu); 3.28797s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = multi-rational map consisting of one single rational map │ │ │ │ source variety: 5-dimensional subvariety of PP^8 cut out by 5 │ │ │ │ hypersurfaces of degree 2 │ │ │ │ target variety: GG(1,4) ⊂ PP^9 │ │ │ │ │ │ │ │ o3 : MultirationalMap (rational map from X to GG(1,4)) │ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ @@ -82,15 +82,15 @@ │ │ │ │ │ │ o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time f = unirationalParametrization X;
│ │ │ - -- used 1.17109s (cpu); 0.597804s (thread); 0s (gc)
│ │ │ + -- used 1.74488s (cpu); 0.826696s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : degreeSequence f
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = specialCubicFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time f = unirationalParametrization X;
│ │ │ │ - -- used 1.17109s (cpu); 0.597804s (thread); 0s (gc)
│ │ │ │ + -- used 1.74488s (cpu); 0.826696s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │ │  i5 : degreeSequence f
│ │ │ │  
│ │ │ │  o5 = {[10]}
│ │ │ │  
│ │ │ │  o5 : List
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/example-output/_graph_lp__Mixed__Graph_rp.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │                                b => {a, c}
│ │ │                                c => {b}
│ │ │  
│ │ │  o2 : HashTable
│ │ │  
│ │ │  i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Digraph, Graph, Bigraph}
│ │ │ +o3 = {Graph, Bigraph, Digraph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_graph_lp__Mixed__Graph_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  o2 : HashTable
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Digraph, Graph, Bigraph}
│ │ │ +o3 = {Graph, Bigraph, Digraph}
│ │ │  
│ │ │  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 = {Digraph, Graph, Bigraph}
│ │ │ │ +o3 = {Graph, Bigraph, Digraph}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_i_x_e_d_G_r_a_p_h -- a graph that has undirected, directed and bidirected edges
│ │ ├── ./usr/share/doc/Macaulay2/Style/example-output/_generate__Grammar.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 3455701143666534588
│ │ │  
│ │ │  i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10066-0/0
│ │ │ +o1 = /tmp/M2-10106-0/0
│ │ │  
│ │ │  i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10066-0/0.in
│ │ │ +o2 = /tmp/M2-10106-0/0.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/M2-10066-0/0
│ │ │ + -- generating /tmp/M2-10106-0/0
│ │ │  
│ │ │  i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.25.11. Do not modify this file manually.
│ │ │  
│ │ │        This is an example file for the generateGrammar method!
│ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ ├── ./usr/share/doc/Macaulay2/Style/html/_generate__Grammar.html
│ │ │ @@ -82,22 +82,22 @@
│ │ │            
    The function demarkf indicates how the elements of each of the lists will be demarked in the resulting file.   The file outfile will then be generated, replacing each of these strings as indicated above.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -143,15 +143,15 @@ │ │ │ @M2KEYWORDS@ │ │ │ } │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10066-0/0
    │ │ │ +o1 = /tmp/M2-10106-0/0 │ │ │
    │ │ │ 
    i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10066-0/0.in
    │ │ │ +o2 = /tmp/M2-10106-0/0.in │ │ │
    │ │ │ 
    i3 : template << "@M2BANNER@" << endl << endl;
    │ │ │
    │ │ │ 
    i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10066-0/0
    │ │ │ + -- generating /tmp/M2-10106-0/0 │ │ │
    │ │ │ 
    i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.25.11. Do not modify this file manually.
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,18 +26,18 @@
│ │ │ │      * @M2CONSTANTS@, for a list of Macaulay2 symbols and packages.
│ │ │ │      * @M2STRINGS@, for a regular expression that matches Macaulay2 strings.
│ │ │ │  The function demarkf indicates how the elements of each of the lists will be
│ │ │ │  demarked in the resulting file. The file outfile will then be generated,
│ │ │ │  replacing each of these strings as indicated above.
│ │ │ │  i1 : outfile = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10066-0/0
│ │ │ │ +o1 = /tmp/M2-10106-0/0
│ │ │ │  i2 : template = outfile | ".in"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10066-0/0.in
│ │ │ │ +o2 = /tmp/M2-10106-0/0.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/M2-10066-0/0
│ │ │ │ + -- generating /tmp/M2-10106-0/0
│ │ │ │  i12 : get outfile
│ │ │ │  
│ │ │ │  o12 = Auto-generated for Macaulay2-1.25.11. Do not modify this file manually.
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ │ │        List of keywords: {
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out
│ │ │ @@ -31,15 +31,15 @@
│ │ │  o5 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i6 : isHomogeneous P
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │  i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.243965s (cpu); 0.155289s (thread); 0s (gc)
│ │ │ + -- used 0.343605s (cpu); 0.194413s (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 0.0338193s (cpu); 0.0338249s (thread); 0s (gc)
│ │ │ + -- used 0.0383245s (cpu); 0.0383277s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html
│ │ │ @@ -141,15 +141,15 @@
│ │ │  
│ │ │  o6 = false
    │ │ │
    │ │ │ 
    i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.243965s (cpu); 0.155289s (thread); 0s (gc)
│ │ │ + -- used 0.343605s (cpu); 0.194413s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
    │ │ │
    │ │ │ 
    i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o9 = true
    │ │ │
    │ │ │ 
    i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.0338193s (cpu); 0.0338249s (thread); 0s (gc)
│ │ │ + -- used 0.0383245s (cpu); 0.0383277s (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 0.243965s (cpu); 0.155289s (thread); 0s (gc) │ │ │ │ + -- used 0.343605s (cpu); 0.194413s (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 0.0338193s (cpu); 0.0338249s (thread); 0s (gc) │ │ │ │ + -- used 0.0383245s (cpu); 0.0383277s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 : Ideal of QQ[x..z] │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_y_m_b_P_o_w_e_r_P_r_i_m_e_P_o_s_C_h_a_r │ │ │ │ ********** WWaayyss ttoo uussee ssyymmbboolliiccPPoowweerr:: ********** │ │ │ │ * symbolicPower(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ o3 = 0 <-- E <-- 0 │ │ │ │ │ │ -1 0 1 │ │ │ │ │ │ o3 : ChainComplex │ │ │ │ │ │ i4 : time T=tateExtension W; │ │ │ - -- used 0.118757s (cpu); 0.118758s (thread); 0s (gc) │ │ │ + -- used 0.156976s (cpu); 0.153547s (thread); 0s (gc) │ │ │ │ │ │ i5 : cohomologyMatrix(T,-{3,3},{3,3}) │ │ │ │ │ │ o5 = | 8h 4h 0 4 8 12 16 | │ │ │ | 6h 3h 0 3 6 9 12 | │ │ │ | 4h 2h 0 2 4 6 8 | │ │ │ | 2h h 0 1 2 3 4 | │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html │ │ │ @@ -92,15 +92,15 @@ │ │ │ │ │ │ o3 : ChainComplex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time T=tateExtension W;
│ │ │ - -- used 0.118757s (cpu); 0.118758s (thread); 0s (gc)
    │ │ │ + -- used 0.156976s (cpu); 0.153547s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │               1
│ │ │ │  o3 = 0  <-- E  <-- 0
│ │ │ │  
│ │ │ │       -1     0      1
│ │ │ │  
│ │ │ │  o3 : ChainComplex
│ │ │ │  i4 : time T=tateExtension W;
│ │ │ │ - -- used 0.118757s (cpu); 0.118758s (thread); 0s (gc)
│ │ │ │ + -- used 0.156976s (cpu); 0.153547s (thread); 0s (gc)
│ │ │ │  i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │ │  
│ │ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ │       | 6h  3h  0 3  6  9  12 |
│ │ │ │       | 4h  2h  0 2  4  6  8  |
│ │ │ │       | 2h  h   0 1  2  3  4  |
│ │ │ │       | 0   0   0 0  0  0  0  |
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out
│ │ │ @@ -63,20 +63,20 @@
│ │ │  o15 : Ideal of R
│ │ │  
│ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 0.840982s (cpu); 0.654651s (thread); 0s (gc)
│ │ │ + -- used 1.09701s (cpu); 0.777006s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
│ │ │  
│ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.46291s (cpu); 2.11269s (thread); 0s (gc)
│ │ │ + -- used 3.04296s (cpu); 2.38122s (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 0.00199605s (cpu); 0.00199517s (thread); 0s (gc)
│ │ │ + -- used 0.00249223s (cpu); 0.00248692s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00461441s (cpu); 0.00462066s (thread); 0s (gc)
│ │ │ + -- used 0.00587824s (cpu); 0.00588542s (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 0.0452172s (cpu); 0.0452175s (thread); 0s (gc)
│ │ │ + -- used 0.0341707s (cpu); 0.0341764s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.37069s (cpu); 1.40142s (thread); 0s (gc)
│ │ │ + -- used 3.04643s (cpu); 1.62192s (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 0.125282s (cpu); 0.0858467s (thread); 0s (gc)
│ │ │ + -- used 0.212553s (cpu); 0.124895s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.155489s (cpu); 0.0965796s (thread); 0s (gc)
│ │ │ + -- used 0.210233s (cpu); 0.116497s (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 0.881446s (cpu); 0.69684s (thread); 0s (gc)
│ │ │ + -- used 1.1462s (cpu); 0.870316s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.347943s (cpu); 0.24112s (thread); 0s (gc)
│ │ │ + -- used 0.467624s (cpu); 0.297134s (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 0.117837s (cpu); 0.0598711s (thread); 0s (gc)
│ │ │ + -- used 0.165014s (cpu); 0.0807181s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
│ │ │  
│ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.225288s (cpu); 0.168828s (thread); 0s (gc)
│ │ │ + -- used 0.304724s (cpu); 0.22157s (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 0.353165s (cpu); 0.184626s (thread); 0s (gc)
│ │ │ + -- used 0.473545s (cpu); 0.224835s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.390436s (cpu); 0.235485s (thread); 0s (gc)
│ │ │ + -- used 0.5423s (cpu); 0.304902s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html
│ │ │ @@ -226,23 +226,23 @@
│ │ │  
│ │ │  o16 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 0.840982s (cpu); 0.654651s (thread); 0s (gc)
│ │ │ + -- used 1.09701s (cpu); 0.777006s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.46291s (cpu); 2.11269s (thread); 0s (gc)
│ │ │ + -- used 3.04296s (cpu); 2.38122s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i19 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  i15 : I2 = ideal(x^20*y^100, x + z^100);
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ │ - -- used 0.840982s (cpu); 0.654651s (thread); 0s (gc)
│ │ │ │ + -- used 1.09701s (cpu); 0.777006s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 : Ideal of R
│ │ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ │ - -- used 2.46291s (cpu); 2.11269s (thread); 0s (gc)
│ │ │ │ + -- used 3.04296s (cpu); 2.38122s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 : Ideal of R
│ │ │ │  i19 : J1 == J2
│ │ │ │  
│ │ │ │  o19 = true
│ │ │ │  For legacy reasons, the last ideal in the list can be specified separately,
│ │ │ │  using frobeniusRoot(e, \{a_1,\ldots,a_n\}, \{I_1,\ldots,I_n\}, I). The last
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html
│ │ │ @@ -96,23 +96,23 @@
│ │ │              
│ │ │                
    i4 : R = S/(ker g);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00199605s (cpu); 0.00199517s (thread); 0s (gc)
│ │ │ + -- used 0.00249223s (cpu); 0.00248692s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00461441s (cpu); 0.00462066s (thread); 0s (gc)
│ │ │ + -- used 0.00587824s (cpu); 0.00588542s (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 0.00199605s (cpu); 0.00199517s (thread); 0s (gc)
│ │ │ │ + -- used 0.00249223s (cpu); 0.00248692s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ │ - -- used 0.00461441s (cpu); 0.00462066s (thread); 0s (gc)
│ │ │ │ + -- used 0.00587824s (cpu); 0.00588542s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │ │  i8 : isCohenMacaulay(R)
│ │ │ │  
│ │ │ │  o8 = false
│ │ │ │  The function isCohenMacaulay considers $R$ as a quotient of a polynomial ring,
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html
│ │ │ @@ -214,23 +214,23 @@
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i20 : time isFInjective(R)
│ │ │ - -- used 0.0452172s (cpu); 0.0452175s (thread); 0s (gc)
│ │ │ + -- used 0.0341707s (cpu); 0.0341764s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
    
│ │ │              
    
│ │ │                
    i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.37069s (cpu); 1.40142s (thread); 0s (gc)
│ │ │ + -- used 3.04643s (cpu); 1.62192s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If the option AtOrigin (default value false) is set to true, isFInjective will only check $F$-injectivity at the origin. Otherwise, it will check $F$-injectivity globally. Note that checking $F$-injectivity at the origin can be slower than checking it globally. Consider the following example of a non-$F$-injective ring.
    
│ │ │ @@ -240,23 +240,23 @@
│ │ │              
│ │ │                
    i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time isFInjective(R)
│ │ │ - -- used 0.125282s (cpu); 0.0858467s (thread); 0s (gc)
│ │ │ + -- used 0.212553s (cpu); 0.124895s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.155489s (cpu); 0.0965796s (thread); 0s (gc)
│ │ │ + -- used 0.210233s (cpu); 0.116497s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If the option AssumeCM (default value false) is set to true, then isFInjective only checks the Frobenius action on top cohomology (which is typically much faster). Note that it can give an incorrect answer if the non-injective Frobenius occurs in a lower degree.  Consider the example of the cone over a supersingular elliptic curve times $\mathbb{P}^1$.
    
│ │ │ @@ -283,23 +283,23 @@
│ │ │              
│ │ │                
    i28 : R = S/(ker f);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : time isFInjective(R)
│ │ │ - -- used 0.881446s (cpu); 0.69684s (thread); 0s (gc)
│ │ │ + -- used 1.1462s (cpu); 0.870316s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.347943s (cpu); 0.24112s (thread); 0s (gc)
│ │ │ + -- used 0.467624s (cpu); 0.297134s (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 0.0452172s (cpu); 0.0452175s (thread); 0s (gc)
│ │ │ │ + -- used 0.0341707s (cpu); 0.0341764s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = true
│ │ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ │ - -- used 2.37069s (cpu); 1.40142s (thread); 0s (gc)
│ │ │ │ + -- used 3.04643s (cpu); 1.62192s (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 0.125282s (cpu); 0.0858467s (thread); 0s (gc)
│ │ │ │ + -- used 0.212553s (cpu); 0.124895s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ │ - -- used 0.155489s (cpu); 0.0965796s (thread); 0s (gc)
│ │ │ │ + -- used 0.210233s (cpu); 0.116497s (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 0.881446s (cpu); 0.69684s (thread); 0s (gc)
│ │ │ │ + -- used 1.1462s (cpu); 0.870316s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = false
│ │ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ │ - -- used 0.347943s (cpu); 0.24112s (thread); 0s (gc)
│ │ │ │ + -- used 0.467624s (cpu); 0.297134s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  If the option AssumedReduced is set to true (its default behavior), then the
│ │ │ │  bottom local cohomology is avoided (this means the Frobenius action on the top
│ │ │ │  potentially nonzero Ext is not computed).
│ │ │ │  If the option AssumeNormal (default value false) is set to true, then the
│ │ │ │  bottom two local cohomology modules (or, rather, their duals) need not be
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html
│ │ │ @@ -273,23 +273,23 @@
│ │ │                
    i26 : debugLevel = 1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.117837s (cpu); 0.0598711s (thread); 0s (gc)
│ │ │ + -- used 0.165014s (cpu); 0.0807181s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.225288s (cpu); 0.168828s (thread); 0s (gc)
│ │ │ + -- used 0.304724s (cpu); 0.22157s (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 0.117837s (cpu); 0.0598711s (thread); 0s (gc)
│ │ │ │ + -- used 0.165014s (cpu); 0.0807181s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = false
│ │ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ │ - -- used 0.225288s (cpu); 0.168828s (thread); 0s (gc)
│ │ │ │ + -- used 0.304724s (cpu); 0.22157s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = true
│ │ │ │  i29 : debugLevel = 0;
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_e_s_t_I_d_e_a_l -- compute a test ideal in a Q-Gorenstein ring
│ │ │ │      * _i_s_F_R_a_t_i_o_n_a_l -- whether a ring is F-rational
│ │ │ │  ********** WWaayyss ttoo uussee iissFFRReegguullaarr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html
│ │ │ @@ -255,25 +255,25 @@
│ │ │          
    
│ │ │            
    It is often more efficient to pass a list, as opposed to finding a common denominator and passing a single element, since testIdeal can do things in a more intelligent way for such a list.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.353165s (cpu); 0.184626s (thread); 0s (gc)
│ │ │ + -- used 0.473545s (cpu); 0.224835s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
    
│ │ │                
    i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.390436s (cpu); 0.235485s (thread); 0s (gc)
│ │ │ + -- used 0.5423s (cpu); 0.304902s (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 0.353165s (cpu); 0.184626s (thread); 0s (gc)
│ │ │ │ + -- used 0.473545s (cpu); 0.224835s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = ideal (y, x)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ │ - -- used 0.390436s (cpu); 0.235485s (thread); 0s (gc)
│ │ │ │ + -- used 0.5423s (cpu); 0.304902s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = ideal (y, x)
│ │ │ │  
│ │ │ │  o24 : Ideal of R
│ │ │ │  The option AssumeDomain (default value false) is used when finding a test
│ │ │ │  element. The option FrobeniusRootStrategy (default value Substitution) is
│ │ │ │  passed to internal _f_r_o_b_e_n_i_u_s_R_o_o_t calls.
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out
│ │ │ @@ -2,17 +2,15 @@
│ │ │  
│ │ │  i1 : S = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ -                  ((0, 1), 1) => null
│ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out
│ │ │ @@ -2,16 +2,16 @@
│ │ │  
│ │ │  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) => null}
│ │ │ -                  ((0, 2), 0) => null
│ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ +                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out
│ │ │ @@ -2,16 +2,18 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ +                                   3
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │ -o3 = LineageTable{((1, 2), 0) => -c      }
│ │ │ +                  ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │ @@ -20,16 +22,17 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : minimize T
│ │ │  
│ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │ -o4 = LineageTable{((1, 2), 0) => c }
│ │ │ +                  ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out
│ │ │ @@ -2,18 +2,18 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │                                       2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ +                                   3
│ │ │ +                  ((0, 2), 0) => -c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -24,16 +24,16 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ +                  ((0, 2), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out
│ │ │ @@ -6,24 +6,60 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : allowableThreads  = 4;
│ │ │  
│ │ │  i4 : H = tgb I
│ │ │  
│ │ │ -                                      4 7      3 7
│ │ │ -o4 = LineageTable{((1, 2), 1) => - 30y z  - 48y z }
│ │ │ -                                    3 8      2 7
│ │ │ -                  ((1, 2), 2) => 26y z  - 35y z
│ │ │ +                                         3 8      2 9
│ │ │ +o4 = LineageTable{((0, 1), (1, 2)) => 25y z  - 36y z   }
│ │ │ +                                           3 9      3 8
│ │ │ +                  ((0, 1), (1, 3)) => - 39y z  + 24y z
│ │ │ +                                         3 4      2 4
│ │ │ +                  ((0, 1), (2, 3)) => 25y z  + 40y z
│ │ │ +                                   4 4      3 6
│ │ │ +                  ((0, 1), 2) => 9y z  - 27y z
│ │ │ +                                      3 9      3 8
│ │ │ +                  ((0, 1), 3) => - 19y z  + 35y z
│ │ │ +                                         2 5      2 4
│ │ │ +                  ((0, 2), (0, 1)) => 40y z  + 22y z
│ │ │ +                                         2 7     2 5
│ │ │ +                  ((0, 2), (1, 2)) => 22y z  + 5y z
│ │ │ +                                         2 6      2 4
│ │ │ +                  ((0, 2), (1, 3)) => 25y z  + 24y z
│ │ │ +                                      2 14     2 12
│ │ │ +                  ((0, 2), 1) => - 19y z   + 6y z
│ │ │                                            2 4
│ │ │ -                  ((1, 3), (1, 2)) => -38y z
│ │ │ -                                 3 4     2 5
│ │ │ -                  (0, 1) => - 19y z  - 9y z
│ │ │ -                               2 6     2 4
│ │ │ -                  (0, 2) => 22y z  + 9y z
│ │ │ +                  ((0, 3), (1, 2)) => -18y z
│ │ │ +                                      4 11      3 14
│ │ │ +                  ((1, 2), 1) => - 14y z   + 43y z
│ │ │ +                                     4 8      3 10
│ │ │ +                  ((1, 2), 3) => - 9y z  + 27y z
│ │ │ +                                        3 16     3 11
│ │ │ +                  ((1, 3), (1, 2)) => 9y z   - 7y z
│ │ │ +                                     3 13      3 12
│ │ │ +                  ((1, 3), 1) => - 8y z   + 36y z
│ │ │ +                                      4 6      3 10
│ │ │ +                  ((1, 3), 2) => - 23y z  - 34y z
│ │ │ +                                      3 12      3 8
│ │ │ +                  ((1, 3), 3) => - 12y z   + 32y z
│ │ │ +                                        4 4      2 8
│ │ │ +                  ((2, 3), (1, 2)) => 8y z  - 40y z
│ │ │ +                                           4 4      3 6
│ │ │ +                  ((2, 3), (1, 3)) => - 40y z  + 19y z
│ │ │ +                                     3 12
│ │ │ +                  ((2, 3), 1) => -14y z
│ │ │ +                                     4 5     4 4
│ │ │ +                  ((2, 3), 2) => - 2y z  + 9y z
│ │ │ +                                     4 5      3 6
│ │ │ +                  ((2, 3), 3) => - 2y z  + 27y z
│ │ │ +                                 5 2      3 4
│ │ │ +                  (0, 1) => - 25y z  - 19y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │                                 5      2 4
│ │ │                    (0, 3) => 28y z - 2y z
│ │ │                                 5 6      4 5
│ │ │                    (1, 2) => 19y z  - 45y z
│ │ │                                   7 2      5 5
│ │ │                    (1, 3) => - 39y z  + 19y z
│ │ │                                 5 2     2 4
│ │ │ @@ -37,16 +73,16 @@
│ │ │                             3         3
│ │ │                    3 => 9x*y z + 10x*y
│ │ │  
│ │ │  o4 : LineageTable
│ │ │  
│ │ │  i5 : H#(0,1)
│ │ │  
│ │ │ -          3 4     2 5
│ │ │ -o5 = - 19y z  - 9y z
│ │ │ +          5 2      3 4
│ │ │ +o5 = - 25y z  - 19y z
│ │ │  
│ │ │  o5 : R
│ │ │  
│ │ │  i6 : QQ[a..d];
│ │ │  
│ │ │  i7 : f0 = a*b-c^2;
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html
│ │ │ @@ -73,17 +73,15 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ -                  ((0, 1), 1) => null
│ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,17 +12,15 @@
│ │ │ │  Gr\"obner basis is minimized. Lineages of non-minimal Gr\"obner basis elements
│ │ │ │  that were added to the basis during the distributed computation are saved, with
│ │ │ │  the corresponding entry in the table being null.
│ │ │ │  i1 : S = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │ │  
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ -                  ((0, 1), 1) => null
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html
│ │ │ @@ -86,16 +86,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{((0, 1), 0) => null}
│ │ │ -                  ((0, 2), 0) => null
│ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ +                  ((0, 2), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,16 +19,16 @@
│ │ │ │  This simple function just returns the Gr\"obner basis computed with threaded
│ │ │ │  Gr\"obner basis function _t_g_b in the expected Macaulay2 format, so that further
│ │ │ │  computation are one step easier to set up.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -o3 = LineageTable{((0, 1), 0) => null}
│ │ │ │ -                  ((0, 2), 0) => null
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ │ +                  ((0, 2), 1) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html
│ │ │ @@ -82,16 +82,18 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ +                                   3
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │ -o3 = LineageTable{((1, 2), 0) => -c      }
│ │ │ +                  ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │ @@ -103,16 +105,17 @@
│ │ │  o3 : LineageTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : minimize T
│ │ │  
│ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │ -o4 = LineageTable{((1, 2), 0) => c }
│ │ │ +                  ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,16 +19,18 @@
│ │ │ │  minimal generators of the ideal generated by the leading terms of the values of
│ │ │ │  H. If the values of H constitute a Gr\"obner basis of the ideal they generate,
│ │ │ │  this method returns a minimal Gr\"obner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ +                                   3
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │                                     2
│ │ │ │ -o3 = LineageTable{((1, 2), 0) => -c      }
│ │ │ │ +                  ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │                                  2
│ │ │ │                    0 => a*b*c + c
│ │ │ │ @@ -36,16 +38,17 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : minimize T
│ │ │ │  
│ │ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │ -o4 = LineageTable{((1, 2), 0) => c }
│ │ │ │ +                  ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html
│ │ │ @@ -82,18 +82,18 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │                                       2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ +                                   3
│ │ │ +                  ((0, 2), 0) => -c
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -107,16 +107,16 @@
│ │ │  o3 : LineageTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ +                  ((0, 2), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,18 +20,18 @@
│ │ │ │  remainder on the division by the remaining values H.
│ │ │ │  If values H constitute a Gr\"obner basis of the ideal they generate, this
│ │ │ │  method returns a reduced Gr\"obner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │ │  
│ │ │ │ -                                        3
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │                                       2
│ │ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ │ +o3 = LineageTable{((0, 1), 0) => -a*c    }
│ │ │ │ +                                   3
│ │ │ │ +                  ((0, 2), 0) => -c
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -41,16 +41,16 @@
│ │ │ │                    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
│ │ │ │ +o4 = LineageTable{((0, 1), 0) => null}
│ │ │ │ +                  ((0, 2), 0) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html
│ │ │ @@ -95,24 +95,60 @@
│ │ │                
    i3 : allowableThreads  = 4;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : H = tgb I
│ │ │  
│ │ │ -                                      4 7      3 7
│ │ │ -o4 = LineageTable{((1, 2), 1) => - 30y z  - 48y z }
│ │ │ -                                    3 8      2 7
│ │ │ -                  ((1, 2), 2) => 26y z  - 35y z
│ │ │ +                                         3 8      2 9
│ │ │ +o4 = LineageTable{((0, 1), (1, 2)) => 25y z  - 36y z   }
│ │ │ +                                           3 9      3 8
│ │ │ +                  ((0, 1), (1, 3)) => - 39y z  + 24y z
│ │ │ +                                         3 4      2 4
│ │ │ +                  ((0, 1), (2, 3)) => 25y z  + 40y z
│ │ │ +                                   4 4      3 6
│ │ │ +                  ((0, 1), 2) => 9y z  - 27y z
│ │ │ +                                      3 9      3 8
│ │ │ +                  ((0, 1), 3) => - 19y z  + 35y z
│ │ │ +                                         2 5      2 4
│ │ │ +                  ((0, 2), (0, 1)) => 40y z  + 22y z
│ │ │ +                                         2 7     2 5
│ │ │ +                  ((0, 2), (1, 2)) => 22y z  + 5y z
│ │ │ +                                         2 6      2 4
│ │ │ +                  ((0, 2), (1, 3)) => 25y z  + 24y z
│ │ │ +                                      2 14     2 12
│ │ │ +                  ((0, 2), 1) => - 19y z   + 6y z
│ │ │                                            2 4
│ │ │ -                  ((1, 3), (1, 2)) => -38y z
│ │ │ -                                 3 4     2 5
│ │ │ -                  (0, 1) => - 19y z  - 9y z
│ │ │ -                               2 6     2 4
│ │ │ -                  (0, 2) => 22y z  + 9y z
│ │ │ +                  ((0, 3), (1, 2)) => -18y z
│ │ │ +                                      4 11      3 14
│ │ │ +                  ((1, 2), 1) => - 14y z   + 43y z
│ │ │ +                                     4 8      3 10
│ │ │ +                  ((1, 2), 3) => - 9y z  + 27y z
│ │ │ +                                        3 16     3 11
│ │ │ +                  ((1, 3), (1, 2)) => 9y z   - 7y z
│ │ │ +                                     3 13      3 12
│ │ │ +                  ((1, 3), 1) => - 8y z   + 36y z
│ │ │ +                                      4 6      3 10
│ │ │ +                  ((1, 3), 2) => - 23y z  - 34y z
│ │ │ +                                      3 12      3 8
│ │ │ +                  ((1, 3), 3) => - 12y z   + 32y z
│ │ │ +                                        4 4      2 8
│ │ │ +                  ((2, 3), (1, 2)) => 8y z  - 40y z
│ │ │ +                                           4 4      3 6
│ │ │ +                  ((2, 3), (1, 3)) => - 40y z  + 19y z
│ │ │ +                                     3 12
│ │ │ +                  ((2, 3), 1) => -14y z
│ │ │ +                                     4 5     4 4
│ │ │ +                  ((2, 3), 2) => - 2y z  + 9y z
│ │ │ +                                     4 5      3 6
│ │ │ +                  ((2, 3), 3) => - 2y z  + 27y z
│ │ │ +                                 5 2      3 4
│ │ │ +                  (0, 1) => - 25y z  - 19y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │                                 5      2 4
│ │ │                    (0, 3) => 28y z - 2y z
│ │ │                                 5 6      4 5
│ │ │                    (1, 2) => 19y z  - 45y z
│ │ │                                   7 2      5 5
│ │ │                    (1, 3) => - 39y z  + 19y z
│ │ │                                 5 2     2 4
│ │ │ @@ -135,16 +171,16 @@
│ │ │            
    Note that the keys in the hash table are strings, and the keys of input polynomials are 0..#L, as in the following example.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : H#(0,1)
│ │ │  
│ │ │ -          3 4     2 5
│ │ │ -o5 = - 19y z  - 9y z
│ │ │ +          5 2      3 4
│ │ │ +o5 = - 25y z  - 19y z
│ │ │  
│ │ │  o5 : R
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Some may be curious  how tgb works.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,24 +26,60 @@
│ │ │ │  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 7      3 7
│ │ │ │ -o4 = LineageTable{((1, 2), 1) => - 30y z  - 48y z }
│ │ │ │ -                                    3 8      2 7
│ │ │ │ -                  ((1, 2), 2) => 26y z  - 35y z
│ │ │ │ +                                         3 8      2 9
│ │ │ │ +o4 = LineageTable{((0, 1), (1, 2)) => 25y z  - 36y z   }
│ │ │ │ +                                           3 9      3 8
│ │ │ │ +                  ((0, 1), (1, 3)) => - 39y z  + 24y z
│ │ │ │ +                                         3 4      2 4
│ │ │ │ +                  ((0, 1), (2, 3)) => 25y z  + 40y z
│ │ │ │ +                                   4 4      3 6
│ │ │ │ +                  ((0, 1), 2) => 9y z  - 27y z
│ │ │ │ +                                      3 9      3 8
│ │ │ │ +                  ((0, 1), 3) => - 19y z  + 35y z
│ │ │ │ +                                         2 5      2 4
│ │ │ │ +                  ((0, 2), (0, 1)) => 40y z  + 22y z
│ │ │ │ +                                         2 7     2 5
│ │ │ │ +                  ((0, 2), (1, 2)) => 22y z  + 5y z
│ │ │ │ +                                         2 6      2 4
│ │ │ │ +                  ((0, 2), (1, 3)) => 25y z  + 24y z
│ │ │ │ +                                      2 14     2 12
│ │ │ │ +                  ((0, 2), 1) => - 19y z   + 6y z
│ │ │ │                                            2 4
│ │ │ │ -                  ((1, 3), (1, 2)) => -38y z
│ │ │ │ -                                 3 4     2 5
│ │ │ │ -                  (0, 1) => - 19y z  - 9y z
│ │ │ │ -                               2 6     2 4
│ │ │ │ -                  (0, 2) => 22y z  + 9y z
│ │ │ │ +                  ((0, 3), (1, 2)) => -18y z
│ │ │ │ +                                      4 11      3 14
│ │ │ │ +                  ((1, 2), 1) => - 14y z   + 43y z
│ │ │ │ +                                     4 8      3 10
│ │ │ │ +                  ((1, 2), 3) => - 9y z  + 27y z
│ │ │ │ +                                        3 16     3 11
│ │ │ │ +                  ((1, 3), (1, 2)) => 9y z   - 7y z
│ │ │ │ +                                     3 13      3 12
│ │ │ │ +                  ((1, 3), 1) => - 8y z   + 36y z
│ │ │ │ +                                      4 6      3 10
│ │ │ │ +                  ((1, 3), 2) => - 23y z  - 34y z
│ │ │ │ +                                      3 12      3 8
│ │ │ │ +                  ((1, 3), 3) => - 12y z   + 32y z
│ │ │ │ +                                        4 4      2 8
│ │ │ │ +                  ((2, 3), (1, 2)) => 8y z  - 40y z
│ │ │ │ +                                           4 4      3 6
│ │ │ │ +                  ((2, 3), (1, 3)) => - 40y z  + 19y z
│ │ │ │ +                                     3 12
│ │ │ │ +                  ((2, 3), 1) => -14y z
│ │ │ │ +                                     4 5     4 4
│ │ │ │ +                  ((2, 3), 2) => - 2y z  + 9y z
│ │ │ │ +                                     4 5      3 6
│ │ │ │ +                  ((2, 3), 3) => - 2y z  + 27y z
│ │ │ │ +                                 5 2      3 4
│ │ │ │ +                  (0, 1) => - 25y z  - 19y z
│ │ │ │ +                              5 3     2 4
│ │ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ │                                 5      2 4
│ │ │ │                    (0, 3) => 28y z - 2y z
│ │ │ │                                 5 6      4 5
│ │ │ │                    (1, 2) => 19y z  - 45y z
│ │ │ │                                   7 2      5 5
│ │ │ │                    (1, 3) => - 39y z  + 19y z
│ │ │ │                                 5 2     2 4
│ │ │ │ @@ -63,16 +99,16 @@
│ │ │ │  polynomial $(g, I_1)$, where $g$ is calculated from the S-pair $(I_0, I_2)$.
│ │ │ │  For this reason, we say that the key communicates the "lineage" of the
│ │ │ │  resulting polynomial. (See _T_h_r_e_a_d_e_d_G_B.)
│ │ │ │  Note that the keys in the hash table are strings, and the keys of input
│ │ │ │  polynomials are 0..#L, as in the following example.
│ │ │ │  i5 : H#(0,1)
│ │ │ │  
│ │ │ │ -          3 4     2 5
│ │ │ │ -o5 = - 19y z  - 9y z
│ │ │ │ +          5 2      3 4
│ │ │ │ +o5 = - 25y z  - 19y z
│ │ │ │  
│ │ │ │  o5 : R
│ │ │ │  Some may be curious how tgb works.
│ │ │ │  The starting basis $L$ (the input list L or L=gens I) populates the entries
│ │ │ │  numbered $0$ through $n-1$ of a mutable hash table $G$, where $n$ is the length
│ │ │ │  of $L$. The method creates all possible S-polynomials of $L$ and schedules
│ │ │ │  their reduction with respect to $G$ as tasks. Throughout the computation, every
│ │ ├── ./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 1.08495s (cpu); 0.766604s (thread); 0s (gc)
│ │ │ + -- used 1.5627s (cpu); 0.998747s (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 4.73932s (cpu); 3.19995s (thread); 0s (gc)
│ │ │ + -- used 6.03485s (cpu); 3.63222s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 1.08495s (cpu); 0.766604s (thread); 0s (gc)
│ │ │ + -- used 1.5627s (cpu); 0.998747s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -150,15 +150,15 @@ │ │ │ The dual variety has degree = 45, and codimension = 1 │ │ │ Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28} │ │ │ Polar Degrees: {45, 98, 81, 28} │ │ │ ED Degree = 252 │ │ │ │ │ │ 5 4 3 2 │ │ │ Chern-Mather Class: 20h + 23h + 31h + 28h │ │ │ - -- used 4.73932s (cpu); 3.19995s (thread); 0s (gc) │ │ │ + -- used 6.03485s (cpu); 3.63222s (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 1.08495s (cpu); 0.766604s (thread); 0s (gc) │ │ │ │ + -- used 1.5627s (cpu); 0.998747s (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 4.73932s (cpu); 3.19995s (thread); 0s (gc) │ │ │ │ + -- used 6.03485s (cpu); 3.63222s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 252 │ │ │ │ │ │ │ │ o5 : QQ │ │ │ │ ********** WWaayyss ttoo uussee eeddDDeegg:: ********** │ │ │ │ * edDeg(Matrix) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ @@ -17,15 +17,15 @@ │ │ │ | -1 1 2 -1 -1 1 -1 1 0 0 | │ │ │ | 1 0 -1 0 0 0 0 0 0 0 | │ │ │ │ │ │ 4 10 │ │ │ o2 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i3 : elapsedTime Ts = allTriangulations(A, Fine => true); │ │ │ - -- .146605s elapsed │ │ │ + -- .104481s elapsed │ │ │ │ │ │ i4 : select(Ts, T -> isStar T) │ │ │ │ │ │ o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, │ │ │ ------------------------------------------------------------------------ │ │ │ 1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3, │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -50,14 +50,14 @@ │ │ │ i7 : T = regularFineTriangulation A │ │ │ │ │ │ o7 = triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, 1, 6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8}, {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6, 7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2, 3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}} │ │ │ │ │ │ o7 : Triangulation │ │ │ │ │ │ i8 : elapsedTime Ts2 = generateTriangulations T; │ │ │ - -- 1.329s elapsed │ │ │ + -- 1.02847s elapsed │ │ │ │ │ │ i9 : #Ts2 == #Ts │ │ │ │ │ │ o9 = true │ │ │ │ │ │ i10 : │ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out │ │ │ @@ -21,87 +21,15 @@ │ │ │ │ │ │ o3 = triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}} │ │ │ │ │ │ o3 : Triangulation │ │ │ │ │ │ i4 : Ts1 = generateTriangulations A -- list of Triangulation's. │ │ │ │ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation │ │ │ - ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ - ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation │ │ │ - ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ - ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation │ │ │ - ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ - ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ - ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -227,275 +155,275 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 5, 6, 7}}} │ │ │ - │ │ │ -o4 : List │ │ │ - │ │ │ -i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets │ │ │ - │ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, │ │ │ + {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, │ │ │ + {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ + triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, │ │ │ + {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ + {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, │ │ │ + {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, │ │ │ + {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, │ │ │ + {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, │ │ │ + triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, │ │ │ + {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, │ │ │ + {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, │ │ │ + {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, │ │ │ + {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, │ │ │ + {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ + triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, │ │ │ + {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, │ │ │ + {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, │ │ │ + {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, │ │ │ + {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ + {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, │ │ │ + {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, │ │ │ + {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, │ │ │ + {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ + {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7}, │ │ │ + {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, │ │ │ + triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, │ │ │ + {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, │ │ │ + {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4}, │ │ │ + {2, 4, 6, 7}}} │ │ │ + │ │ │ +o4 : List │ │ │ + │ │ │ +i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets │ │ │ + │ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ + {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, │ │ │ + {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, │ │ │ + {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, │ │ │ + {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, │ │ │ + {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, │ │ │ + {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, │ │ │ + {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, │ │ │ + {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, │ │ │ + {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, │ │ │ + {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, │ │ │ + {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7}, │ │ │ + {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}}, │ │ │ + {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ + {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, │ │ │ + {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, │ │ │ + {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, │ │ │ + {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, │ │ │ + {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ + {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, │ │ │ + {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, │ │ │ + {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, │ │ │ + {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, │ │ │ + {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ + {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, │ │ │ + {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, │ │ │ + {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, │ │ │ + {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, │ │ │ + {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, │ │ │ + {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6}, │ │ │ + {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, │ │ │ + {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, │ │ │ + {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ + {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, │ │ │ + {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, │ │ │ + {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7}, │ │ │ + {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, │ │ │ + {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ + {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, │ │ │ + {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, │ │ │ + {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6}, │ │ │ + {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6}, │ │ │ + {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, │ │ │ + {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ + {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, │ │ │ + {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, │ │ │ + {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, │ │ │ + {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 4, 5, 6}, {2, 5, 6, 7}}} │ │ │ - │ │ │ -o5 : List │ │ │ - │ │ │ -i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations │ │ │ - │ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ + {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ + {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation │ │ │ + {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, │ │ │ + {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, │ │ │ + {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, │ │ │ + {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ + {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ + {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, │ │ │ + {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, │ │ │ + {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ + {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ + {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, │ │ │ + {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation │ │ │ + {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, │ │ │ + {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ + {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ + {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ + {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, │ │ │ + {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, │ │ │ + {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ + {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ + {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, │ │ │ + {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ + {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ + {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, │ │ │ + {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, │ │ │ + {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation │ │ │ + {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6}, │ │ │ + {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, │ │ │ + {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ + {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ + {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ + {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ + {1, 4, 5, 7}, {2, 4, 6, 7}}} │ │ │ + │ │ │ +o5 : List │ │ │ + │ │ │ +i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations │ │ │ + │ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -621,93 +549,93 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 5, 6, 7}}} │ │ │ - │ │ │ -o6 : List │ │ │ - │ │ │ -i7 : Ts4 = generateTriangulations tri -- list of Triangulations │ │ │ - │ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ + {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ + {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, │ │ │ + triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, │ │ │ + {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, │ │ │ + {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ + {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, │ │ │ + {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ + {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, │ │ │ + triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, │ │ │ + {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ + {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ + {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, │ │ │ + {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation │ │ │ + {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, │ │ │ + triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ + {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ + {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, │ │ │ + {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, │ │ │ + {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ + {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, │ │ │ + {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ + {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, │ │ │ + {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, │ │ │ + {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation │ │ │ + {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6}, │ │ │ + triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, │ │ │ + {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ + {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ - triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ + {2, 4, 6, 7}}} │ │ │ + │ │ │ +o6 : List │ │ │ + │ │ │ +i7 : Ts4 = generateTriangulations tri -- list of Triangulations │ │ │ + │ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7}, │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -833,15 +761,87 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation │ │ │ ------------------------------------------------------------------------ │ │ │ {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6}, │ │ │ ------------------------------------------------------------------------ │ │ │ - {2, 5, 6, 7}}} │ │ │ + {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ + ------------------------------------------------------------------------ │ │ │ + {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation │ │ │ + ------------------------------------------------------------------------ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation │ │ │ + ------------------------------------------------------------------------ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ + ------------------------------------------------------------------------ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation │ │ │ + ------------------------------------------------------------------------ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, │ │ │ + ------------------------------------------------------------------------ │ │ │ + {2, 4, 6, 7}}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : all(Ts4, isFine) │ │ │ │ │ │ o8 = true │ │ │ │ │ │ @@ -858,191 +858,193 @@ │ │ │ o11 = Tally{false => 66} │ │ │ true => 8 │ │ │ │ │ │ o11 : Tally │ │ │ │ │ │ i12 : Ts4/gkzVector │ │ │ │ │ │ - 16 16 4 8 4 20 8 8 4 8 4 8 8 │ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 8 20 16 16 4 16 4 4 8 4 16 4 16 │ │ │ - 4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 20 4 4 20 16 8 16 4 4 20 4 4 │ │ │ - 4, --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 20 16 16 4 4 8 4 4 16 8 16 16 4 │ │ │ - --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 16 4 8 4 16 4 16 8 8 8 8 8 8 │ │ │ - --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 8 8 4 20 8 8 16 4 4 16 16 4 4 16 │ │ │ - -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 16 8 4 4 20 20 4 4 20 20 │ │ │ - --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, │ │ │ - 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 4 4 8 16 16 4 4 20 20 4 4 │ │ │ - -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 20 20 4 8 16 4 4 16 8 4 8 8 20 │ │ │ - --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 8 8 8 4 20 20 4 4 20 4 20 20 4 4 20 │ │ │ - 4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 20 4 20 4 4 20 20 4 4 20 8 4 4 │ │ │ - --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 16 16 8 4 4 16 16 4 16 16 4 8 4 │ │ │ - --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 20 8 8 8 8 8 8 20 4 4 8 8 20 │ │ │ - 4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 8 4 16 4 8 16 4 20 8 8 8 16 4 4 │ │ │ - -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 16 8 16 4 16 4 16 4 8 20 8 4 │ │ │ - 8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 8 8 8 8 20 4 4 16 4 16 4 16 20 8 │ │ │ - -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 8 4 8 20 4 8 8 8 8 8 8 4 20 │ │ │ - -, -, 4, 8, -}, {--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 16 16 4 4 8 8 4 8 8 20 20 4 4 │ │ │ - {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 20 20 4 4 20 4 20 20 4 8 │ │ │ - 4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, │ │ │ - 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 8 20 4 8 16 4 16 8 4 4 16 16 8 │ │ │ - 4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 4 4 8 16 16 4 4 20 20 4 8 8 │ │ │ - -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 8 8 8 8 8 20 8 8 4 16 8 4 16 │ │ │ - -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 4 4 8 16 4 16 16 4 16 4 16 4 4 16 │ │ │ - -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 4 16 4 16 8 8 8 8 8 8 20 8 8 8 4 │ │ │ - -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 4 8 4 16 16 20 4 4 20 4 4 16 │ │ │ - {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 4 16 16 8 16 4 16 4 8 8 20 8 4 │ │ │ - -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 4 8 20 8 8 8 4 16 16 4 8 8 20 8 │ │ │ - {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 4 4 8 20 8 8 8 8 8 8 8 8 4 16 │ │ │ - 8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 16 4 4 16 20 4 8 8 8 8 4 8 20 8 │ │ │ - --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ - ----------------------------------------------------------------------- │ │ │ - 4}} │ │ │ + 20 4 4 20 8 4 4 16 16 8 4 │ │ │ +o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 16 16 4 16 16 4 8 4 20 8 8 8 │ │ │ + 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -}, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 8 8 20 4 4 8 8 20 8 4 16 4 │ │ │ + {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 16 4 20 8 8 8 16 4 4 16 8 16 │ │ │ + -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 16 4 16 4 8 20 8 4 8 8 8 8 20 │ │ │ + 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 4 16 4 16 4 16 20 8 8 4 8 20 │ │ │ + -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 8 8 8 8 8 8 4 20 16 16 4 4 8 │ │ │ + 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 4 8 8 20 20 4 4 20 20 4 │ │ │ + 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 20 4 20 20 4 8 8 20 4 8 │ │ │ + 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4}, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 16 4 16 8 4 4 16 16 8 4 4 8 │ │ │ + {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 16 16 4 4 20 20 4 8 8 8 8 8 8 │ │ │ + --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -}, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 20 8 8 4 16 8 4 16 4 4 8 16 │ │ │ + {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 16 16 4 16 4 16 4 4 16 4 16 4 16 │ │ │ + -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --}, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 8 8 8 8 8 20 8 8 8 4 4 8 4 │ │ │ + {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 16 16 20 4 4 20 4 4 16 4 16 16 │ │ │ + --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4}, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 16 4 16 4 8 8 20 8 4 4 8 20 │ │ │ + {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 8 8 4 16 16 4 8 8 20 8 4 4 │ │ │ + 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 20 8 8 8 8 8 8 8 8 4 16 16 4 4 │ │ │ + 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 16 20 4 8 8 8 8 4 8 20 8 16 16 │ │ │ + --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 8 4 20 8 8 4 8 4 8 8 8 20 │ │ │ + -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --}, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 16 16 4 16 4 4 8 4 16 4 16 20 4 │ │ │ + {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 20 16 8 16 4 4 20 4 4 20 │ │ │ + -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4}, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 16 16 4 4 8 4 4 16 8 16 16 4 16 │ │ │ + {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 8 4 16 4 16 8 8 8 8 8 8 8 │ │ │ + -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 4 20 8 8 16 4 4 16 16 4 4 16 16 8 │ │ │ + -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 4 20 20 4 4 20 20 4 4 │ │ │ + 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-, │ │ │ + 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 16 16 4 4 20 20 4 4 20 20 │ │ │ + 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 8 16 4 4 16 8 4 8 8 20 8 │ │ │ + -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 8 8 4 20 20 4 4 20 4 20 20 4 4 20 20 4 20 4 │ │ │ + -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -, │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ + ----------------------------------------------------------------------- │ │ │ + 4 20 │ │ │ + -, --}} │ │ │ + 3 3 │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : volume convexHull A -- 8 │ │ │ │ │ │ o13 = 8 │ │ │ │ │ │ o13 : QQ │ │ │ │ │ │ i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8) │ │ │ │ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ + {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation │ │ │ + {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation │ │ │ ----------------------------------------------------------------------- │ │ │ - {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, │ │ │ + {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, │ │ │ + {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, │ │ │ + {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}}, │ │ │ + {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, │ │ │ ----------------------------------------------------------------------- │ │ │ - triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, │ │ │ + triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ + {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation │ │ │ + {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ ----------------------------------------------------------------------- │ │ │ - {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ + {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ {2, 4, 6, 7}}} │ │ │ │ │ │ o14 : List │ │ │ │ │ │ i15 : stars2 = select(Ts4, isStar) │ │ │ │ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ + {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation │ │ │ + {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation │ │ │ ----------------------------------------------------------------------- │ │ │ - {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, │ │ │ + {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, │ │ │ + {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, │ │ │ + {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}}, │ │ │ + {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, │ │ │ ----------------------------------------------------------------------- │ │ │ - triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, │ │ │ + triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ + {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation │ │ │ + {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ ----------------------------------------------------------------------- │ │ │ - {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ + {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, │ │ │ ----------------------------------------------------------------------- │ │ │ {2, 4, 6, 7}}} │ │ │ │ │ │ o15 : List │ │ │ │ │ │ i16 : stars1 == stars2 │ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html │ │ │ @@ -117,87 +117,15 @@ │ │ │ o3 : Triangulation │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : Ts1 = generateTriangulations A -- list of Triangulation's.
│ │ │  
│ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -323,281 +251,281 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o4 : List
    │ │ │ - │ │ │ - │ │ │ - │ │ │ - │ │ │ - 
    i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ -
│ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │ +     {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o4 : List
    │ │ │ + │ │ │ + │ │ │ + │ │ │ + │ │ │ + 
    i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets
│ │ │ +
│ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6},
│ │ │ +     {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7},
│ │ │ +     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7},
│ │ │ +     {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6},
│ │ │ +     {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6},
│ │ │ +     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7},
│ │ │ +     {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +     {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7},
│ │ │ +     {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7},
│ │ │ +     {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}},
│ │ │ +     {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o5 : List
    │ │ │ - │ │ │ - │ │ │ - │ │ │ - │ │ │ - 
    i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ -
│ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {1, 4, 5, 7}, {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o5 : List
    │ │ │ + │ │ │ + │ │ │ + │ │ │ + │ │ │ + 
    i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations
│ │ │ +
│ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -723,96 +651,96 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ -
│ │ │ -o6 : List
    │ │ │ - │ │ │ - │ │ │ - │ │ │ - │ │ │ - 
    i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ -
│ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7},
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5},
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7},
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7},
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6},
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7},
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6},
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6},
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5},
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}},
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7},
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4},
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6},
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6},
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5},
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │ +     {2, 4, 6, 7}}}
│ │ │ +
│ │ │ +o6 : List
    │ │ │ + │ │ │ + │ │ │ + │ │ │ + │ │ │ + 
    i7 : Ts4 = generateTriangulations tri -- list of Triangulations
│ │ │ +
│ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7},
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -938,15 +866,87 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │       {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation
│ │ │       ------------------------------------------------------------------------
│ │ │       {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, 5, 6, 7}}}
│ │ │ +     {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7},
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     {2, 4, 6, 7}}}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : all(Ts4, isFine)
│ │ │ @@ -978,131 +978,133 @@
│ │ │  o11 : Tally
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : Ts4/gkzVector
│ │ │  
│ │ │ -        16     16  4     8  4       20     8  8     4  8          4  8     8 
│ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ -         3      3  3     3  3        3     3  3     3  3          3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20       16  16  4  16  4  4       8        4     16  4  16      
│ │ │ -      4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4,
│ │ │ -         3   3        3   3  3   3  3  3       3        3      3  3   3      
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20  4  4  20          16        8  16  4  4          20  4        4 
│ │ │ -      4, --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -,
│ │ │ -          3  3  3   3           3        3   3  3  3           3  3        3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20       16  16  4     4        8       4  4  16  8        16    16  4 
│ │ │ -      --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -,
│ │ │ -       3        3   3  3     3        3       3  3   3  3         3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16     4        8    4  16     4     16  8       8  8     8  8     8 
│ │ │ -      --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -,
│ │ │ -       3     3        3    3   3     3      3  3       3  3     3  3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8       8  4     20  8     8    16  4  4        16  16  4       4  16 
│ │ │ -      -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --,
│ │ │ -      3       3  3      3  3     3     3  3  3         3   3  3       3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  8        4    4  20              20  4    4        20  20       
│ │ │ -      --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4,
│ │ │ -       3  3        3    3   3               3  3    3         3   3       
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  16  4          4     20  20     4             4 
│ │ │ -      -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -,
│ │ │ -      3    3        3   3   3  3          3      3   3     3             3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20  20  4          8        16     4  4  16    8        4  8  8  20 
│ │ │ -      --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --,
│ │ │ -       3   3  3          3         3     3  3   3    3        3  3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -           8     8  8     4  20       20  4  4  20  4  20  20  4    4  20 
│ │ │ -      4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --,
│ │ │ -           3     3  3     3   3        3  3  3   3  3   3   3  3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      20  4  20  4  4  20    20        4  4        20       8  4     4    
│ │ │ -      --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4,
│ │ │ -       3  3   3  3  3   3     3        3  3         3       3  3     3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16  16    8        4     4  16  16    4  16     16     4  8       4    
│ │ │ -      --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4,
│ │ │ -       3   3    3        3     3   3   3    3   3      3     3  3       3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20     8  8  8    8  8  8     20        4    4        8     8  20 
│ │ │ -      4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --,
│ │ │ -          3     3  3  3    3  3  3      3        3    3        3     3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8    4     16  4     8  16          4  20     8     8  8    16  4  4 
│ │ │ -      -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -,
│ │ │ -      3    3      3  3     3   3          3   3     3     3  3     3  3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         16        8    16     4  16  4  16     4       8  20  8  4       
│ │ │ -      8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4,
│ │ │ -          3        3     3     3   3  3   3     3       3   3  3  3       
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8    8  8     8     20  4       4  16     4  16     4  16       20  8 
│ │ │ -      -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -,
│ │ │ -      3    3  3     3      3  3       3   3     3   3     3   3        3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8  4        8    20        4  8  8  8          8  8  8  4        20  
│ │ │ -      -, -, 4, 8, -}, {--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --},
│ │ │ -      3  3        3     3        3  3  3  3          3  3  3  3         3  
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -       16  16     4     4  8          8  4     8  8     20       20     4  4 
│ │ │ -      {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -,
│ │ │ -        3   3     3     3  3          3  3     3  3      3        3     3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         20       20  4              4  20       4  20        20  4       8 
│ │ │ -      4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-,
│ │ │ -          3        3  3              3   3       3   3         3  3       3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -         8  20     4  8          16  4  16  8        4    4     16  16     8 
│ │ │ -      4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -,
│ │ │ -         3   3     3  3           3  3   3  3        3    3      3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4          4  8     16  16     4    4     20        20     4    8  8 
│ │ │ -      -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -,
│ │ │ -      3          3  3      3   3     3    3      3         3     3    3  3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      8        8  8  8    8  20  8     8        4       16  8     4  16    
│ │ │ -      -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8,
│ │ │ -      3        3  3  3    3   3  3     3        3        3  3     3   3    
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4    4        8  16  4  16       16  4     16  4     16  4    4     16 
│ │ │ -      -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --,
│ │ │ -      3    3        3   3  3   3        3  3      3  3      3  3    3      3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  4     16       8  8  8  8  8  8       20  8     8     8  4     
│ │ │ -      -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8},
│ │ │ -      3   3  3      3       3  3  3  3  3  3        3  3     3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     4  16     16    20     4        4     20       4  4  16 
│ │ │ -      {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --,
│ │ │ -          3  3     3   3      3     3     3        3      3       3  3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4  16  16          8  16     4     16  4    8  8     20     8  4     
│ │ │ -      -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4},
│ │ │ -      3   3   3          3   3     3      3  3    3  3      3     3  3     
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -          4  8     20     8  8       8  4     16     16  4    8  8  20     8 
│ │ │ -      {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -,
│ │ │ -          3  3      3     3  3       3  3      3      3  3    3  3   3     3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -            4    4        8     20  8  8    8     8  8  8  8     8    4  16 
│ │ │ -      8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --,
│ │ │ -            3    3        3      3  3  3    3     3  3  3  3     3    3   3 
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      16        4  4  16       20  4     8  8     8    8        4  8  20  8
│ │ │ -      --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -,
│ │ │ -       3        3  3   3        3  3     3  3     3    3        3  3   3  3
│ │ │ -      -----------------------------------------------------------------------
│ │ │ -      4}}
│ │ │ +        20        4  4        20       8  4     4     16  16    8        4 
│ │ │ +o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -,
│ │ │ +         3        3  3         3       3  3     3      3   3    3        3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16  16    4  16     16     4  8       4        20     8  8  8  
│ │ │ +      4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -},
│ │ │ +         3   3   3    3   3      3     3  3       3         3     3  3  3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       8  8  8     20        4    4        8     8  20  8    4     16  4    
│ │ │ +      {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4,
│ │ │ +       3  3  3      3        3    3        3     3   3  3    3      3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  16          4  20     8     8  8    16  4  4     16        8    16 
│ │ │ +      -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--,
│ │ │ +      3   3          3   3     3     3  3     3  3  3      3        3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  16  4  16     4       8  20  8  4        8    8  8     8     20 
│ │ │ +      4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --,
│ │ │ +         3   3  3   3     3       3   3  3  3        3    3  3     3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4       4  16     4  16     4  16       20  8  8  4        8    20    
│ │ │ +      -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4,
│ │ │ +      3       3   3     3   3     3   3        3  3  3  3        3     3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         4  8  8  8          8  8  8  4        20    16  16     4     4  8 
│ │ │ +      4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -,
│ │ │ +         3  3  3  3          3  3  3  3         3     3   3     3     3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +              8  4     8  8     20       20     4  4     20       20  4    
│ │ │ +      8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4,
│ │ │ +              3  3     3  3      3        3     3  3      3        3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +               4  20       4  20        20  4       8     8  20     4  8     
│ │ │ +      4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4},
│ │ │ +               3   3       3   3         3  3       3     3   3     3  3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          16  4  16  8        4    4     16  16     8  4          4  8    
│ │ │ +      {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8,
│ │ │ +           3  3   3  3        3    3      3   3     3  3          3  3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16  16     4    4     20        20     4    8  8  8        8  8  8  
│ │ │ +      --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -},
│ │ │ +       3   3     3    3      3         3     3    3  3  3        3  3  3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       8  20  8     8        4       16  8     4  16     4    4        8  16 
│ │ │ +      {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --,
│ │ │ +       3   3  3     3        3        3  3     3   3     3    3        3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  16       16  4     16  4     16  4    4     16  4  16  4     16  
│ │ │ +      -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --},
│ │ │ +      3   3        3  3      3  3      3  3    3      3  3   3  3      3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          8  8  8  8  8  8       20  8     8     8  4          4  8     4 
│ │ │ +      {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -,
│ │ │ +          3  3  3  3  3  3        3  3     3     3  3          3  3     3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16     16    20     4        4     20       4  4  16  4  16  16     
│ │ │ +      --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4},
│ │ │ +       3      3     3     3        3      3       3  3   3  3   3   3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          8  16     4     16  4    8  8     20     8  4          4  8     20 
│ │ │ +      {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --,
│ │ │ +          3   3     3      3  3    3  3      3     3  3          3  3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8  8       8  4     16     16  4    8  8  20     8        4    4    
│ │ │ +      4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4,
│ │ │ +         3  3       3  3      3      3  3    3  3   3     3        3    3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         8     20  8  8    8     8  8  8  8     8    4  16  16        4  4 
│ │ │ +      8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -,
│ │ │ +         3      3  3  3    3     3  3  3  3     3    3   3   3        3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      16       20  4     8  8     8    8        4  8  20  8       16     16 
│ │ │ +      --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --,
│ │ │ +       3        3  3     3  3     3    3        3  3   3  3        3      3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4     8  4       20     8  8     4  8          4  8     8     8  20  
│ │ │ +      -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --},
│ │ │ +      3     3  3        3     3  3     3  3          3  3     3     3   3  
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +          16  16  4  16  4  4       8        4     16  4  16          20  4 
│ │ │ +      {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -,
│ │ │ +           3   3  3   3  3  3       3        3      3  3   3           3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  20          16        8  16  4  4          20  4        4  20     
│ │ │ +      -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4},
│ │ │ +      3   3           3        3   3  3  3           3  3        3   3     
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +       16  16  4     4        8       4  4  16  8        16    16  4  16    
│ │ │ +      {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4,
│ │ │ +        3   3  3     3        3       3  3   3  3         3     3  3   3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4        8    4  16     4     16  8       8  8     8  8     8  8      
│ │ │ +      -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4,
│ │ │ +      3        3    3   3     3      3  3       3  3     3  3     3  3      
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  4     20  8     8    16  4  4        16  16  4       4  16  16  8 
│ │ │ +      -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -,
│ │ │ +      3  3      3  3     3     3  3  3         3   3  3       3   3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +            4    4  20              20  4    4        20  20        4    4 
│ │ │ +      8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-,
│ │ │ +            3    3   3               3  3    3         3   3        3    3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +            8  16  16  4          4     20  20     4             4  20  20 
│ │ │ +      4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --,
│ │ │ +            3   3   3  3          3      3   3     3             3   3   3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4          8        16     4  4  16    8        4  8  8  20       8    
│ │ │ +      -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8,
│ │ │ +      3          3         3     3  3   3    3        3  3  3   3       3    
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      8  8     4  20       20  4  4  20  4  20  20  4    4  20  20  4  20  4 
│ │ │ +      -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -,
│ │ │ +      3  3     3   3        3  3  3   3  3   3   3  3    3   3   3  3   3  3 
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +      4  20
│ │ │ +      -, --}}
│ │ │ +      3   3
│ │ │  
│ │ │  o12 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : volume convexHull A -- 8
│ │ │ @@ -1112,66 +1114,66 @@
│ │ │  o13 : QQ
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8)
│ │ │  
│ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o14 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : stars2 = select(Ts4, isStar)
│ │ │  
│ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation
│ │ │ +      {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7},
│ │ │ +      {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │ +      {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}},
│ │ │ +      {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7},
│ │ │ +      triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7},
│ │ │ +      {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation
│ │ │ +      {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7},
│ │ │ +      {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7},
│ │ │        -----------------------------------------------------------------------
│ │ │        {2, 4, 6, 7}}}
│ │ │  
│ │ │  o15 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -57,87 +57,15 @@ │ │ │ │ │ │ │ │ o3 = triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, │ │ │ │ 4, 5, 6}, {3, 5, 6, 7}} │ │ │ │ │ │ │ │ o3 : Triangulation │ │ │ │ i4 : Ts1 = generateTriangulations A -- list of Triangulation's. │ │ │ │ │ │ │ │ -o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ │ +o4 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ @@ -263,273 +191,273 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 5, 6, 7}}} │ │ │ │ - │ │ │ │ -o4 : List │ │ │ │ -i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets │ │ │ │ - │ │ │ │ -o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, │ │ │ │ + {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, │ │ │ │ + {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, │ │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 3, 5, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ │ + triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, │ │ │ │ + {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ + {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, │ │ │ │ + {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, │ │ │ │ + {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, │ │ │ │ + {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, │ │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, │ │ │ │ + triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, │ │ │ │ + {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, │ │ │ │ + {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, │ │ │ │ + {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, │ │ │ │ + {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, │ │ │ │ + {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ + triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, │ │ │ │ + {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, │ │ │ │ + {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, │ │ │ │ + {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, │ │ │ │ + {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ │ + {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, │ │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, │ │ │ │ + {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 4, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, │ │ │ │ + {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, │ │ │ │ + {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ │ + {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 7}, │ │ │ │ + {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, {1, 2, 3, 7}}, │ │ │ │ + triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, │ │ │ │ + {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, │ │ │ │ + {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, │ │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4}, │ │ │ │ + {2, 4, 6, 7}}} │ │ │ │ + │ │ │ │ +o4 : List │ │ │ │ +i5 : Ts2 = generateTriangulations(A, T) -- list of list of subsets │ │ │ │ + │ │ │ │ +o5 = {{{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ + {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, │ │ │ │ + {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, │ │ │ │ + {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, │ │ │ │ + {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 2, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, │ │ │ │ + {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, │ │ │ │ + {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ + {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 7}, │ │ │ │ + {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, │ │ │ │ + {1, 2, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 5}, │ │ │ │ + {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 3, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, │ │ │ │ + {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 2, 7}, │ │ │ │ + {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}}, │ │ │ │ + {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ │ + {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, │ │ │ │ + {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 6}, │ │ │ │ + {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, │ │ │ │ + {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, │ │ │ │ + {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ + {1, 2, 3, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, │ │ │ │ + {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, │ │ │ │ + {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, │ │ │ │ + {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 2, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 4}, │ │ │ │ + {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ + {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, │ │ │ │ + {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, │ │ │ │ + {1, 4, 5, 7}, {2, 3, 4, 6}, {3, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, │ │ │ │ + {0, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 5, 6}, │ │ │ │ + {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, │ │ │ │ + {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 6}, │ │ │ │ + {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, {1, 3, 4, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, {1, 2, 6, 7}, │ │ │ │ + {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, │ │ │ │ + {1, 2, 3, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ │ + {0, 1, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 6}, {1, 3, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, │ │ │ │ + {{0, 1, 2, 4}, {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, │ │ │ │ + {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 6, 7}, │ │ │ │ + {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, │ │ │ │ + {1, 2, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ │ + {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 4}, │ │ │ │ + {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 5}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, {3, 4, 6, 7}}, │ │ │ │ + {0, 5, 6, 7}}, {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 6}, {0, 3, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 6}, │ │ │ │ + {0, 4, 6, 7}, {1, 4, 5, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 6}, │ │ │ │ + {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, │ │ │ │ + {0, 4, 5, 6}, {1, 2, 3, 5}, {2, 3, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ │ + {0, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, │ │ │ │ + {3, 4, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, │ │ │ │ + {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, │ │ │ │ + {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 4, 5, 6}, {2, 5, 6, 7}}} │ │ │ │ - │ │ │ │ -o5 : List │ │ │ │ -i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations │ │ │ │ - │ │ │ │ -o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ │ + {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ │ + {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation │ │ │ │ + {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, │ │ │ │ + {2, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, │ │ │ │ + {1, 4, 5, 6}, {1, 5, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, │ │ │ │ + {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ + {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, │ │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ + {0, 4, 6, 7}}, {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, │ │ │ │ + {1, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, │ │ │ │ + {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 2, 5}, {0, 2, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ + {0, 4, 5, 6}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, │ │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ │ + {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, │ │ │ │ + {1, 4, 6, 7}}, {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, {0, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation │ │ │ │ + {1, 3, 6, 7}, {1, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, │ │ │ │ + {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ │ + {0, 2, 5, 6}, {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ + {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ + {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, │ │ │ │ + {2, 5, 6, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, │ │ │ │ + {3, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ │ + {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ │ + {0, 2, 4, 5}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, │ │ │ │ + {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ + {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ │ + {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {2, 3, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, │ │ │ │ + {2, 4, 5, 7}, {2, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, │ │ │ │ + {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, {{0, 1, 3, 4}, {0, 2, 3, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ + {0, 3, 4, 6}, {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, {{0, 1, 3, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6}, │ │ │ │ + {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, │ │ │ │ + {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ + {1, 5, 6, 7}}, {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, {1, 3, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ │ + {1, 4, 5, 7}, {1, 4, 6, 7}}, {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ + {0, 4, 5, 6}, {3, 5, 6, 7}}, {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ │ + {1, 4, 5, 7}, {2, 4, 6, 7}}} │ │ │ │ + │ │ │ │ +o5 : List │ │ │ │ +i6 : Ts3 = generateTriangulations triangulation(A, T) -- list of Triangulations │ │ │ │ + │ │ │ │ +o6 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ @@ -655,92 +583,92 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 5, 6, 7}}} │ │ │ │ - │ │ │ │ -o6 : List │ │ │ │ -i7 : Ts4 = generateTriangulations tri -- list of Triangulations │ │ │ │ - │ │ │ │ -o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ │ + {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ │ + {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation │ │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, {0, 4, 5, 7}, │ │ │ │ + triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, │ │ │ │ + {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, │ │ │ │ + {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ + {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, {1, 2, 3, 7}, │ │ │ │ + {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ │ + {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, │ │ │ │ + triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, {0, 2, 3, 6}, │ │ │ │ + {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ + {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ │ + {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, │ │ │ │ + {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation │ │ │ │ + {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, {2, 4, 5, 6}, │ │ │ │ + triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ │ + {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ + {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, {2, 3, 5, 6}, │ │ │ │ + {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 6}, │ │ │ │ + {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ │ + {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, │ │ │ │ + {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ + {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, {3, 4, 5, 7}}, │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, {1, 3, 4, 7}, │ │ │ │ + {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 4}, │ │ │ │ + {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ + {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 6}, │ │ │ │ + triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 6}, │ │ │ │ + {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, │ │ │ │ + {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, triangulation │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ │ + {2, 4, 6, 7}}} │ │ │ │ + │ │ │ │ +o6 : List │ │ │ │ +i7 : Ts4 = generateTriangulations tri -- list of Triangulations │ │ │ │ + │ │ │ │ +o7 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {0, 5, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {0, 2, 3, 6}, {0, 3, 6, 7}, {0, 4, 5, 6}, {0, 5, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, {1, 4, 5, 7}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ @@ -866,15 +794,87 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {2, 3, 4, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {0, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 6}, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, 5, 6, 7}}} │ │ │ │ + {2, 5, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + triangulation {{0, 1, 3, 5}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 3, 5, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 4, 5, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {2, 4, 6, 7}}, triangulation {{0, 1, 2, 5}, {0, 2, 5, 6}, {0, 4, 5, 6}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {1, 2, 3, 7}, {1, 2, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + triangulation {{0, 1, 3, 6}, {0, 1, 4, 6}, {0, 2, 3, 6}, {1, 3, 6, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation {{0, 1, 3, 6}, {0, 1, 5, 6}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {0, 2, 3, 6}, {0, 4, 5, 6}, {1, 3, 6, 7}, {1, 5, 6, 7}}, triangulation │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 7}, {0, 3, 6, 7}, {0, 4, 5, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {0, 4, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 5, 6}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {0, 4, 5, 6}, {2, 3, 5, 7}, {2, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {1, 2, 3, 7}, {1, 2, 4, 5}, {1, 2, 5, 7}, {2, 4, 5, 6}, {2, 5, 6, 7}}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + triangulation {{0, 1, 2, 5}, {0, 2, 4, 5}, {1, 2, 3, 5}, {2, 3, 5, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {2, 4, 5, 6}, {2, 5, 6, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {0, 3, 4, 6}, {1, 3, 4, 5}, {3, 4, 5, 7}, {3, 4, 6, 7}}, triangulation │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 4, 5}, {0, 3, 4, 6}, {3, 4, 5, 6}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {3, 5, 6, 7}}, triangulation {{0, 1, 3, 5}, {0, 2, 3, 5}, {0, 2, 4, 5}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {1, 2, 3, 6}, {1, 2, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 4, 6}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {3, 4, 5, 6}, {3, 5, 6, 7}}, triangulation {{0, 1, 2, 4}, {1, 2, 3, 5}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {1, 2, 4, 5}, {2, 3, 5, 7}, {2, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 4}, {0, 3, 4, 5}, {2, 3, 4, 7}, {2, 4, 6, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {3, 4, 5, 7}}, triangulation {{0, 1, 3, 4}, {0, 2, 3, 6}, {0, 3, 4, 6}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {1, 3, 4, 7}, {1, 4, 5, 7}, {3, 4, 6, 7}}, triangulation {{0, 1, 3, 4}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {0, 2, 3, 4}, {1, 3, 4, 7}, {1, 4, 5, 7}, {2, 3, 4, 7}, {2, 4, 6, 7}}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, {1, 2, 3, 7}, {1, 2, 6, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {1, 4, 5, 6}, {1, 5, 6, 7}}, triangulation {{0, 1, 2, 6}, {0, 1, 4, 6}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {1, 2, 3, 6}, {1, 3, 6, 7}, {1, 4, 5, 7}, {1, 4, 6, 7}}, triangulation │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {{0, 1, 3, 5}, {0, 2, 3, 6}, {0, 3, 5, 6}, {0, 4, 5, 6}, {3, 5, 6, 7}}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + triangulation {{0, 1, 2, 4}, {1, 2, 3, 7}, {1, 2, 4, 7}, {1, 4, 5, 7}, │ │ │ │ + ------------------------------------------------------------------------ │ │ │ │ + {2, 4, 6, 7}}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : all(Ts4, isFine) │ │ │ │ │ │ │ │ o8 = true │ │ │ │ i9 : all(Ts4, isStar) │ │ │ │ │ │ │ │ @@ -886,188 +886,190 @@ │ │ │ │ │ │ │ │ o11 = Tally{false => 66} │ │ │ │ true => 8 │ │ │ │ │ │ │ │ o11 : Tally │ │ │ │ i12 : Ts4/gkzVector │ │ │ │ │ │ │ │ - 16 16 4 8 4 20 8 8 4 8 4 8 8 │ │ │ │ -o12 = {{--, 4, --, -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 8 20 16 16 4 16 4 4 8 4 16 4 16 │ │ │ │ - 4, -, --}, {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 20 4 4 20 16 8 16 4 4 20 4 4 │ │ │ │ - 4, --, -, -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 20 16 16 4 4 8 4 4 16 8 16 16 4 │ │ │ │ - --, 4}, {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 16 4 8 4 16 4 16 8 8 8 8 8 8 │ │ │ │ - --, 4, -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 8 8 4 20 8 8 16 4 4 16 16 4 4 16 │ │ │ │ - -}, {4, -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 16 8 4 4 20 20 4 4 20 20 │ │ │ │ - --, -, 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 4 4 8 16 16 4 4 20 20 4 4 │ │ │ │ - -}, {-, 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 20 20 4 8 16 4 4 16 8 4 8 8 20 │ │ │ │ - --, --, -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 8 8 8 4 20 20 4 4 20 4 20 20 4 4 20 │ │ │ │ - 4}, {-, 8, -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 20 4 20 4 4 20 20 4 4 20 8 4 4 │ │ │ │ - --, -, --, -, -, --}, {--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 16 16 8 4 4 16 16 4 16 16 4 8 4 │ │ │ │ - --, --}, {-, 8, 4, -, 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 20 8 8 8 8 8 8 20 4 4 8 8 20 │ │ │ │ - 4, --, 8, -, -, -}, {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 8 4 16 4 8 16 4 20 8 8 8 16 4 4 │ │ │ │ - -}, {-, 8, --, -, 4, -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 16 8 16 4 16 4 16 4 8 20 8 4 │ │ │ │ - 8, --, 4, 4, -}, {--, 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 8 8 8 8 20 4 4 16 4 16 4 16 20 8 │ │ │ │ - -}, {-, -, 8, -, 4, --, -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 8 4 8 20 4 8 8 8 8 8 8 4 20 │ │ │ │ - -, -, 4, 8, -}, {--, 4, 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 16 16 4 4 8 8 4 8 8 20 20 4 4 │ │ │ │ - {--, --, 4, -, 4, -, -, 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 20 20 4 4 20 4 20 20 4 8 │ │ │ │ - 4, --, 4}, {--, -, 4, 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 8 20 4 8 16 4 16 8 4 4 16 16 8 │ │ │ │ - 4, -, --, 8, -, -, 4}, {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 4 4 8 16 16 4 4 20 20 4 8 8 │ │ │ │ - -, 4}, {4, -, -, 8, --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 8 8 8 8 8 20 8 8 4 16 8 4 16 │ │ │ │ - -, 8, 8, -, -, -}, {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 4 4 8 16 4 16 16 4 16 4 16 4 4 16 │ │ │ │ - -}, {-, 8, 4, -, --, -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 4 16 4 16 8 8 8 8 8 8 20 8 8 8 4 │ │ │ │ - -, --, -, 4, --}, {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 4 8 4 16 16 20 4 4 20 4 4 16 │ │ │ │ - {8, -, -, 4, -, --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 4 16 16 8 16 4 16 4 8 8 20 8 4 │ │ │ │ - -, --, --, 4}, {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 4 8 20 8 8 8 4 16 16 4 8 8 20 8 │ │ │ │ - {4, -, -, 8, --, 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 4 4 8 20 8 8 8 8 8 8 8 8 4 16 │ │ │ │ - 8, 4, -}, {-, 4, 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 16 4 4 16 20 4 8 8 8 8 4 8 20 8 │ │ │ │ - --, 4, 8, -, -, --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, │ │ │ │ - 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ - ----------------------------------------------------------------------- │ │ │ │ - 4}} │ │ │ │ + 20 4 4 20 8 4 4 16 16 8 4 │ │ │ │ +o12 = {{--, 4, 4, -, -, 4, 4, --}, {8, -, -, 4, -, 4, --, --}, {-, 8, 4, -, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 16 16 4 16 16 4 8 4 20 8 8 8 │ │ │ │ + 4, -, --, --}, {-, --, 4, --, 8, -, -, 4}, {-, 4, 4, --, 8, -, -, -}, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 8 8 20 4 4 8 8 20 8 4 16 4 │ │ │ │ + {-, -, -, 8, --, 4, 4, -}, {-, 8, 4, -, 4, -, --, -}, {-, 8, --, -, 4, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 16 4 20 8 8 8 16 4 4 16 8 16 │ │ │ │ + -, --, 4}, {4, -, --, 4, -, 8, -, -}, {--, -, -, 8, --, 4, 4, -}, {--, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 16 4 16 4 8 20 8 4 8 8 8 8 20 │ │ │ │ + 4, -, --, -, --, 8, -}, {4, -, --, -, -, 8, 4, -}, {-, -, 8, -, 4, --, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 4 16 4 16 4 16 20 8 8 4 8 20 │ │ │ │ + -, 4}, {-, --, 8, -, --, 4, -, --}, {4, --, -, -, -, 4, 8, -}, {--, 4, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 8 8 8 8 8 8 4 20 16 16 4 4 8 │ │ │ │ + 4, -, -, -, -, 8}, {8, -, -, -, -, 4, 4, --}, {--, --, 4, -, 4, -, -, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 4 8 8 20 20 4 4 20 20 4 │ │ │ │ + 8}, {8, -, -, 4, -, -, 4, --}, {4, --, 4, -, -, 4, --, 4}, {--, -, 4, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 20 4 20 20 4 8 8 20 4 8 │ │ │ │ + 4, 4, 4, -, --}, {4, -, --, 4, 4, --, -, 4}, {-, 4, -, --, 8, -, -, 4}, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 16 4 16 8 4 4 16 16 8 4 4 8 │ │ │ │ + {4, --, -, --, -, 4, 8, -}, {-, 4, --, --, 8, -, -, 4}, {4, -, -, 8, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 16 16 4 4 20 20 4 8 8 8 8 8 8 │ │ │ │ + --, --, 4, -}, {-, 4, --, 4, 4, --, 4, -}, {-, -, -, 8, 8, -, -, -}, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 20 8 8 4 16 8 4 16 4 4 8 16 │ │ │ │ + {-, --, -, 4, -, 4, 8, -}, {4, --, -, 4, -, --, 8, -}, {-, 8, 4, -, --, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 16 16 4 16 4 16 4 4 16 4 16 4 16 │ │ │ │ + -, --, 4}, {--, -, 4, --, -, 8, --, -}, {-, 8, --, -, --, -, 4, --}, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 8 8 8 8 8 20 8 8 8 4 4 8 4 │ │ │ │ + {8, -, -, -, -, -, -, 8}, {--, -, 4, -, 4, -, -, 8}, {8, -, -, 4, -, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 16 16 20 4 4 20 4 4 16 4 16 16 │ │ │ │ + --, 4, --}, {--, 4, -, 4, 4, -, 4, --}, {8, -, -, --, -, --, --, 4}, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 16 4 16 4 8 8 20 8 4 4 8 20 │ │ │ │ + {4, -, --, 4, -, 8, --, -}, {-, -, 4, --, 8, -, -, 4}, {4, -, -, 8, --, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 8 8 4 16 16 4 8 8 20 8 4 4 │ │ │ │ + 4, -, -}, {4, -, -, 8, --, 4, --, -}, {-, -, --, 4, -, 8, 4, -}, {-, 4, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 20 8 8 8 8 8 8 8 8 4 16 16 4 4 │ │ │ │ + 8, -, 4, --, -, -}, {-, 8, -, -, -, -, 8, -}, {-, --, --, 4, 8, -, -, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 16 20 4 8 8 8 8 4 8 20 8 16 16 │ │ │ │ + --}, {4, --, -, 4, -, -, 8, -}, {-, 4, 8, -, -, --, -, 4}, {--, 4, --, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 8 4 20 8 8 4 8 4 8 8 8 20 │ │ │ │ + -, 4, -, -, 8}, {--, 4, -, -, 4, -, -, 8}, {8, -, -, 4, -, 4, -, --}, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 16 16 4 16 4 4 8 4 16 4 16 20 4 │ │ │ │ + {4, --, --, -, --, -, -, 8}, {-, 4, 8, -, 4, --, -, --}, {4, 4, --, -, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 20 16 8 16 4 4 20 4 4 20 │ │ │ │ + -, --, 4, 4}, {--, 4, 4, -, --, -, -, 8}, {4, --, -, 4, 4, -, --, 4}, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 16 16 4 4 8 4 4 16 8 16 16 4 16 │ │ │ │ + {--, --, -, 4, -, 4, 8, -}, {8, -, -, --, -, 4, 4, --}, {--, -, --, 4, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 8 4 16 4 16 8 8 8 8 8 8 8 │ │ │ │ + -, 8, 4, -}, {-, --, 8, -, 4, --, -, 4}, {-, -, 8, -, -, 8, -, -}, {4, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 4 20 8 8 16 4 4 16 16 4 4 16 16 8 │ │ │ │ + -, -, 8, --, -, 4, -}, {--, -, -, 8, 4, --, --, -}, {4, -, --, --, -, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 4 20 20 4 4 20 20 4 4 │ │ │ │ + 8, 4, -}, {-, --, 4, 4, 4, 4, --, -}, {-, 4, 4, --, --, 4, 4, -}, {-, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 16 16 4 4 20 20 4 4 20 20 │ │ │ │ + 4, 8, -, --, --, -, 4}, {4, -, 4, --, --, 4, -, 4}, {4, 4, -, --, --, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 8 16 4 4 16 8 4 8 8 20 8 │ │ │ │ + -, 4, 4}, {-, 4, 4, --, 8, -, -, --}, {-, 8, 4, -, -, -, --, 4}, {-, 8, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 8 8 4 20 20 4 4 20 4 20 20 4 4 20 20 4 20 4 │ │ │ │ + -, -, 4, -, --, 4}, {--, -, -, --, -, --, --, -}, {-, --, --, -, --, -, │ │ │ │ + 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 4 20 │ │ │ │ + -, --}} │ │ │ │ + 3 3 │ │ │ │ │ │ │ │ o12 : List │ │ │ │ i13 : volume convexHull A -- 8 │ │ │ │ │ │ │ │ o13 = 8 │ │ │ │ │ │ │ │ o13 : QQ │ │ │ │ i14 : stars1 = select(Ts4, t -> (gkzVector t)#-1 == 8) │ │ │ │ │ │ │ │ -o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ │ +o14 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ │ + {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation │ │ │ │ + {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, │ │ │ │ + {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, │ │ │ │ + {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, │ │ │ │ + {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}}, │ │ │ │ + {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, │ │ │ │ + triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ │ + {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation │ │ │ │ + {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ │ + {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ {2, 4, 6, 7}}} │ │ │ │ │ │ │ │ o14 : List │ │ │ │ i15 : stars2 = select(Ts4, isStar) │ │ │ │ │ │ │ │ -o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ │ +o15 = {triangulation {{0, 1, 2, 7}, {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {1, 2, 3, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, │ │ │ │ + {0, 4, 6, 7}, {1, 2, 3, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation │ │ │ │ + {0, 2, 6, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, │ │ │ │ + {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {2, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, │ │ │ │ + {0, 4, 6, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {0, 2, 4, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, │ │ │ │ + {0, 2, 4, 7}, {0, 4, 5, 7}, {2, 4, 6, 7}}, triangulation {{0, 1, 2, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {0, 1, 5, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}, {1, 2, 3, 7}}, │ │ │ │ + {0, 1, 5, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, {1, 2, 3, 7}, {2, 4, 6, 7}}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, {0, 2, 6, 7}, {0, 4, 6, 7}, │ │ │ │ + triangulation {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 6, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {1, 2, 3, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 3, 7}, {0, 1, 5, 7}, │ │ │ │ + {0, 4, 6, 7}, {1, 4, 5, 7}}, triangulation {{0, 1, 2, 7}, {0, 1, 4, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {0, 2, 3, 7}, {0, 2, 6, 7}, {0, 4, 5, 7}, {0, 4, 6, 7}}, triangulation │ │ │ │ + {0, 2, 4, 7}, {1, 2, 3, 7}, {1, 4, 5, 7}, {2, 4, 6, 7}}, triangulation │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {{0, 1, 3, 7}, {0, 1, 5, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {0, 4, 5, 7}, │ │ │ │ + {{0, 1, 3, 7}, {0, 1, 4, 7}, {0, 2, 3, 7}, {0, 2, 4, 7}, {1, 4, 5, 7}, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ {2, 4, 6, 7}}} │ │ │ │ │ │ │ │ o15 : List │ │ │ │ i16 : stars1 == stars2 │ │ │ │ │ │ │ │ o16 = true │ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ @@ -150,15 +150,15 @@ │ │ │ 4 10 │ │ │ o2 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .146605s elapsed
    │ │ │ + -- .104481s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o7 : Triangulation
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 1.329s elapsed
    │ │ │ + -- 1.02847s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ | 0 0 0 1 0 0 -1 0 0 0 | │ │ │ │ | -1 1 2 -1 -1 1 -1 1 0 0 | │ │ │ │ | 1 0 -1 0 0 0 0 0 0 0 | │ │ │ │ │ │ │ │ 4 10 │ │ │ │ o2 : Matrix ZZ <-- ZZ │ │ │ │ i3 : elapsedTime Ts = allTriangulations(A, Fine => true); │ │ │ │ - -- .146605s elapsed │ │ │ │ + -- .104481s elapsed │ │ │ │ i4 : select(Ts, T -> isStar T) │ │ │ │ │ │ │ │ o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 5, 7, 9}, {0, 3, 5, 8, 9}, {0, 4, 6, 8, 9}, {0, 5, 6, 7, 9}, {0, 5, 6, │ │ │ │ @@ -86,15 +86,15 @@ │ │ │ │ 6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8}, │ │ │ │ {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6, │ │ │ │ 7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2, │ │ │ │ 3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}} │ │ │ │ │ │ │ │ o7 : Triangulation │ │ │ │ i8 : elapsedTime Ts2 = generateTriangulations T; │ │ │ │ - -- 1.329s elapsed │ │ │ │ + -- 1.02847s elapsed │ │ │ │ i9 : #Ts2 == #Ts │ │ │ │ │ │ │ │ o9 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _P_o_l_y_h_e_d_r_a -- for computations with convex polyhedra, cones, and fans │ │ │ │ * _T_o_p_c_o_m -- interface to selected functions from topcom package │ │ │ │ * _R_e_f_l_e_x_i_v_e_P_o_l_y_t_o_p_e_s_D_B -- simple access to Kreuzer-Skarke database of │ │ ├── ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump │ │ │ @@ -1,8 +1,8 @@ │ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Jan 19 15:01:18 2026 │ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Mon Jan 19 15:01:19 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ #:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=7 │ │ │ cm90Rm9ydw== │ │ ├── ./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 0.914506s (cpu); 0.607753s (thread); 0s (gc) │ │ │ + -- used 1.326s (cpu); 0.72383s (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 0.649867s (cpu); 0.43703s (thread); 0s (gc) │ │ │ + -- used 0.963005s (cpu); 0.534015s (thread); 0s (gc) │ │ │ │ │ │ i7 : sum G │ │ │ │ │ │ o7 = | t_1t_16 │ │ │ | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+ │ │ │ | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t │ │ │ | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2- │ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ @@ -71,15 +71,15 @@ │ │ │ │ │ │ │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -98,15 +98,15 @@ │ │ │ │ │ │
    │ │ │ 
    i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 0.914506s (cpu); 0.607753s (thread); 0s (gc)
    │ │ │ + -- used 1.326s (cpu); 0.72383s (thread); 0s (gc) │ │ │
    │ │ │ 
    i4 : T=ring first G;
    │ │ │
    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ - -- used 0.649867s (cpu); 0.43703s (thread); 0s (gc)
    │ │ │ + -- used 0.963005s (cpu); 0.534015s (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 0.914506s (cpu); 0.607753s (thread); 0s (gc)
│ │ │ │ + -- used 1.326s (cpu); 0.72383s (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 0.649867s (cpu); 0.43703s (thread); 0s (gc)
│ │ │ │ + -- used 0.963005s (cpu); 0.534015s (thread); 0s (gc)
│ │ │ │  i7 : sum G
│ │ │ │  
│ │ │ │  o7 = | t_1t_16
│ │ │ │       | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
│ │ │ │       | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
│ │ │ │       | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Basic__Divisor.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 18380296066161043289
│ │ │  
│ │ │  i1 : R = QQ[x,y,z];
│ │ │  
│ │ │  i2 : D = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o2 = 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) + 4/3*Div(y) + 2*Div(z)
│ │ │  
│ │ │  o4 : QWeilDivisor on R
│ │ │  
│ │ │  i5 : 0.6*D
│ │ │  
│ │ │ -o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x)
│ │ │ +o5 = .6*Div(x) + 1.2*Div(y) + 1.8*Div(z)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Basic__Divisor_sp_pl_sp__Basic__Divisor.out
│ │ │ @@ -1,32 +1,32 @@
│ │ │  -- -*- M2-comint -*- hash: 11085051886200177329
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = Div(x) + 3*Div(y) + 2*Div(z)
│ │ │ +o2 = 3*Div(y) + 2*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)})
│ │ │  
│ │ │ -o3 = -5*Div(x) + -2*Div(z) + 3*Div(y)
│ │ │ +o3 = -2*Div(z) + 3*Div(y) + -5*Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : D1 + D2
│ │ │  
│ │ │ -o4 = -4*Div(x) + 6*Div(y)
│ │ │ +o4 = 6*Div(y) + -4*Div(x)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : D1 - D2
│ │ │  
│ │ │ -o5 = 6*Div(x) + 4*Div(z)
│ │ │ +o5 = 4*Div(z) + 6*Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : R = QQ[x,y];
│ │ │  
│ │ │  i7 : D1 = divisor({3, 1}, {ideal(x), ideal(y)})
│ │ │  
│ │ │ @@ -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
│ │ │ @@ -16,21 +16,21 @@
│ │ │  
│ │ │  o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
│ │ │  
│ │ │  i5 : 8*D
│ │ │  
│ │ │ -o5 = -8*Div(x+y) + 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
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___O__O_sp__R__Weil__Divisor.out
│ │ │ @@ -90,15 +90,15 @@
│ │ │                                2
│ │ │  o15 : R-module, submodule of R
│ │ │  
│ │ │  i16 : R = ZZ/11[x,y];
│ │ │  
│ │ │  i17 : D = divisor(x*y/(x+y))
│ │ │  
│ │ │ -o17 = -Div(x+y) + Div(x) + Div(y)
│ │ │ +o17 = Div(x) + Div(y) + -Div(x+y)
│ │ │  
│ │ │  o17 : WeilDivisor on R
│ │ │  
│ │ │  i18 : divisor(OO(D))
│ │ │  
│ │ │  o18 = 0, the zero divisor
│ │ ├── ./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) + 2*Div(y) + -Div(z)
│ │ │ +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 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │ +o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
│ │ │  
│ │ │  i3 : ceiling( D )
│ │ │  
│ │ │ -o3 = Div(x, z) + 2*Div(y, z)
│ │ │ +o3 = 2*Div(y, z) + Div(x, 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 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │ +o5 = -.7*Div(y, z) + .3*Div(x, 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 = 0*Div(y) + -2*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 0*Div(y) + -2*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : cleanSupport(D)
│ │ │  
│ │ │ -o3 = -2*Div(z) + Div(x)
│ │ │ +o3 = Div(x) + -2*Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out
│ │ │ @@ -68,21 +68,21 @@
│ │ │  
│ │ │  o15 : WeilDivisor on A
│ │ │  
│ │ │  i16 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │ +o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │  
│ │ │  o17 : QWeilDivisor on R
│ │ │  
│ │ │  i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │  
│ │ │ -o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │ +o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │  
│ │ │  o18 : QWeilDivisor on R
│ │ │  
│ │ │  i19 : R = ZZ/11[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i20 : D = divisor({1.1, -3.14159}, {ideal(x,u), ideal(x, v)}, CoefficientType=>RR)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out
│ │ │ @@ -44,51 +44,51 @@
│ │ │  i10 : J = m^9;
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : M = J*R^1;
│ │ │  
│ │ │  i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.16902s (cpu); 0.118168s (thread); 0s (gc)
│ │ │ + -- used 0.187844s (cpu); 0.0825385s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.584267s (cpu); 0.58425s (thread); 0s (gc)
│ │ │ + -- used 0.556403s (cpu); 0.556414s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.574295s (cpu); 0.510554s (thread); 0s (gc)
│ │ │ + -- used 0.74175s (cpu); 0.629369s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00294085s (cpu); 0.00294153s (thread); 0s (gc)
│ │ │ + -- used 0.00343641s (cpu); 0.00344385s (thread); 0s (gc)
│ │ │  
│ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00228041s (cpu); 0.00228141s (thread); 0s (gc)
│ │ │ + -- used 0.00276856s (cpu); 0.00277332s (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 0.301857s (cpu); 0.177561s (thread); 0s (gc)
│ │ │ + -- used 0.418123s (cpu); 0.206925s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
│ │ │  
│ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.0121946s (cpu); 0.0122024s (thread); 0s (gc)
│ │ │ + -- used 0.00736563s (cpu); 0.00737385s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i23 : J = ideal(x,y);
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Cartier.out
│ │ │ @@ -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 = 2*Div(y, z) + Div(x, z)
│ │ │ +o14 = Div(x, z) + 2*Div(y, 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__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 = Div(x, z) + 2*Div(y, z)
│ │ │ +o12 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o12 : WeilDivisor on R
│ │ │  
│ │ │  i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ +o13 = 1/2*Div(y, z) + 3/4*Div(x, 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
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Reduced.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 6263371580478090172
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D1 = divisor(x^2 * y^3 * z)
│ │ │  
│ │ │ -o2 = 3*Div(y) + Div(z) + 2*Div(x)
│ │ │ +o2 = 2*Div(x) + 3*Div(y) + Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor(x * y * z)
│ │ │  
│ │ │ -o3 = Div(y) + Div(z) + Div(x)
│ │ │ +o3 = Div(x) + Div(y) + Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : isReduced( D1 )
│ │ │  
│ │ │  o4 = false
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__S__N__C.out
│ │ │ @@ -48,15 +48,15 @@
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : R = QQ[x, y];
│ │ │  
│ │ │  i14 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o14 = Div(y) + Div(x) + Div(x+y)
│ │ │ +o14 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o14 : WeilDivisor on R
│ │ │  
│ │ │  i15 : isSNC( D, IsGraded => true )
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Weil__Divisor.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 9088327857146195664
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D1 = divisor({1/1, 2/2, -6/3}, {ideal(x), ideal(y), ideal(z)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o2 = Div(x) + Div(y) + -2*Div(z)
│ │ │ +o2 = Div(y) + -2*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({1/2, 3/4, 5/6}, {ideal(y), ideal(z), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o3 = 5/6*Div(x) + 1/2*Div(y) + 3/4*Div(z)
│ │ │ +o3 = 1/2*Div(y) + 3/4*Div(z) + 5/6*Div(x)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : isWeilDivisor( D1 )
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_map__To__Projective__Space.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  o3 : RingMap R <-- QQ[YY ..YY ]
│ │ │                          1    2
│ │ │  
│ │ │  i4 : R = ZZ/7[x,y,z];
│ │ │  
│ │ │  i5 : D = divisor(x*y)
│ │ │  
│ │ │ -o5 = Div(x) + Div(y)
│ │ │ +o5 = Div(y) + Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │  
│ │ │               ZZ            2             2        2
│ │ │  o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z })
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.out
│ │ │ @@ -6,21 +6,21 @@
│ │ │  
│ │ │  i3 : f = map(T, R, {a^3, a^2*b, a*b^2, b^3});
│ │ │  
│ │ │  o3 : RingMap T <-- R
│ │ │  
│ │ │  i4 : D = divisor(y*z)
│ │ │  
│ │ │ -o4 = 3*Div(z, y, x) + 3*Div(w, z, y)
│ │ │ +o4 = 3*Div(w, z, y) + 3*Div(z, y, x)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : pullback(f, D, Strategy=>Primes)
│ │ │  
│ │ │ -o5 = 3*Div(a) + 3*Div(b)
│ │ │ +o5 = 3*Div(b) + 3*Div(a)
│ │ │  
│ │ │  o5 : WeilDivisor on T
│ │ │  
│ │ │  i6 : pullback(f, D, Strategy=>Sheaves)
│ │ │  
│ │ │  o6 = 3*Div(b) + 3*Div(a)
│ │ │  
│ │ │ @@ -36,18 +36,18 @@
│ │ │  
│ │ │  i10 : D = divisor(x*y*(x+y));
│ │ │  
│ │ │  o10 : WeilDivisor on R
│ │ │  
│ │ │  i11 : D1 = pullback(f, D)
│ │ │  
│ │ │ -o11 = Div(a+1) + Div(a) + 3*Div(b)
│ │ │ +o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │  
│ │ │  o11 : WeilDivisor on S
│ │ │  
│ │ │  i12 : f^* D
│ │ │  
│ │ │ -o12 = Div(a+1) + Div(a) + 3*Div(b)
│ │ │ +o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │  
│ │ │  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 0.278001s (cpu); 0.214596s (thread); 0s (gc)
│ │ │ + -- used 0.358927s (cpu); 0.254018s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.528559s (cpu); 0.402293s (thread); 0s (gc)
│ │ │ + -- used 0.501157s (cpu); 0.40115s (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 0.237147s (cpu); 0.127066s (thread); 0s (gc)
│ │ │ + -- used 0.394066s (cpu); 0.176743s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 6.57025s (cpu); 4.65331s (thread); 0s (gc)
│ │ │ + -- used 6.71642s (cpu); 5.05609s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
│ │ │  
│ │ │  i31 : J1 == J2
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 5.83537s (cpu); 4.82046s (thread); 0s (gc)
│ │ │ + -- used 6.94151s (cpu); 5.40388s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.886189s (cpu); 0.630433s (thread); 0s (gc)
│ │ │ + -- used 0.673858s (cpu); 0.449784s (thread); 0s (gc)
│ │ │  
│ │ │  i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i35 : I = ideal(x,u);
│ │ │  
│ │ │  o35 : Ideal of R
│ │ │  
│ │ │  i36 : J = I^20;
│ │ │  
│ │ │  o36 : Ideal of R
│ │ │  
│ │ │  i37 : M = I^20*R^1;
│ │ │  
│ │ │  i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 1.18031s (cpu); 0.565503s (thread); 0s (gc)
│ │ │ + -- used 1.45959s (cpu); 0.497626s (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 0.256434s (cpu); 0.110057s (thread); 0s (gc)
│ │ │ + -- used 0.0165079s (cpu); 0.0165102s (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 0.240022s (cpu); 0.109141s (thread); 0s (gc)
│ │ │ + -- used 0.34902s (cpu); 0.106931s (thread); 0s (gc)
│ │ │  
│ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00676458s (cpu); 0.00676817s (thread); 0s (gc)
│ │ │ + -- used 0.00802442s (cpu); 0.00802475s (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 0.034296s (cpu); 0.0342949s (thread); 0s (gc)
│ │ │ + -- used 0.038347s (cpu); 0.0383441s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.207227s (cpu); 0.144933s (thread); 0s (gc)
│ │ │ + -- used 0.268761s (cpu); 0.169687s (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 0.0297901s (cpu); 0.0297947s (thread); 0s (gc)
│ │ │ + -- used 0.03856s (cpu); 0.0385645s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.146748s (cpu); 0.0814116s (thread); 0s (gc)
│ │ │ + -- used 0.197218s (cpu); 0.104885s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : J1 == J2
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ring_lp__Basic__Divisor_rp.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 5006859181202351713
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : ring( D )
│ │ │  
│ │ │  o3 = R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__Q__Weil__Divisor.out
│ │ │ @@ -16,12 +16,12 @@
│ │ │  
│ │ │  o4 = Div(x)
│ │ │  
│ │ │  o4 : QWeilDivisor on R
│ │ │  
│ │ │  i5 : F = divisor({3, 0, -2}, {ideal(x), ideal(y), ideal(x+y)})
│ │ │  
│ │ │ -o5 = 3*Div(x) + 0*Div(y) + -2*Div(x+y)
│ │ │ +o5 = -2*Div(x+y) + 3*Div(x) + 0*Div(y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 :
│ │ ├── ./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
│ │ │ @@ -72,15 +72,15 @@
│ │ │                
    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
│ │ │ @@ -94,24 +94,24 @@
│ │ │  o3 : HashTable
    │ │ │
    │ │ │ 
    i4 : (2/3)*D
│ │ │  
│ │ │ -o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x)
│ │ │ +o4 = 2/3*Div(x) + 4/3*Div(y) + 2*Div(z)
│ │ │  
│ │ │  o4 : QWeilDivisor on R
    │ │ │
    │ │ │ 
    i5 : 0.6*D
│ │ │  
│ │ │ -o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x)
│ │ │ +o5 = .6*Div(x) + 1.2*Div(y) + 1.8*Div(z)
│ │ │  
│ │ │  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) + 4/3*Div(y) + 2*Div(z) │ │ │ │ │ │ │ │ o4 : QWeilDivisor on R │ │ │ │ i5 : 0.6*D │ │ │ │ │ │ │ │ -o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x) │ │ │ │ +o5 = .6*Div(x) + 1.2*Div(y) + 1.8*Div(z) │ │ │ │ │ │ │ │ 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 │ │ │ @@ -80,42 +80,42 @@ │ │ │ 
    i1 : R = QQ[x, y, z];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = Div(x) + 3*Div(y) + 2*Div(z)
│ │ │ +o2 = 3*Div(y) + 2*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)})
│ │ │  
│ │ │ -o3 = -5*Div(x) + -2*Div(z) + 3*Div(y)
│ │ │ +o3 = -2*Div(z) + 3*Div(y) + -5*Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : D1 + D2
│ │ │  
│ │ │ -o4 = -4*Div(x) + 6*Div(y)
│ │ │ +o4 = 6*Div(y) + -4*Div(x)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : D1 - D2
│ │ │  
│ │ │ -o5 = 6*Div(x) + 4*Div(z)
│ │ │ +o5 = 4*Div(z) + 6*Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    We can also add or subtract divisors with different coefficients.

    │ │ │ @@ -190,42 +190,42 @@ │ │ │ 
    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 {} │ │ │ │ @@ -16,30 +16,30 @@ │ │ │ │ * Outputs: │ │ │ │ o an instance of the type _B_a_s_i_c_D_i_v_i_s_o_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ We can add or subtract two divisors: │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)}) │ │ │ │ │ │ │ │ -o2 = Div(x) + 3*Div(y) + 2*Div(z) │ │ │ │ +o2 = 3*Div(y) + 2*Div(z) + Div(x) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)}) │ │ │ │ │ │ │ │ -o3 = -5*Div(x) + -2*Div(z) + 3*Div(y) │ │ │ │ +o3 = -2*Div(z) + 3*Div(y) + -5*Div(x) │ │ │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ i4 : D1 + D2 │ │ │ │ │ │ │ │ -o4 = -4*Div(x) + 6*Div(y) │ │ │ │ +o4 = 6*Div(y) + -4*Div(x) │ │ │ │ │ │ │ │ o4 : WeilDivisor on R │ │ │ │ i5 : D1 - D2 │ │ │ │ │ │ │ │ -o5 = 6*Div(x) + 4*Div(z) │ │ │ │ +o5 = 4*Div(z) + 6*Div(x) │ │ │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ We can also add or subtract divisors with different coefficients. │ │ │ │ i6 : R = QQ[x,y]; │ │ │ │ i7 : D1 = divisor({3, 1}, {ideal(x), ideal(y)}) │ │ │ │ │ │ │ │ o7 = 3*Div(x) + Div(y) │ │ │ │ @@ -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 │ │ │ @@ -103,24 +103,24 @@ │ │ │ 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
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,20 +27,20 @@
│ │ │ │  CoefficientType=>RR)
│ │ │ │  
│ │ │ │  o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y)
│ │ │ │  
│ │ │ │  o4 : RWeilDivisor on R
│ │ │ │  i5 : 8*D
│ │ │ │  
│ │ │ │ -o5 = -8*Div(x+y) + 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
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___O__O_sp__R__Weil__Divisor.html
│ │ │ @@ -227,15 +227,15 @@
│ │ │                
    i16 : R = ZZ/11[x,y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : D = divisor(x*y/(x+y))
│ │ │  
│ │ │ -o17 = -Div(x+y) + Div(x) + Div(y)
│ │ │ +o17 = Div(x) + Div(y) + -Div(x+y)
│ │ │  
│ │ │  o17 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : divisor(OO(D))
│ │ │ ├── html2text {}
│ │ │ │ @@ -100,15 +100,15 @@
│ │ │ │  Note that you can call the divisor constructor on the module you construct, but
│ │ │ │  it will only produce a divisor up to linear equivalence (which can mean
│ │ │ │  different things depending on whether or not you are keeping track of the
│ │ │ │  grading).
│ │ │ │  i16 : R = ZZ/11[x,y];
│ │ │ │  i17 : D = divisor(x*y/(x+y))
│ │ │ │  
│ │ │ │ -o17 = -Div(x+y) + Div(x) + Div(y)
│ │ │ │ +o17 = Div(x) + Div(y) + -Div(x+y)
│ │ │ │  
│ │ │ │  o17 : WeilDivisor on R
│ │ │ │  i18 : divisor(OO(D))
│ │ │ │  
│ │ │ │  o18 = 0, the zero divisor
│ │ │ │  
│ │ │ │  o18 : WeilDivisor on R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_apply__To__Coefficients.html
│ │ │ @@ -82,24 +82,24 @@
│ │ │                
    i1 : R = QQ[x, y, z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor(x*y^2/z)
│ │ │  
│ │ │ -o2 = Div(x) + 2*Div(y) + -Div(z)
│ │ │ +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) + 2*Div(y) + -Div(z) │ │ │ │ +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 │ │ │ @@ -78,24 +78,24 @@ │ │ │ 
    i1 : R = QQ[x, y, z] / ideal(x *y - z^2);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D = divisor({1/2, 4/3}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │ +o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : ceiling( D )
│ │ │  
│ │ │ -o3 = Div(x, z) + 2*Div(y, z)
│ │ │ +o3 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : floor( D )
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o4 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : E = divisor({0.3, -0.7}, {ideal(x, z), ideal(y,z)}, CoefficientType => RR)
│ │ │  
│ │ │ -o5 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │ +o5 = -.7*Div(y, z) + .3*Div(x, 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 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │ │ +o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │ │  
│ │ │ │  o2 : QWeilDivisor on R
│ │ │ │  i3 : ceiling( D )
│ │ │ │  
│ │ │ │ -o3 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o3 = 2*Div(y, z) + Div(x, 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 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │ │ +o5 = -.7*Div(y, z) + .3*Div(x, 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
│ │ │ @@ -76,24 +76,24 @@
│ │ │                
    i1 : R = QQ[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1,0,-2}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = 0*Div(y) + -2*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 0*Div(y) + -2*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : cleanSupport(D)
│ │ │  
│ │ │ -o3 = -2*Div(z) + Div(x)
│ │ │ +o3 = Div(x) + -2*Div(z)
│ │ │  
│ │ │  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 = 0*Div(y) + -2*Div(z) + Div(x) │ │ │ │ +o2 = Div(x) + 0*Div(y) + -2*Div(z) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : cleanSupport(D) │ │ │ │ │ │ │ │ -o3 = -2*Div(z) + Div(x) │ │ │ │ +o3 = Div(x) + -2*Div(z) │ │ │ │ │ │ │ │ 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 │ │ │ @@ -223,24 +223,24 @@ │ │ │ 
    i16 : R = ZZ/7[x,y];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │ +o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │  
│ │ │  o17 : QWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │  
│ │ │ -o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │ +o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │  
│ │ │  o18 : QWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Or an R-divisor. This time we work in the cone over $P^1 \times P^1$.

    │ │ │ ├── html2text {} │ │ │ │ @@ -103,20 +103,20 @@ │ │ │ │ │ │ │ │ o15 : WeilDivisor on A │ │ │ │ We can construct a Q-divisor as well. Here are two ways to do it (we work in │ │ │ │ $A^2$ this time). │ │ │ │ i16 : R = ZZ/7[x,y]; │ │ │ │ i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ) │ │ │ │ │ │ │ │ -o17 = 2*Div(x) + -1/2*Div(-x^3+y^2) │ │ │ │ +o17 = -1/2*Div(-x^3+y^2) + 2*Div(x) │ │ │ │ │ │ │ │ o17 : QWeilDivisor on R │ │ │ │ i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x) │ │ │ │ │ │ │ │ -o18 = 2*Div(x) + -1/2*Div(-x^3+y^2) │ │ │ │ +o18 = -1/2*Div(-x^3+y^2) + 2*Div(x) │ │ │ │ │ │ │ │ o18 : QWeilDivisor on R │ │ │ │ Or an R-divisor. This time we work in the cone over $P^1 \times P^1$. │ │ │ │ i19 : R = ZZ/11[x,y,u,v]/ideal(x*y-u*v); │ │ │ │ i20 : D = divisor({1.1, -3.14159}, {ideal(x,u), ideal(x, v)}, │ │ │ │ CoefficientType=>RR) │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html │ │ │ @@ -163,43 +163,43 @@ │ │ │ │ │ │ 
    i11 : M = J*R^1;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.16902s (cpu); 0.118168s (thread); 0s (gc)
│ │ │ + -- used 0.187844s (cpu); 0.0825385s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.584267s (cpu); 0.58425s (thread); 0s (gc)
│ │ │ + -- used 0.556403s (cpu); 0.556414s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.574295s (cpu); 0.510554s (thread); 0s (gc)
    │ │ │ + -- used 0.74175s (cpu); 0.629369s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00294085s (cpu); 0.00294153s (thread); 0s (gc)
    │ │ │ + -- used 0.00343641s (cpu); 0.00344385s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00228041s (cpu); 0.00228141s (thread); 0s (gc)
│ │ │ + -- used 0.00276856s (cpu); 0.00277332s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │ @@ -223,23 +223,23 @@ │ │ │ │ │ │ o19 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.301857s (cpu); 0.177561s (thread); 0s (gc)
│ │ │ + -- used 0.418123s (cpu); 0.206925s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.0121946s (cpu); 0.0122024s (thread); 0s (gc)
│ │ │ + -- used 0.00736563s (cpu); 0.00737385s (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 0.16902s (cpu); 0.118168s (thread); 0s (gc) │ │ │ │ + -- used 0.187844s (cpu); 0.0825385s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of R │ │ │ │ i13 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.584267s (cpu); 0.58425s (thread); 0s (gc) │ │ │ │ + -- used 0.556403s (cpu); 0.556414s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ i14 : time dualize(M, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.574295s (cpu); 0.510554s (thread); 0s (gc) │ │ │ │ + -- used 0.74175s (cpu); 0.629369s (thread); 0s (gc) │ │ │ │ i15 : time dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00294085s (cpu); 0.00294153s (thread); 0s (gc) │ │ │ │ + -- used 0.00343641s (cpu); 0.00344385s (thread); 0s (gc) │ │ │ │ i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00228041s (cpu); 0.00228141s (thread); 0s (gc) │ │ │ │ + -- used 0.00276856s (cpu); 0.00277332s (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 0.301857s (cpu); 0.177561s (thread); 0s (gc) │ │ │ │ + -- used 0.418123s (cpu); 0.206925s (thread); 0s (gc) │ │ │ │ │ │ │ │ o20 : Ideal of R │ │ │ │ i21 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.0121946s (cpu); 0.0122024s (thread); 0s (gc) │ │ │ │ + -- used 0.00736563s (cpu); 0.00737385s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 : Ideal of R │ │ │ │ KnownDomain is an option for dualize. If it is false (default is true), then │ │ │ │ the computer will first check whether the ring is a domain, if it is not then │ │ │ │ it will revert to ModuleStrategy. If KnownDomain is set to true for a non- │ │ │ │ domain, then the function can return an incorrect answer. │ │ │ │ i22 : R = QQ[x,y]/ideal(x*y); │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Cartier.html │ │ │ @@ -132,15 +132,15 @@ │ │ │ 
    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 )
│ │ │ @@ -181,15 +181,15 @@
│ │ │                
    i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o14 = 2*Div(y, z) + Div(x, z)
│ │ │ +o14 = Div(x, z) + 2*Div(y, 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 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o14 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : isCartier(D, IsGraded => true)
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  The output value of this function is stored in the divisor's cache with the
│ │ │ │  value of the last IsGraded option. If you change the IsGraded option, the value
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Homogeneous_lp__Basic__Divisor_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │                
    i1 : R = QQ[x, y, z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1, 2, 3}, {ideal(x * y - z^2), ideal(y * z - x^2), ideal(x * z - y^2)})
│ │ │  
│ │ │ -o2 = 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__Q__Cartier.html
│ │ │ @@ -164,24 +164,24 @@
│ │ │                
    i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o12 = Div(x, z) + 2*Div(y, z)
│ │ │ +o12 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o12 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ +o13 = 1/2*Div(y, z) + 3/4*Div(x, 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 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o12 = 2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o12 : WeilDivisor on R
│ │ │ │  i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ │ +o13 = 1/2*Div(y, z) + 3/4*Div(x, 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
│ │ │ @@ -82,24 +82,24 @@
│ │ │                
    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)
│ │ │ ├── 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
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Reduced.html
│ │ │ @@ -76,24 +76,24 @@
│ │ │                
    i1 : R = QQ[x, y, z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D1 = divisor(x^2 * y^3 * z)
│ │ │  
│ │ │ -o2 = 3*Div(y) + Div(z) + 2*Div(x)
│ │ │ +o2 = 2*Div(x) + 3*Div(y) + Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : D2 = divisor(x * y * z)
│ │ │  
│ │ │ -o3 = Div(y) + Div(z) + Div(x)
│ │ │ +o3 = Div(x) + Div(y) + Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : isReduced( D1 )
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,20 +12,20 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns true if the divisor is reduced (all coefficients equal to
│ │ │ │  1), otherwise it returns false.
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D1 = divisor(x^2 * y^3 * z)
│ │ │ │  
│ │ │ │ -o2 = 3*Div(y) + Div(z) + 2*Div(x)
│ │ │ │ +o2 = 2*Div(x) + 3*Div(y) + Div(z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor(x * y * z)
│ │ │ │  
│ │ │ │ -o3 = Div(y) + Div(z) + Div(x)
│ │ │ │ +o3 = Div(x) + Div(y) + Div(z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : isReduced( D1 )
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : isReduced( D2 )
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__S__N__C.html
│ │ │ @@ -175,15 +175,15 @@
│ │ │                
    i13 : R = QQ[x, y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o14 = Div(y) + Div(x) + Div(x+y)
│ │ │ +o14 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o14 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : isSNC( D, IsGraded => true )
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │  o11 : WeilDivisor on R
│ │ │ │  i12 : isSNC( D, IsGraded => true )
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : R = QQ[x, y];
│ │ │ │  i14 : D = divisor(x*y*(x+y))
│ │ │ │  
│ │ │ │ -o14 = Div(y) + Div(x) + Div(x+y)
│ │ │ │ +o14 = Div(x+y) + Div(y) + Div(x)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : isSNC( D, IsGraded => true )
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  i16 : R = QQ[x,y,z];
│ │ │ │  i17 : D = divisor(x*y*(x+y))
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Weil__Divisor.html
│ │ │ @@ -76,24 +76,24 @@
│ │ │                
    i1 : R = QQ[x, y, z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D1 = divisor({1/1, 2/2, -6/3}, {ideal(x), ideal(y), ideal(z)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o2 = Div(x) + Div(y) + -2*Div(z)
│ │ │ +o2 = Div(y) + -2*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : D2 = divisor({1/2, 3/4, 5/6}, {ideal(y), ideal(z), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o3 = 5/6*Div(x) + 1/2*Div(y) + 3/4*Div(z)
│ │ │ +o3 = 1/2*Div(y) + 3/4*Div(z) + 5/6*Div(x)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : isWeilDivisor( D1 )
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,21 +13,21 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Check if a rational/real divisor is a Weil divisor
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D1 = divisor({1/1, 2/2, -6/3}, {ideal(x), ideal(y), ideal(z)},
│ │ │ │  CoefficientType=>QQ)
│ │ │ │  
│ │ │ │ -o2 = Div(x) + Div(y) + -2*Div(z)
│ │ │ │ +o2 = Div(y) + -2*Div(z) + Div(x)
│ │ │ │  
│ │ │ │  o2 : QWeilDivisor on R
│ │ │ │  i3 : D2 = divisor({1/2, 3/4, 5/6}, {ideal(y), ideal(z), ideal(x)},
│ │ │ │  CoefficientType=>QQ)
│ │ │ │  
│ │ │ │ -o3 = 5/6*Div(x) + 1/2*Div(y) + 3/4*Div(z)
│ │ │ │ +o3 = 1/2*Div(y) + 3/4*Div(z) + 5/6*Div(x)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : isWeilDivisor( D1 )
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : isWeilDivisor( D2 )
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_map__To__Projective__Space.html
│ │ │ @@ -112,15 +112,15 @@
│ │ │                
    i4 : R = ZZ/7[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : D = divisor(x*y)
│ │ │  
│ │ │ -o5 = Div(x) + Div(y)
│ │ │ +o5 = Div(y) + Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  
│ │ │ │  o3 : RingMap R <-- QQ[YY ..YY ]
│ │ │ │                          1    2
│ │ │ │  The user may also specify the variable name of the new projective space.
│ │ │ │  i4 : R = ZZ/7[x,y,z];
│ │ │ │  i5 : D = divisor(x*y)
│ │ │ │  
│ │ │ │ -o5 = Div(x) + Div(y)
│ │ │ │ +o5 = Div(y) + Div(x)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │ │  
│ │ │ │               ZZ            2             2        2
│ │ │ │  o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z })
│ │ │ │                7  1   6
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.html
│ │ │ @@ -94,24 +94,24 @@
│ │ │  o3 : RingMap T <-- R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : D = divisor(y*z)
│ │ │  
│ │ │ -o4 = 3*Div(z, y, x) + 3*Div(w, z, y)
│ │ │ +o4 = 3*Div(w, z, y) + 3*Div(z, y, x)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : pullback(f, D, Strategy=>Primes)
│ │ │  
│ │ │ -o5 = 3*Div(a) + 3*Div(b)
│ │ │ +o5 = 3*Div(b) + 3*Div(a)
│ │ │  
│ │ │  o5 : WeilDivisor on T
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : pullback(f, D, Strategy=>Sheaves)
│ │ │ @@ -150,24 +150,24 @@
│ │ │  o10 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : D1 = pullback(f, D)
│ │ │  
│ │ │ -o11 = Div(a+1) + Div(a) + 3*Div(b)
│ │ │ +o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │  
│ │ │  o11 : WeilDivisor on S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : f^* D
│ │ │  
│ │ │ -o12 = Div(a+1) + Div(a) + 3*Div(b)
│ │ │ +o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │  
│ │ │  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(z, y, x) + 3*Div(w, z, y)
│ │ │ │ +o4 = 3*Div(w, z, y) + 3*Div(z, y, x)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : pullback(f, D, Strategy=>Primes)
│ │ │ │  
│ │ │ │ -o5 = 3*Div(a) + 3*Div(b)
│ │ │ │ +o5 = 3*Div(b) + 3*Div(a)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on T
│ │ │ │  i6 : pullback(f, D, Strategy=>Sheaves)
│ │ │ │  
│ │ │ │  o6 = 3*Div(b) + 3*Div(a)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on T
│ │ │ │ @@ -53,20 +53,20 @@
│ │ │ │  
│ │ │ │  o9 : RingMap S <-- R
│ │ │ │  i10 : D = divisor(x*y*(x+y));
│ │ │ │  
│ │ │ │  o10 : WeilDivisor on R
│ │ │ │  i11 : D1 = pullback(f, D)
│ │ │ │  
│ │ │ │ -o11 = Div(a+1) + Div(a) + 3*Div(b)
│ │ │ │ +o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ │  
│ │ │ │  o11 : WeilDivisor on S
│ │ │ │  i12 : f^* D
│ │ │ │  
│ │ │ │ -o12 = Div(a+1) + Div(a) + 3*Div(b)
│ │ │ │ +o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ │  
│ │ │ │  o12 : WeilDivisor on S
│ │ │ │  As illustrated by the previous example, the same functionality can also be
│ │ │ │  accomplished by f^* (which creates a function which sends a divisor $D$ to $f^*
│ │ │ │  D$).
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _P_r_i_m_e_s -- a value for the option Strategy for the pullback method
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexify.html
│ │ │ @@ -267,23 +267,23 @@
│ │ │  
│ │ │  o22 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time reflexify(J);
│ │ │ - -- used 0.278001s (cpu); 0.214596s (thread); 0s (gc)
│ │ │ + -- used 0.358927s (cpu); 0.254018s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.528559s (cpu); 0.402293s (thread); 0s (gc)
    
│ │ │ + -- used 0.501157s (cpu); 0.40115s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Because of this, there are two strategies for computing a reflexification (at least if the module embeds as an ideal).
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -319,26 +319,26 @@
│ │ │              
│ │ │                
    i28 : M = J*R^1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.237147s (cpu); 0.127066s (thread); 0s (gc)
│ │ │ + -- used 0.394066s (cpu); 0.176743s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 6.57025s (cpu); 4.65331s (thread); 0s (gc)
│ │ │ + -- used 6.71642s (cpu); 5.05609s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │ @@ -348,21 +348,21 @@
│ │ │  
│ │ │  o31 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 5.83537s (cpu); 4.82046s (thread); 0s (gc)
    
│ │ │ + -- used 6.94151s (cpu); 5.40388s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.886189s (cpu); 0.630433s (thread); 0s (gc)
    
│ │ │ + -- used 0.673858s (cpu); 0.449784s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -389,15 +389,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : M = I^20*R^1;
    
│ │ │              
    
│ │ │                
    i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 1.18031s (cpu); 0.565503s (thread); 0s (gc)
│ │ │ + -- used 1.45959s (cpu); 0.497626s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -406,15 +406,15 @@
│ │ │  
│ │ │  o38 : Ideal of R
    
│ │ │              
    
│ │ │                
    i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.256434s (cpu); 0.110057s (thread); 0s (gc)
│ │ │ + -- used 0.0165079s (cpu); 0.0165102s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -423,21 +423,21 @@
│ │ │  
│ │ │  o39 : Ideal of R
    
│ │ │              
    
│ │ │                
    i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.240022s (cpu); 0.109141s (thread); 0s (gc)
    
│ │ │ + -- used 0.34902s (cpu); 0.106931s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00676458s (cpu); 0.00676817s (thread); 0s (gc)
    
│ │ │ + -- used 0.00802442s (cpu); 0.00802475s (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 0.278001s (cpu); 0.214596s (thread); 0s (gc)
│ │ │ │ + -- used 0.358927s (cpu); 0.254018s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time reflexify(J*R^1);
│ │ │ │ - -- used 0.528559s (cpu); 0.402293s (thread); 0s (gc)
│ │ │ │ + -- used 0.501157s (cpu); 0.40115s (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 0.237147s (cpu); 0.127066s (thread); 0s (gc)
│ │ │ │ + -- used 0.394066s (cpu); 0.176743s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2            2     9       9   11
│ │ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │ │  
│ │ │ │  o29 : Ideal of R
│ │ │ │  i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ │ - -- used 6.57025s (cpu); 4.65331s (thread); 0s (gc)
│ │ │ │ + -- used 6.71642s (cpu); 5.05609s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2            2     9       9   11
│ │ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │ │  
│ │ │ │  o30 : Ideal of R
│ │ │ │  i31 : J1 == J2
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ │ - -- used 5.83537s (cpu); 4.82046s (thread); 0s (gc)
│ │ │ │ + -- used 6.94151s (cpu); 5.40388s (thread); 0s (gc)
│ │ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.886189s (cpu); 0.630433s (thread); 0s (gc)
│ │ │ │ + -- used 0.673858s (cpu); 0.449784s (thread); 0s (gc)
│ │ │ │  However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
│ │ │ │  i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
│ │ │ │  i35 : I = ideal(x,u);
│ │ │ │  
│ │ │ │  o35 : Ideal of R
│ │ │ │  i36 : J = I^20;
│ │ │ │  
│ │ │ │  o36 : Ideal of R
│ │ │ │  i37 : M = I^20*R^1;
│ │ │ │  i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ │ - -- used 1.18031s (cpu); 0.565503s (thread); 0s (gc)
│ │ │ │ + -- used 1.45959s (cpu); 0.497626s (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 0.256434s (cpu); 0.110057s (thread); 0s (gc)
│ │ │ │ + -- used 0.0165079s (cpu); 0.0165102s (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 0.240022s (cpu); 0.109141s (thread); 0s (gc)
│ │ │ │ + -- used 0.34902s (cpu); 0.106931s (thread); 0s (gc)
│ │ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.00676458s (cpu); 0.00676817s (thread); 0s (gc)
│ │ │ │ + -- used 0.00802442s (cpu); 0.00802475s (thread); 0s (gc)
│ │ │ │  For ideals, if KnownDomain is false (default value is true), then the function
│ │ │ │  will check whether it is a domain. If it is a domain (or assumed to be a
│ │ │ │  domain), it will reflexify using a strategy which can speed up computation, if
│ │ │ │  not it will compute using a sometimes slower method which is essentially
│ │ │ │  reflexifying it as a module.
│ │ │ │  Consider the following example showing the importance of making the correct
│ │ │ │  assumption about the ring being a domain.
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexive__Power.html
│ │ │ @@ -124,30 +124,30 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.034296s (cpu); 0.0342949s (thread); 0s (gc)
│ │ │ + -- used 0.038347s (cpu); 0.0383441s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.207227s (cpu); 0.144933s (thread); 0s (gc)
│ │ │ + -- used 0.268761s (cpu); 0.169687s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : J20a == J20b
│ │ │ @@ -171,23 +171,23 @@
│ │ │  
│ │ │  o12 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0297901s (cpu); 0.0297947s (thread); 0s (gc)
│ │ │ + -- used 0.03856s (cpu); 0.0385645s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.146748s (cpu); 0.0814116s (thread); 0s (gc)
│ │ │ + -- used 0.197218s (cpu); 0.104885s (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 0.034296s (cpu); 0.0342949s (thread); 0s (gc)
│ │ │ │ + -- used 0.038347s (cpu); 0.0383441s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : I20 = I^20;
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time J20b = reflexify(I20);
│ │ │ │ - -- used 0.207227s (cpu); 0.144933s (thread); 0s (gc)
│ │ │ │ + -- used 0.268761s (cpu); 0.169687s (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 0.0297901s (cpu); 0.0297947s (thread); 0s (gc)
│ │ │ │ + -- used 0.03856s (cpu); 0.0385645s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.146748s (cpu); 0.0814116s (thread); 0s (gc)
│ │ │ │ + -- used 0.197218s (cpu); 0.104885s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : J1 == J2
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_f_l_e_x_i_f_y -- calculate the double dual of an ideal or module Hom(Hom(M,
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_ring_lp__Basic__Divisor_rp.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : ring( D )
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _r_i_n_g,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns the ambient ring of a divisor.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : ring( D )
│ │ │ │  
│ │ │ │  o3 = R
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__Q__Weil__Divisor.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  o4 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : F = divisor({3, 0, -2}, {ideal(x), ideal(y), ideal(x+y)})
│ │ │  
│ │ │ -o5 = 3*Div(x) + 0*Div(y) + -2*Div(x+y)
│ │ │ +o5 = -2*Div(x+y) + 3*Div(x) + 0*Div(y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -24,15 +24,15 @@ │ │ │ │ i4 : toQWeilDivisor(E) │ │ │ │ │ │ │ │ o4 = Div(x) │ │ │ │ │ │ │ │ o4 : QWeilDivisor on R │ │ │ │ i5 : F = divisor({3, 0, -2}, {ideal(x), ideal(y), ideal(x+y)}) │ │ │ │ │ │ │ │ -o5 = 3*Div(x) + 0*Div(y) + -2*Div(x+y) │ │ │ │ +o5 = -2*Div(x+y) + 3*Div(x) + 0*Div(y) │ │ │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_o_W_e_i_l_D_i_v_i_s_o_r -- create a Weil divisor from a Q or R-divisor │ │ │ │ * _t_o_R_W_e_i_l_D_i_v_i_s_o_r -- create a R-divisor from a Q or Weil divisor │ │ │ │ ********** WWaayyss ttoo uussee ttooQQWWeeiillDDiivviissoorr:: ********** │ │ │ │ * toQWeilDivisor(QWeilDivisor) │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_trim_lp__Basic__Divisor_rp.html │ │ │ @@ -88,24 +88,24 @@ │ │ │ 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/example-output/_factor__Weyl__Algebra.out
│ │ │ @@ -4,19 +4,19 @@
│ │ │  
│ │ │  o1 = R
│ │ │  
│ │ │  o1 : PolynomialRing, 1 differential variable(s)
│ │ │  
│ │ │  i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │  
│ │ │ -                                    2         3       2                  2  
│ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ +        3       2                                      3                  
│ │ │ +o2 = {(x dx + 3x  + x*dx - 1)(dx)(x - 1)(x + 1), (dx)(x dx + x*dx - 2)(x -
│ │ │       ------------------------------------------------------------------------
│ │ │ -               3                2         3       2                         
│ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ +                  3       2                  2             3                2
│ │ │ +     1)(x + 1), (x dx + 3x  - x*dx + 1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x 
│ │ │       ------------------------------------------------------------------------
│ │ │ -                 3
│ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ +                                         2
│ │ │ +     + 1), (x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1)}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylAlgebras/html/_factor__Weyl__Algebra.html
│ │ │ @@ -84,22 +84,22 @@
│ │ │  o1 : PolynomialRing, 1 differential variable(s)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx)
│ │ │  
│ │ │ -                                    2         3       2                  2  
│ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1), (x dx + 3x  - x*dx + 1)(dx)(x  +
│ │ │ +        3       2                                      3                  
│ │ │ +o2 = {(x dx + 3x  + x*dx - 1)(dx)(x - 1)(x + 1), (dx)(x dx + x*dx - 2)(x -
│ │ │       ------------------------------------------------------------------------
│ │ │ -               3                2         3       2                         
│ │ │ -     1), (dx)(x dx - x*dx + 2)(x  + 1), (x dx + 3x  + x*dx - 1)(dx)(x - 1)(x
│ │ │ +                  3       2                  2             3                2
│ │ │ +     1)(x + 1), (x dx + 3x  - x*dx + 1)(dx)(x  + 1), (dx)(x dx - x*dx + 2)(x 
│ │ │       ------------------------------------------------------------------------
│ │ │ -                 3
│ │ │ -     + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)}
│ │ │ +                                         2
│ │ │ +     + 1), (x*dx - 1)(dx)(x - 1)(x + 1)(x  + 1)}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    To reduce their number, two factorisations are considered equivalent if they can be related by (1) switching commuting irreducible factors or (2) switching monomials and degree 0 factors; a normal order is chosen where commuting factors are sorted, and monomials are pushed to the right/left if they're differential/not.

    │ │ │ ├── html2text {} │ │ │ │ @@ -20,22 +20,22 @@ │ │ │ │ i1 : R = makeWA(QQ[x]) │ │ │ │ │ │ │ │ o1 = R │ │ │ │ │ │ │ │ o1 : PolynomialRing, 1 differential variable(s) │ │ │ │ i2 : factorWA(x^5*dx^2+7*x^4*dx+8*x^3-x*dx^2+dx) │ │ │ │ │ │ │ │ - 2 3 2 2 │ │ │ │ -o2 = {(x*dx - 1)(dx)(x - 1)(x + 1)(x + 1), (x dx + 3x - x*dx + 1)(dx)(x + │ │ │ │ + 3 2 3 │ │ │ │ +o2 = {(x dx + 3x + x*dx - 1)(dx)(x - 1)(x + 1), (dx)(x dx + x*dx - 2)(x - │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 3 2 3 2 │ │ │ │ - 1), (dx)(x dx - x*dx + 2)(x + 1), (x dx + 3x + x*dx - 1)(dx)(x - 1)(x │ │ │ │ + 3 2 2 3 2 │ │ │ │ + 1)(x + 1), (x dx + 3x - x*dx + 1)(dx)(x + 1), (dx)(x dx - x*dx + 2)(x │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 3 │ │ │ │ - + 1), (dx)(x dx + x*dx - 2)(x - 1)(x + 1)} │ │ │ │ + 2 │ │ │ │ + + 1), (x*dx - 1)(dx)(x - 1)(x + 1)(x + 1)} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ To reduce their number, two factorisations are considered equivalent if they │ │ │ │ can be related by (1) switching commuting irreducible factors or (2) switching │ │ │ │ monomials and degree 0 factors; a normal order is chosen where commuting │ │ │ │ factors are sorted, and monomials are pushed to the right/left if they're │ │ │ │ differential/not. │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out │ │ │ @@ -122,15 +122,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 1.88418s (cpu); 1.08874s (thread); 0s (gc) │ │ │ + -- used 2.5226s (cpu); 1.33026s (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 4.52784s (cpu); 2.81927s (thread); 0s (gc) │ │ │ + -- used 7.99619s (cpu); 3.39016s (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 5.26973s (cpu); 3.24364s (thread); 0s (gc) │ │ │ + -- used 9.19769s (cpu); 3.67156s (thread); 0s (gc) │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o27 : List │ │ │ │ │ │ i28 : peek last ms │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html │ │ │ @@ -292,15 +292,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 1.88418s (cpu); 1.08874s (thread); 0s (gc) │ │ │ + -- used 2.5226s (cpu); 1.33026s (thread); 0s (gc) │ │ │ │ │ │ o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o23 : List │ │ │ │ │ │ │ │ │ │ │ │ @@ -318,15 +318,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 4.52784s (cpu); 2.81927s (thread); 0s (gc) │ │ │ + -- used 7.99619s (cpu); 3.39016s (thread); 0s (gc) │ │ │ │ │ │ o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o25 : List │ │ │ │ │ │ │ │ │ │ │ │ @@ -344,15 +344,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 5.26973s (cpu); 3.24364s (thread); 0s (gc) │ │ │ + -- used 9.19769s (cpu); 3.67156s (thread); 0s (gc) │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o27 : List │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -184,15 +184,15 @@ │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ - -- used 1.88418s (cpu); 1.08874s (thread); 0s (gc) │ │ │ │ + -- used 2.5226s (cpu); 1.33026s (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 4.52784s (cpu); 2.81927s (thread); 0s (gc) │ │ │ │ + -- used 7.99619s (cpu); 3.39016s (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 5.26973s (cpu); 3.24364s (thread); 0s (gc) │ │ │ │ + -- used 9.19769s (cpu); 3.67156s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ │ │ o27 : List │ │ │ │ i28 : peek last ms │ │ │ │ │ │ │ │ o28 = MutableHashTable{0 => {ideal (P, M1)} } │ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/example-output/___Installation_spand_sp__Configuration_spof_spgfan__Interface.out │ │ │ @@ -19,15 +19,15 @@ │ │ │ i4 : prefixDirectory | currentLayout#"programs" │ │ │ │ │ │ o4 = /usr/x86_64-Linux- │ │ │ Debian-forky/libexec/Macaulay2/bin/ │ │ │ │ │ │ i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true); │ │ │ -- warning: reloading gfanInterface; recreate instances of types from this package │ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-17026-0/172 │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22501-0/172 │ │ │ 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/M2-17026-0/172 │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-17026-0/174 │ │ │ +using temporary file /tmp/M2-22501-0/172 │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22501-0/174 │ │ │ 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/M2-17026-0/174 │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-17026-0/176 │ │ │ +using temporary file /tmp/M2-22501-0/174 │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22501-0/176 │ │ │ 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/M2-17026-0/176 │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-17026-0/178 │ │ │ +using temporary file /tmp/M2-22501-0/176 │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22501-0/178 │ │ │ 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/M2-17026-0/178 │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-17026-0/180 │ │ │ +using temporary file /tmp/M2-22501-0/178 │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22501-0/180 │ │ │ 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/M2-17026-0/180 │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-17026-0/182 │ │ │ +using temporary file /tmp/M2-22501-0/180 │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22501-0/182 │ │ │ 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/M2-17026-0/182 │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-17026-0/184 │ │ │ +using temporary file /tmp/M2-22501-0/182 │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22501-0/184 │ │ │ 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/M2-17026-0/184 │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-17026-0/186 │ │ │ +using temporary file /tmp/M2-22501-0/184 │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22501-0/186 │ │ │ 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/M2-17026-0/186 │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-17026-0/188 │ │ │ +using temporary file /tmp/M2-22501-0/186 │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22501-0/188 │ │ │ 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/M2-17026-0/188 │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-17026-0/190 │ │ │ +using temporary file /tmp/M2-22501-0/188 │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22501-0/190 │ │ │ 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/M2-17026-0/190 │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-17026-0/192 │ │ │ +using temporary file /tmp/M2-22501-0/190 │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22501-0/192 │ │ │ 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/M2-17026-0/192 │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-17026-0/194 │ │ │ +using temporary file /tmp/M2-22501-0/192 │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22501-0/194 │ │ │ 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/M2-17026-0/194 │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-17026-0/196 │ │ │ +using temporary file /tmp/M2-22501-0/194 │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22501-0/196 │ │ │ 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/M2-17026-0/196 │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-17026-0/198 │ │ │ +using temporary file /tmp/M2-22501-0/196 │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22501-0/198 │ │ │ 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/M2-17026-0/198 │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-17026-0/200 │ │ │ +using temporary file /tmp/M2-22501-0/198 │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22501-0/200 │ │ │ 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/M2-17026-0/200 │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-17026-0/202 │ │ │ +using temporary file /tmp/M2-22501-0/200 │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22501-0/202 │ │ │ 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/M2-17026-0/202 │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-17026-0/204 │ │ │ +using temporary file /tmp/M2-22501-0/202 │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22501-0/204 │ │ │ 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/M2-17026-0/204 │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-17026-0/206 │ │ │ +using temporary file /tmp/M2-22501-0/204 │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22501-0/206 │ │ │ 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/M2-17026-0/206 │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-17026-0/208 │ │ │ +using temporary file /tmp/M2-22501-0/206 │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22501-0/208 │ │ │ This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials. │ │ │ Options: │ │ │ --vectorinput: │ │ │ Read in a list of point configurations instead of a polynomial ring and a list of polynomials. │ │ │ --cyclic value: │ │ │ Use cyclic-n example instead of reading input. │ │ │ --noon value: │ │ │ @@ -249,44 +249,44 @@ │ │ │ Use Katsura-n example instead of reading input. │ │ │ --gaukwa value: │ │ │ Use Gaukwa-n example instead of reading input. │ │ │ --eco value: │ │ │ Use Eco-n example instead of reading input. │ │ │ -j value: │ │ │ Number of threads │ │ │ -using temporary file /tmp/M2-17026-0/208 │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-17026-0/210 │ │ │ +using temporary file /tmp/M2-22501-0/208 │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22501-0/210 │ │ │ This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets. │ │ │ Options: │ │ │ -s: │ │ │ Sort output by degree. │ │ │ │ │ │ -using temporary file /tmp/M2-17026-0/210 │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-17026-0/212 │ │ │ +using temporary file /tmp/M2-22501-0/210 │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22501-0/212 │ │ │ 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/M2-17026-0/212 │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-17026-0/214 │ │ │ +using temporary file /tmp/M2-22501-0/212 │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22501-0/214 │ │ │ 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/M2-17026-0/214 │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-17026-0/216 │ │ │ +using temporary file /tmp/M2-22501-0/214 │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22501-0/216 │ │ │ This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution. │ │ │ Options: │ │ │ --codimension: │ │ │ Compute only the codimension of the resultant fan and return. │ │ │ │ │ │ --symmetry: │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry. │ │ │ @@ -299,25 +299,25 @@ │ │ │ │ │ │ --vectorinput: │ │ │ Read in a list of point configurations instead of a polynomial ring and a list of polynomials. │ │ │ │ │ │ --projection: │ │ │ Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates. │ │ │ │ │ │ -using temporary file /tmp/M2-17026-0/216 │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-17026-0/218 │ │ │ +using temporary file /tmp/M2-22501-0/216 │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22501-0/218 │ │ │ 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/M2-17026-0/218 │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-17026-0/220 │ │ │ +using temporary file /tmp/M2-22501-0/218 │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22501-0/220 │ │ │ This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}. │ │ │ Options: │ │ │ --unimodular: │ │ │ Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan │ │ │ --scale value: │ │ │ Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future. │ │ │ --restrictingfan value: │ │ │ @@ -326,70 +326,70 @@ │ │ │ --symmetry: │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry. │ │ │ │ │ │ --nocones: │ │ │ Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used. │ │ │ --interrupt value: │ │ │ Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans). │ │ │ -using temporary file /tmp/M2-17026-0/220 │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-17026-0/222 │ │ │ +using temporary file /tmp/M2-22501-0/220 │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22501-0/222 │ │ │ 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/M2-17026-0/222 │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-17026-0/224 │ │ │ +using temporary file /tmp/M2-22501-0/222 │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22501-0/224 │ │ │ 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/M2-17026-0/224 │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-17026-0/226 │ │ │ +using temporary file /tmp/M2-22501-0/224 │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22501-0/226 │ │ │ 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/M2-17026-0/226 │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-17026-0/228 │ │ │ +using temporary file /tmp/M2-22501-0/226 │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22501-0/228 │ │ │ This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan. │ │ │ Options: │ │ │ --restrict: │ │ │ Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant. │ │ │ --symmetry: │ │ │ Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input. │ │ │ │ │ │ -using temporary file /tmp/M2-17026-0/228 │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-17026-0/230 │ │ │ +using temporary file /tmp/M2-22501-0/228 │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22501-0/230 │ │ │ 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/M2-17026-0/230 │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-17026-0/232 │ │ │ +using temporary file /tmp/M2-22501-0/230 │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22501-0/232 │ │ │ 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/M2-17026-0/232 │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-17026-0/234 │ │ │ +using temporary file /tmp/M2-22501-0/232 │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22501-0/234 │ │ │ This program evaluates a tropical polynomial function in a given set of points. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-17026-0/234 │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-17026-0/236 │ │ │ +using temporary file /tmp/M2-22501-0/234 │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22501-0/236 │ │ │ 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/M2-17026-0/236 │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-17026-0/238 │ │ │ +using temporary file /tmp/M2-22501-0/236 │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22501-0/238 │ │ │ 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/M2-17026-0/238 │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-17026-0/240 │ │ │ +using temporary file /tmp/M2-22501-0/238 │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22501-0/240 │ │ │ This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces. │ │ │ Options: │ │ │ --tropicalbasistest: │ │ │ This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.) │ │ │ │ │ │ --tplane: │ │ │ This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed. │ │ │ @@ -401,16 +401,16 @@ │ │ │ Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used. │ │ │ --restrict: │ │ │ Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms. │ │ │ --stable: │ │ │ Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored. │ │ │ --parameters value: │ │ │ With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters. │ │ │ -using temporary file /tmp/M2-17026-0/240 │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-17026-0/242 │ │ │ +using temporary file /tmp/M2-22501-0/240 │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22501-0/242 │ │ │ This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in: │ │ │ │ │ │ Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig: │ │ │ tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 │ │ │ │ │ │ See also │ │ │ │ │ │ @@ -435,48 +435,48 @@ │ │ │ Options: │ │ │ --noMult: │ │ │ Disable the multiplicity computation. │ │ │ -n value: │ │ │ Number of variables that should have negative weight. │ │ │ -c: │ │ │ Only output a list of vectors being the possible choices. │ │ │ -using temporary file /tmp/M2-17026-0/242 │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-17026-0/244 │ │ │ +using temporary file /tmp/M2-22501-0/242 │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22501-0/244 │ │ │ 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/M2-17026-0/244 │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-17026-0/246 │ │ │ +using temporary file /tmp/M2-22501-0/244 │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22501-0/246 │ │ │ This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-17026-0/246 │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-17026-0/248 │ │ │ +using temporary file /tmp/M2-22501-0/246 │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22501-0/248 │ │ │ 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/M2-17026-0/248 │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-17026-0/250 │ │ │ +using temporary file /tmp/M2-22501-0/248 │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22501-0/250 │ │ │ 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/M2-17026-0/250 │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-17026-0/252 │ │ │ +using temporary file /tmp/M2-22501-0/250 │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22501-0/252 │ │ │ This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety. │ │ │ Options: │ │ │ --symmetry: │ │ │ Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster. │ │ │ --symsigns: │ │ │ Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix. │ │ │ --nocones: │ │ │ @@ -484,24 +484,24 @@ │ │ │ --disableSymmetryTest: │ │ │ When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal. │ │ │ │ │ │ --stable: │ │ │ Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric. │ │ │ --interrupt value: │ │ │ Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans). │ │ │ -using temporary file /tmp/M2-17026-0/252 │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-17026-0/254 │ │ │ +using temporary file /tmp/M2-22501-0/252 │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22501-0/254 │ │ │ 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/M2-17026-0/254 │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-17026-0/256 │ │ │ +using temporary file /tmp/M2-22501-0/254 │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22501-0/256 │ │ │ This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n]. │ │ │ Several operations are supported by specifying the appropriate option: │ │ │ (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically), │ │ │ (2) computation of an initial ideal, │ │ │ (3) computation of the Groebner fan, │ │ │ (4) computation of a single Groebner cone. │ │ │ Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive. │ │ │ @@ -521,21 +521,21 @@ │ │ │ --groebnerCone: │ │ │ Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector. │ │ │ -m: │ │ │ For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list. │ │ │ -g: │ │ │ Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if the usual --groebnerFan is too slow. │ │ │ │ │ │ -using temporary file /tmp/M2-17026-0/256 │ │ │ +using temporary file /tmp/M2-22501-0/256 │ │ │ │ │ │ i6 : QQ[x,y]; │ │ │ │ │ │ i7 : gfan {x,y}; │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-17026-0/258 │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22501-0/258 │ │ │ Q[x1,x2] │ │ │ {{ │ │ │ x2, │ │ │ x1} │ │ │ } │ │ │ -using temporary file /tmp/M2-17026-0/258 │ │ │ +using temporary file /tmp/M2-22501-0/258 │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html │ │ │ @@ -109,15 +109,15 @@ │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │   -- warning: reloading gfanInterface; recreate instances of types from this package
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-17026-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22501-0/172
│ │ │  This is a program for computing all reduced Groebner bases of a polynomial ideal. It takes the ring and a generating set for the ideal as input. By default the enumeration is done by an almost memoryless reverse search. If the ideal is symmetric the symmetry option is useful and enumeration will be done up to symmetry using a breadth first search. The program needs a starting Groebner basis to do its computations. If the -g option is not specified it will compute one using Buchberger's algorithm.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if it takes too much time to compute the starting (standard degree lexicographic) Groebner basis and the input is already a Groebner basis.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │ @@ -128,16 +128,16 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-17026-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-17026-0/174
│ │ │ +using temporary file /tmp/M2-22501-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22501-0/174
│ │ │  This program computes a reduced lexicographic Groebner basis of the polynomial ideal given as input. The default behavior is to use Buchberger's algorithm. The ordering of the variables is $a>b>c...$ (assuming that the ring is Q[a,b,c,...]).
│ │ │  Options:
│ │ │  -w:
│ │ │   Compute a Groebner basis with respect to a degree lexicographic order with $a>b>c...$ instead. The degrees are given by a weight vector which is read from the input after the generating set has been read.
│ │ │  
│ │ │  -r:
│ │ │   Use the reverse lexicographic order (or the reverse lexicographic order as a tie breaker if -w is used). The input must be homogeneous if the pure reverse lexicographic order is chosen. Ignored if -W is used.
│ │ │ @@ -146,69 +146,69 @@
│ │ │   Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored.
│ │ │  
│ │ │  -g:
│ │ │   Do a generic Groebner walk. The input must be homogeneous and must be a minimal Groebner basis with respect to the reverse lexicographic term order. The target term order is always lexicographic. The -W option must be used.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-17026-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-17026-0/176
│ │ │ +using temporary file /tmp/M2-22501-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22501-0/176
│ │ │  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/M2-17026-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-17026-0/178
│ │ │ +using temporary file /tmp/M2-22501-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22501-0/178
│ │ │  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/M2-17026-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-17026-0/180
│ │ │ +using temporary file /tmp/M2-22501-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22501-0/180
│ │ │  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/M2-17026-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-17026-0/182
│ │ │ +using temporary file /tmp/M2-22501-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22501-0/182
│ │ │  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/M2-17026-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-17026-0/184
│ │ │ +using temporary file /tmp/M2-22501-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22501-0/184
│ │ │  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/M2-17026-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-17026-0/186
│ │ │ +using temporary file /tmp/M2-22501-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22501-0/186
│ │ │  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/M2-17026-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-17026-0/188
│ │ │ +using temporary file /tmp/M2-22501-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22501-0/188
│ │ │  This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[x,y]{y-1}
│ │ │  z
│ │ │  Output:
│ │ │  Q[x,y,z]{y-z}
│ │ │ @@ -216,30 +216,30 @@
│ │ │  -i:
│ │ │   Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it.
│ │ │  -w:
│ │ │   Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials.
│ │ │  
│ │ │  -H:
│ │ │   Let the name of the new variable be H rather than reading in a name from the input.
│ │ │ -using temporary file /tmp/M2-17026-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-17026-0/190
│ │ │ +using temporary file /tmp/M2-22501-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22501-0/190
│ │ │  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/M2-17026-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-17026-0/192
│ │ │ +using temporary file /tmp/M2-22501-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22501-0/192
│ │ │  This is a program for doing interactive walks in the Groebner fan of an ideal. The input is a Groebner basis defining the starting Groebner cone of the walk. The program will list all flippable facets of the Groebner cone and ask the user to choose one. The user types in the index (number) of the facet in the list. The program will walk through the selected facet and display the new Groebner basis and a list of new facet normals for the user to choose from. Since the program reads the user's choices through the the standard input it is recommended not to redirect the standard input for this program.
│ │ │  Options:
│ │ │  -L:
│ │ │   Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi.
│ │ │  
│ │ │  -x:
│ │ │   Exit immediately.
│ │ │ @@ -254,57 +254,57 @@
│ │ │   Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis.
│ │ │  
│ │ │  -W:
│ │ │   Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone.
│ │ │  
│ │ │  --tropical:
│ │ │   Traverse a tropical variety interactively.
│ │ │ -using temporary file /tmp/M2-17026-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-17026-0/194
│ │ │ +using temporary file /tmp/M2-22501-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22501-0/194
│ │ │  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/M2-17026-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-17026-0/196
│ │ │ +using temporary file /tmp/M2-22501-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22501-0/196
│ │ │  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/M2-17026-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-17026-0/198
│ │ │ +using temporary file /tmp/M2-22501-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22501-0/198
│ │ │  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/M2-17026-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-17026-0/200
│ │ │ +using temporary file /tmp/M2-22501-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22501-0/200
│ │ │  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/M2-17026-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-17026-0/202
│ │ │ +using temporary file /tmp/M2-22501-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22501-0/202
│ │ │  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/M2-17026-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-17026-0/204
│ │ │ +using temporary file /tmp/M2-22501-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22501-0/204
│ │ │  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/M2-17026-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-17026-0/206
│ │ │ +using temporary file /tmp/M2-22501-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22501-0/206
│ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │  Options:
│ │ │  -r value:
│ │ │   Specify r.
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │ @@ -319,16 +319,16 @@
│ │ │   Do nothing but produce symmetry generators for the Pluecker ideal.
│ │ │  --symmetry:
│ │ │   Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names).
│ │ │  --parametrize:
│ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials.
│ │ │  --ultrametric:
│ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ -using temporary file /tmp/M2-17026-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-17026-0/208
│ │ │ +using temporary file /tmp/M2-22501-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22501-0/208
│ │ │  This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials.
│ │ │  Options:
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  --cyclic value:
│ │ │   Use cyclic-n example instead of reading input.
│ │ │  --noon value:
│ │ │ @@ -339,44 +339,44 @@
│ │ │   Use Katsura-n example instead of reading input.
│ │ │  --gaukwa value:
│ │ │   Use Gaukwa-n example instead of reading input.
│ │ │  --eco value:
│ │ │   Use Eco-n example instead of reading input.
│ │ │  -j value:
│ │ │   Number of threads
│ │ │ -using temporary file /tmp/M2-17026-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-17026-0/210
│ │ │ +using temporary file /tmp/M2-22501-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22501-0/210
│ │ │  This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets.
│ │ │  Options:
│ │ │  -s:
│ │ │   Sort output by degree.
│ │ │  
│ │ │ -using temporary file /tmp/M2-17026-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-17026-0/212
│ │ │ +using temporary file /tmp/M2-22501-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22501-0/212
│ │ │  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/M2-17026-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-17026-0/214
│ │ │ +using temporary file /tmp/M2-22501-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22501-0/214
│ │ │  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/M2-17026-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-17026-0/216
│ │ │ +using temporary file /tmp/M2-22501-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22501-0/216
│ │ │  This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution.
│ │ │  Options:
│ │ │  --codimension:
│ │ │   Compute only the codimension of the resultant fan and return.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │ @@ -389,25 +389,25 @@
│ │ │  
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  
│ │ │  --projection:
│ │ │   Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │  
│ │ │ -using temporary file /tmp/M2-17026-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-17026-0/218
│ │ │ +using temporary file /tmp/M2-22501-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22501-0/218
│ │ │  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/M2-17026-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-17026-0/220
│ │ │ +using temporary file /tmp/M2-22501-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22501-0/220
│ │ │  This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │  Options:
│ │ │  --unimodular:
│ │ │   Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan
│ │ │  --scale value:
│ │ │   Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future.
│ │ │  --restrictingfan value:
│ │ │ @@ -416,70 +416,70 @@
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │  
│ │ │  --nocones:
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-17026-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-17026-0/222
│ │ │ +using temporary file /tmp/M2-22501-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22501-0/222
│ │ │  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/M2-17026-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-17026-0/224
│ │ │ +using temporary file /tmp/M2-22501-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22501-0/224
│ │ │  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/M2-17026-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-17026-0/226
│ │ │ +using temporary file /tmp/M2-22501-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22501-0/226
│ │ │  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/M2-17026-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-17026-0/228
│ │ │ +using temporary file /tmp/M2-22501-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22501-0/228
│ │ │  This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant.
│ │ │  --symmetry:
│ │ │   Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input.
│ │ │  
│ │ │ -using temporary file /tmp/M2-17026-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-17026-0/230
│ │ │ +using temporary file /tmp/M2-22501-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22501-0/230
│ │ │  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/M2-17026-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-17026-0/232
│ │ │ +using temporary file /tmp/M2-22501-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22501-0/232
│ │ │  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/M2-17026-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-17026-0/234
│ │ │ +using temporary file /tmp/M2-22501-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22501-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-17026-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-17026-0/236
│ │ │ +using temporary file /tmp/M2-22501-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22501-0/236
│ │ │  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/M2-17026-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-17026-0/238
│ │ │ +using temporary file /tmp/M2-22501-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22501-0/238
│ │ │  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/M2-17026-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-17026-0/240
│ │ │ +using temporary file /tmp/M2-22501-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22501-0/240
│ │ │  This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces.
│ │ │  Options:
│ │ │  --tropicalbasistest:
│ │ │   This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.)
│ │ │  
│ │ │  --tplane:
│ │ │   This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed.
│ │ │ @@ -491,16 +491,16 @@
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --restrict:
│ │ │   Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms.
│ │ │  --stable:
│ │ │   Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored.
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-17026-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-17026-0/242
│ │ │ +using temporary file /tmp/M2-22501-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22501-0/242
│ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in:
│ │ │  
│ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 
│ │ │  
│ │ │  See also
│ │ │  
│ │ │ @@ -525,48 +525,48 @@
│ │ │  Options:
│ │ │  --noMult:
│ │ │   Disable the multiplicity computation.
│ │ │  -n value:
│ │ │   Number of variables that should have negative weight.
│ │ │  -c:
│ │ │   Only output a list of vectors being the possible choices.
│ │ │ -using temporary file /tmp/M2-17026-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-17026-0/244
│ │ │ +using temporary file /tmp/M2-22501-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22501-0/244
│ │ │  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/M2-17026-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-17026-0/246
│ │ │ +using temporary file /tmp/M2-22501-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22501-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-17026-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-17026-0/248
│ │ │ +using temporary file /tmp/M2-22501-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22501-0/248
│ │ │  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/M2-17026-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-17026-0/250
│ │ │ +using temporary file /tmp/M2-22501-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22501-0/250
│ │ │  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/M2-17026-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-17026-0/252
│ │ │ +using temporary file /tmp/M2-22501-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22501-0/252
│ │ │  This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster.
│ │ │  --symsigns:
│ │ │   Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix.
│ │ │  --nocones:
│ │ │ @@ -574,24 +574,24 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --stable:
│ │ │   Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-17026-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-17026-0/254
│ │ │ +using temporary file /tmp/M2-22501-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22501-0/254
│ │ │  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/M2-17026-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-17026-0/256
│ │ │ +using temporary file /tmp/M2-22501-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22501-0/256
│ │ │  This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n].
│ │ │  Several operations are supported by specifying the appropriate option:
│ │ │   (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically),
│ │ │   (2) computation of an initial ideal,
│ │ │   (3) computation of the Groebner fan,
│ │ │   (4) computation of a single Groebner cone.
│ │ │  Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive.
│ │ │ @@ -611,32 +611,32 @@
│ │ │  --groebnerCone:
│ │ │   Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector.
│ │ │  -m:
│ │ │   For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list.
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if the usual --groebnerFan is too slow.
│ │ │  
│ │ │ -using temporary file /tmp/M2-17026-0/256
    │ │ │ +using temporary file /tmp/M2-22501-0/256 │ │ │
    │ │ │ 
    i6 : QQ[x,y];
    │ │ │
    │ │ │ 
    i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-17026-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22501-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-17026-0/258
    │ │ │ +using temporary file /tmp/M2-22501-0/258 │ │ │
    │ │ │
    │ │ │

    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/M2-17026-0/172 │ │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-22501-0/172 │ │ │ │ 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/M2-17026-0/172 │ │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-17026-0/174 │ │ │ │ +using temporary file /tmp/M2-22501-0/172 │ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-22501-0/174 │ │ │ │ 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/M2-17026-0/174 │ │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-17026-0/176 │ │ │ │ +using temporary file /tmp/M2-22501-0/174 │ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-22501-0/176 │ │ │ │ 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/M2-17026-0/176 │ │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-17026-0/178 │ │ │ │ +using temporary file /tmp/M2-22501-0/176 │ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-22501-0/178 │ │ │ │ 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/M2-17026-0/178 │ │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-17026-0/180 │ │ │ │ +using temporary file /tmp/M2-22501-0/178 │ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-22501-0/180 │ │ │ │ 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/M2-17026-0/180 │ │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-17026-0/182 │ │ │ │ +using temporary file /tmp/M2-22501-0/180 │ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-22501-0/182 │ │ │ │ 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/M2-17026-0/182 │ │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-17026-0/184 │ │ │ │ +using temporary file /tmp/M2-22501-0/182 │ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-22501-0/184 │ │ │ │ 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/M2-17026-0/184 │ │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-17026-0/186 │ │ │ │ +using temporary file /tmp/M2-22501-0/184 │ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-22501-0/186 │ │ │ │ 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/M2-17026-0/186 │ │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-17026-0/188 │ │ │ │ +using temporary file /tmp/M2-22501-0/186 │ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-22501-0/188 │ │ │ │ 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/M2-17026-0/188 │ │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-17026-0/190 │ │ │ │ +using temporary file /tmp/M2-22501-0/188 │ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-22501-0/190 │ │ │ │ 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/M2-17026-0/190 │ │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-17026-0/192 │ │ │ │ +using temporary file /tmp/M2-22501-0/190 │ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-22501-0/192 │ │ │ │ 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/M2-17026-0/192 │ │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-17026-0/194 │ │ │ │ +using temporary file /tmp/M2-22501-0/192 │ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-22501-0/194 │ │ │ │ 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/M2-17026-0/194 │ │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-17026-0/196 │ │ │ │ +using temporary file /tmp/M2-22501-0/194 │ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-22501-0/196 │ │ │ │ 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/M2-17026-0/196 │ │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-17026-0/198 │ │ │ │ +using temporary file /tmp/M2-22501-0/196 │ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-22501-0/198 │ │ │ │ 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/M2-17026-0/198 │ │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-17026-0/200 │ │ │ │ +using temporary file /tmp/M2-22501-0/198 │ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-22501-0/200 │ │ │ │ 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/M2-17026-0/200 │ │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-17026-0/202 │ │ │ │ +using temporary file /tmp/M2-22501-0/200 │ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-22501-0/202 │ │ │ │ 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/M2-17026-0/202 │ │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-17026-0/204 │ │ │ │ +using temporary file /tmp/M2-22501-0/202 │ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-22501-0/204 │ │ │ │ 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/M2-17026-0/204 │ │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-17026-0/206 │ │ │ │ +using temporary file /tmp/M2-22501-0/204 │ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-22501-0/206 │ │ │ │ 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/M2-17026-0/206 │ │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-17026-0/208 │ │ │ │ +using temporary file /tmp/M2-22501-0/206 │ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-22501-0/208 │ │ │ │ This program computes the mixed volume of the Newton polytopes of a list of │ │ │ │ polynomials. The ring is specified on the input. After this follows the list of │ │ │ │ polynomials. │ │ │ │ Options: │ │ │ │ --vectorinput: │ │ │ │ Read in a list of point configurations instead of a polynomial ring and a list │ │ │ │ of polynomials. │ │ │ │ @@ -388,25 +388,25 @@ │ │ │ │ Use Katsura-n example instead of reading input. │ │ │ │ --gaukwa value: │ │ │ │ Use Gaukwa-n example instead of reading input. │ │ │ │ --eco value: │ │ │ │ Use Eco-n example instead of reading input. │ │ │ │ -j value: │ │ │ │ Number of threads │ │ │ │ -using temporary file /tmp/M2-17026-0/208 │ │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-17026-0/210 │ │ │ │ +using temporary file /tmp/M2-22501-0/208 │ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-22501-0/210 │ │ │ │ This program computes the union of a list of polynomial sets given as input. │ │ │ │ The polynomials must all belong to the same ring. The ring is specified on the │ │ │ │ input. After this follows the list of polynomial sets. │ │ │ │ Options: │ │ │ │ -s: │ │ │ │ Sort output by degree. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-17026-0/210 │ │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-17026-0/212 │ │ │ │ +using temporary file /tmp/M2-22501-0/210 │ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-22501-0/212 │ │ │ │ This program renders a Groebner fan as an xfig file. To be more precise, the │ │ │ │ input is the list of all reduced Groebner bases of an ideal. The output is a │ │ │ │ drawing of the Groebner fan intersected with a triangle. The corners of the │ │ │ │ triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. │ │ │ │ If there are more than three variables in the ring these coordinates are │ │ │ │ extended with zeros. It is possible to shift the 1 entry cyclic with the option │ │ │ │ --shiftVariables. │ │ │ │ @@ -414,16 +414,16 @@ │ │ │ │ -L: │ │ │ │ Make the triangle larger so that the shape of the Groebner region appears. │ │ │ │ --shiftVariables value: │ │ │ │ Shift the positions of the variables in the drawing. For example with the │ │ │ │ value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) │ │ │ │ and top: (0,0,0,1,...). The shifting is done modulo the number of variables in │ │ │ │ the polynomial ring. The default value is 0. │ │ │ │ -using temporary file /tmp/M2-17026-0/212 │ │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-17026-0/214 │ │ │ │ +using temporary file /tmp/M2-22501-0/212 │ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-22501-0/214 │ │ │ │ This program renders a staircase diagram of a monomial initial ideal to an xfig │ │ │ │ file. The input is a Groebner basis of a (not necessarily monomial) polynomial │ │ │ │ ideal. The initial ideal is given by the leading terms in the Groebner basis. │ │ │ │ Using the -m option it is possible to render more than one staircase diagram. │ │ │ │ The program only works for ideals in a polynomial ring with three variables. │ │ │ │ Options: │ │ │ │ -m: │ │ │ │ @@ -436,16 +436,16 @@ │ │ │ │ number is large enough to give a correct picture of the standard monomials. The │ │ │ │ default value is 8. │ │ │ │ │ │ │ │ -w value: │ │ │ │ Width. Specifies the number of staircase diagrams per row in the xfig file. │ │ │ │ The default value is 5. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-17026-0/214 │ │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-17026-0/216 │ │ │ │ +using temporary file /tmp/M2-22501-0/214 │ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-22501-0/216 │ │ │ │ This program computes the resultant fan as defined in "Computing Tropical │ │ │ │ Resultants" by Jensen and Yu. The input is a polynomial ring followed by │ │ │ │ polynomials, whose coefficients are ignored. The output is the fan of │ │ │ │ coefficients such that the input system has a tropical solution. │ │ │ │ Options: │ │ │ │ --codimension: │ │ │ │ Compute only the codimension of the resultant fan and return. │ │ │ │ @@ -473,28 +473,28 @@ │ │ │ │ of polynomials. │ │ │ │ │ │ │ │ --projection: │ │ │ │ Use the projection method to compute the resultant fan. This works only if the │ │ │ │ resultant fan is a hypersurface. If this option is combined with --special, │ │ │ │ then the output fan lives in the subspace of the non-specialized coordinates. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-17026-0/216 │ │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-17026-0/218 │ │ │ │ +using temporary file /tmp/M2-22501-0/216 │ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-22501-0/218 │ │ │ │ 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/M2-17026-0/218 │ │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-17026-0/220 │ │ │ │ +using temporary file /tmp/M2-22501-0/218 │ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-22501-0/220 │ │ │ │ This program computes the secondary fan of a vector configuration. The │ │ │ │ configuration is given as an ordered list of vectors. In order to compute the │ │ │ │ secondary fan of a point configuration an additional coordinate of ones must be │ │ │ │ added. For example {(1,0),(1,1),(1,2),(1,3)}. │ │ │ │ Options: │ │ │ │ --unimodular: │ │ │ │ Use heuristics to search for unimodular triangulation rather than computing │ │ │ │ @@ -523,103 +523,103 @@ │ │ │ │ Tells the program not to output the CONES and MAXIMAL_CONES sections, but │ │ │ │ still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is │ │ │ │ used. │ │ │ │ --interrupt value: │ │ │ │ Interrupt the enumeration after a specified number of facets have been │ │ │ │ computed (works for usual symmetric traversals, but may not work in general for │ │ │ │ non-symmetric traversals or for traversals restricted to fans). │ │ │ │ -using temporary file /tmp/M2-17026-0/220 │ │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-17026-0/222 │ │ │ │ +using temporary file /tmp/M2-22501-0/220 │ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-22501-0/222 │ │ │ │ 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/M2-17026-0/222 │ │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-17026-0/224 │ │ │ │ +using temporary file /tmp/M2-22501-0/222 │ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-22501-0/224 │ │ │ │ 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/M2-17026-0/224 │ │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-17026-0/226 │ │ │ │ +using temporary file /tmp/M2-22501-0/224 │ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-22501-0/226 │ │ │ │ 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/M2-17026-0/226 │ │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-17026-0/228 │ │ │ │ +using temporary file /tmp/M2-22501-0/226 │ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-22501-0/228 │ │ │ │ This program takes a list of reduced Groebner bases and produces the fan of all │ │ │ │ faces of these. In this way by giving the complete list of reduced Groebner │ │ │ │ bases, the Groebner fan can be computed as a polyhedral complex. The option -- │ │ │ │ restrict lets the user choose between computing the Groebner fan or the │ │ │ │ restricted Groebner fan. │ │ │ │ Options: │ │ │ │ --restrict: │ │ │ │ Add an inequality for each coordinate, so that the the cones are restricted to │ │ │ │ the non-negative orthant. │ │ │ │ --symmetry: │ │ │ │ Tell the program to read in generators for a group of symmetries (subgroup of │ │ │ │ $S_n$) after having read in the ring. The output is grouped according to these │ │ │ │ symmetries. Only one representative for each orbit is needed on the input. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-17026-0/228 │ │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-17026-0/230 │ │ │ │ +using temporary file /tmp/M2-22501-0/228 │ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-22501-0/230 │ │ │ │ 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/M2-17026-0/230 │ │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-17026-0/232 │ │ │ │ +using temporary file /tmp/M2-22501-0/230 │ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-22501-0/232 │ │ │ │ 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/M2-17026-0/232 │ │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-17026-0/234 │ │ │ │ +using temporary file /tmp/M2-22501-0/232 │ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-22501-0/234 │ │ │ │ This program evaluates a tropical polynomial function in a given set of points. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-17026-0/234 │ │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-17026-0/236 │ │ │ │ +using temporary file /tmp/M2-22501-0/234 │ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-22501-0/236 │ │ │ │ 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/M2-17026-0/236 │ │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-17026-0/238 │ │ │ │ +using temporary file /tmp/M2-22501-0/236 │ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-22501-0/238 │ │ │ │ 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/M2-17026-0/238 │ │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-17026-0/240 │ │ │ │ +using temporary file /tmp/M2-22501-0/238 │ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-22501-0/240 │ │ │ │ This program computes the set theoretical intersection of a set of tropical │ │ │ │ hypersurfaces (or to be precise, their common refinement as a fan). The input │ │ │ │ is a list of polynomials with each polynomial defining a hypersurface. │ │ │ │ Considering tropical hypersurfaces as fans, the intersection can be computed as │ │ │ │ the common refinement of these. Thus the output is a fan whose support is the │ │ │ │ intersection of the tropical hypersurfaces. │ │ │ │ Options: │ │ │ │ @@ -656,16 +656,16 @@ │ │ │ │ --stable: │ │ │ │ Find the stable intersection of the input polynomials using tropical │ │ │ │ intersection theory. This can be slow. Most other options are ignored. │ │ │ │ --parameters value: │ │ │ │ With this option you can specify how many variables to treat as parameters │ │ │ │ instead of variables. This makes it possible to do computations where the │ │ │ │ coefficient field is the field of rational functions in the parameters. │ │ │ │ -using temporary file /tmp/M2-17026-0/240 │ │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-17026-0/242 │ │ │ │ +using temporary file /tmp/M2-22501-0/240 │ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-22501-0/242 │ │ │ │ This program is part of the Puiseux lifting algorithm implemented in Gfan and │ │ │ │ Singular. The Singular part of the implementation can be found in: │ │ │ │ │ │ │ │ Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig: │ │ │ │ tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, │ │ │ │ 2007 │ │ │ │ │ │ │ │ @@ -693,54 +693,54 @@ │ │ │ │ Options: │ │ │ │ --noMult: │ │ │ │ Disable the multiplicity computation. │ │ │ │ -n value: │ │ │ │ Number of variables that should have negative weight. │ │ │ │ -c: │ │ │ │ Only output a list of vectors being the possible choices. │ │ │ │ -using temporary file /tmp/M2-17026-0/242 │ │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-17026-0/244 │ │ │ │ +using temporary file /tmp/M2-22501-0/242 │ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-22501-0/244 │ │ │ │ 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/M2-17026-0/244 │ │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-17026-0/246 │ │ │ │ +using temporary file /tmp/M2-22501-0/244 │ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-22501-0/246 │ │ │ │ This program computes the multiplicity of a tropical cone given a marked │ │ │ │ reduced Groebner basis for its initial ideal. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-17026-0/246 │ │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-17026-0/248 │ │ │ │ +using temporary file /tmp/M2-22501-0/246 │ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-22501-0/248 │ │ │ │ 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/M2-17026-0/248 │ │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-17026-0/250 │ │ │ │ +using temporary file /tmp/M2-22501-0/248 │ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-22501-0/250 │ │ │ │ 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/M2-17026-0/250 │ │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-17026-0/252 │ │ │ │ +using temporary file /tmp/M2-22501-0/250 │ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-22501-0/252 │ │ │ │ This program computes a polyhedral fan representation of the tropical variety │ │ │ │ of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let │ │ │ │ $\omega$ be a relative interior point of $d$-dimensional Groebner cone │ │ │ │ contained in the tropical variety. The input for this program is a pair of │ │ │ │ marked reduced Groebner bases with respect to the term order represented by │ │ │ │ $\omega$, tie-broken in some way. The first one is for the initial ideal │ │ │ │ $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point │ │ │ │ @@ -770,27 +770,27 @@ │ │ │ │ --stable: │ │ │ │ Traverse the stable intersection or, equivalently, pretend that the │ │ │ │ coefficients are genereric. │ │ │ │ --interrupt value: │ │ │ │ Interrupt the enumeration after a specified number of facets have been │ │ │ │ computed (works for usual symmetric traversals, but may not work in general for │ │ │ │ non-symmetric traversals or for traversals restricted to fans). │ │ │ │ -using temporary file /tmp/M2-17026-0/252 │ │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-17026-0/254 │ │ │ │ +using temporary file /tmp/M2-22501-0/252 │ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-22501-0/254 │ │ │ │ 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/M2-17026-0/254 │ │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-17026-0/256 │ │ │ │ +using temporary file /tmp/M2-22501-0/254 │ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-22501-0/256 │ │ │ │ This program is an experimental implementation of Groebner bases for ideals in │ │ │ │ Z[x_1,...,x_n]. │ │ │ │ Several operations are supported by specifying the appropriate option: │ │ │ │ (1) computation of the reduced Groebner basis with respect to a given vector │ │ │ │ (tiebroken lexicographically), │ │ │ │ (2) computation of an initial ideal, │ │ │ │ (3) computation of the Groebner fan, │ │ │ │ @@ -825,23 +825,23 @@ │ │ │ │ For the operations taking a vector as input, read in a list of vectors │ │ │ │ instead, and perform the operation for each vector in the list. │ │ │ │ -g: │ │ │ │ Tells the program that the input is already a Groebner basis (with the initial │ │ │ │ term of each polynomial being the first ones listed). Use this option if the │ │ │ │ usual --groebnerFan is too slow. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-17026-0/256 │ │ │ │ +using temporary file /tmp/M2-22501-0/256 │ │ │ │ i6 : QQ[x,y]; │ │ │ │ i7 : gfan {x,y}; │ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-17026-0/258 │ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-22501-0/258 │ │ │ │ Q[x1,x2] │ │ │ │ {{ │ │ │ │ x2, │ │ │ │ x1} │ │ │ │ } │ │ │ │ -using temporary file /tmp/M2-17026-0/258 │ │ │ │ +using temporary file /tmp/M2-22501-0/258 │ │ │ │ Finally, if you want to be able to render Groebner fans and monomial staircases │ │ │ │ to .png files, you should install fig2dev. If it is installed in a non-standard │ │ │ │ location, then you may specify its path using _p_r_o_g_r_a_m_P_a_t_h_s. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.25.11+ds/M2/Macaulay2/packages/gfanInterface.m2:2630:0. │ │ ├── ./usr/share/info/AInfinity.info.gz │ │ │ ├── AInfinity.info │ │ │ │ @@ -6133,16 +6133,16 @@ │ │ │ │ 00017f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017f70: 2b0a 7c69 3320 3a20 656c 6170 7365 6454 +.|i3 : elapsedT │ │ │ │ 00017f80: 696d 6520 6275 726b 6552 6573 6f6c 7574 ime burkeResolut │ │ │ │ 00017f90: 696f 6e28 4d2c 2037 2c20 4368 6563 6b20 ion(M, 7, Check │ │ │ │ 00017fa0: 3d3e 2066 616c 7365 2920 2020 2020 2020 => false) │ │ │ │ -00017fb0: 2020 2020 7c0a 7c20 2d2d 2031 2e38 3730 |.| -- 1.870 │ │ │ │ -00017fc0: 3536 7320 656c 6170 7365 6420 2020 2020 56s elapsed │ │ │ │ +00017fb0: 2020 2020 7c0a 7c20 2d2d 2031 2e34 3438 |.| -- 1.448 │ │ │ │ +00017fc0: 3273 2065 6c61 7073 6564 2020 2020 2020 2s elapsed │ │ │ │ 00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017ff0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00018000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018030: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ @@ -6176,15 +6176,15 @@ │ │ │ │ 000181f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018210: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ 00018220: 656c 6170 7365 6454 696d 6520 6275 726b elapsedTime burk │ │ │ │ 00018230: 6552 6573 6f6c 7574 696f 6e28 4d2c 2037 eResolution(M, 7 │ │ │ │ 00018240: 2c20 4368 6563 6b20 3d3e 2074 7275 6529 , Check => true) │ │ │ │ 00018250: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00018260: 2d2d 2031 2e39 3837 3835 7320 656c 6170 -- 1.98785s elap │ │ │ │ +00018260: 2d2d 2031 2e38 3437 3438 7320 656c 6170 -- 1.84748s elap │ │ │ │ 00018270: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 00018280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000182a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000182b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000182c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000182d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./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 3738 3738 |.| -- .07878 │ │ │ │ -00002ec0: 3236 7320 656c 6170 7365 6420 2020 2020 26s elapsed │ │ │ │ +00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3330 3238 |.| -- .03028 │ │ │ │ +00002ec0: 3773 2065 6c61 7073 6564 2020 2020 2020 7s 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 3431 3732 3373 2065 6c61 7073 6564 .641723s elapsed │ │ │ │ +00006420: 2e36 3035 3432 7320 656c 6170 7365 6420 .60542s elapsed │ │ │ │ 00006430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006450: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00006460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006480: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00006490: 0a7c 2020 2020 2020 2020 2020 2020 2030 .| 0 │ │ │ │ @@ -1651,15 +1651,15 @@ │ │ │ │ 00006720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006750: 2d2d 2d2b 0a7c 6931 3720 3a20 656c 6170 ---+.|i17 : elap │ │ │ │ 00006760: 7365 6454 696d 6520 6261 7365 5074 733d sedTime basePts= │ │ │ │ 00006770: 7072 696d 6172 7944 6563 6f6d 706f 7369 primaryDecomposi │ │ │ │ 00006780: 7469 6f6e 2069 6465 616c 2048 3b20 7c0a tion ideal H; |. │ │ │ │ -00006790: 7c20 2d2d 2035 2e39 3632 3536 7320 656c | -- 5.96256s el │ │ │ │ +00006790: 7c20 2d2d 2035 2e33 3839 3636 7320 656c | -- 5.38966s el │ │ │ │ 000067a0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 000067b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000067c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 000067d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000067e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000067f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006800: 2d2d 2d2d 2b0a 7c69 3138 203a 2074 616c ----+.|i18 : tal │ │ │ │ @@ -2608,15 +2608,15 @@ │ │ │ │ 0000a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0000a320: 3134 203a 2065 6c61 7073 6564 5469 6d65 14 : elapsedTime │ │ │ │ 0000a330: 2073 7562 2849 2c48 2920 2020 2020 2020 sub(I,H) │ │ │ │ 0000a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a350: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -0000a360: 3132 3831 3638 7320 656c 6170 7365 6420 128168s elapsed │ │ │ │ +0000a360: 3135 3335 3336 7320 656c 6170 7365 6420 153536s 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 3436 3937 3473 |.| -- .0546974s │ │ │ │ +0000a5e0: 7c0a 7c20 2d2d 202e 3036 3633 3630 3273 |.| -- .0663602s │ │ │ │ 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 2e36 3839 3431 7320 656c 6170 7365 1.68941s elapse │ │ │ │ +0000a920: 2031 2e36 3134 3138 7320 656c 6170 7365 1.61418s 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 3137 3831 7320 2863 7075 293b 0.451781s (cpu); │ │ │ │ -00010f90: 2030 2e33 3733 3035 3173 2028 7468 7265 0.373051s (thre │ │ │ │ +00010f80: 302e 3532 3638 3534 7320 2863 7075 293b 0.526854s (cpu); │ │ │ │ +00010f90: 2030 2e34 3432 3536 3373 2028 7468 7265 0.442563s (thre │ │ │ │ 00010fa0: 6164 293b 2030 7320 2867 6329 7c0a 7c20 ad); 0s (gc)|.| │ │ │ │ 00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00010ff0: 2020 2020 3020 3120 3220 2020 2020 2020 0 1 2 │ │ │ │ 00011000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4403,16 +4403,16 @@ │ │ │ │ 00011320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00011350: 0a7c 6931 3520 3a20 7469 6d65 2062 6574 .|i15 : time bet │ │ │ │ 00011360: 7469 2028 4620 3d20 7075 7265 5265 736f ti (F = pureReso │ │ │ │ 00011370: 6c75 7469 6f6e 2831 312c 342c 7b30 2c32 lution(11,4,{0,2 │ │ │ │ 00011380: 2c34 7d29 2920 207c 0a7c 202d 2d20 7573 ,4})) |.| -- us │ │ │ │ -00011390: 6564 2030 2e34 3935 3634 3373 2028 6370 ed 0.495643s (cp │ │ │ │ -000113a0: 7529 3b20 302e 3431 3837 3737 7320 2874 u); 0.418777s (t │ │ │ │ +00011390: 6564 2030 2e36 3231 3236 3673 2028 6370 ed 0.621266s (cp │ │ │ │ +000113a0: 7529 3b20 302e 3533 3134 3135 7320 2874 u); 0.531415s (t │ │ │ │ 000113b0: 6872 6561 6429 3b20 3073 2028 6763 297c hread); 0s (gc)| │ │ │ │ 000113c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000113d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00011400: 2020 2020 2020 2030 2031 2032 2020 2020 0 1 2 │ │ │ │ 00011410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/Benchmark.info.gz │ │ │ ├── Benchmark.info │ │ │ │ @@ -200,71 +200,76 @@ │ │ │ │ 00000c70: 2d2d 2d2d 2b0a 7c69 3120 3a20 7275 6e42 ----+.|i1 : runB │ │ │ │ 00000c80: 656e 6368 6d61 726b 7320 2272 6573 3339 enchmarks "res39 │ │ │ │ 00000c90: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ 00000ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000cc0: 2020 2020 7c0a 7c2d 2d20 6265 6769 6e6e |.|-- beginn │ │ │ │ 00000cd0: 696e 6720 636f 6d70 7574 6174 696f 6e20 ing computation │ │ │ │ -00000ce0: 4d6f 6e20 4a61 6e20 3139 2031 373a 3335 Mon Jan 19 17:35 │ │ │ │ -00000cf0: 3a30 3120 5554 4320 3230 3236 2020 2020 :01 UTC 2026 │ │ │ │ +00000ce0: 5375 6e20 4a61 6e20 3235 2030 303a 3438 Sun Jan 25 00:48 │ │ │ │ +00000cf0: 3a31 3020 5554 4320 3230 3236 2020 2020 :10 UTC 2026 │ │ │ │ 00000d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000d10: 2020 2020 7c0a 7c2d 2d20 4c69 6e75 7820 |.|-- Linux │ │ │ │ 00000d20: 7362 7569 6c64 2036 2e31 322e 3633 2b64 sbuild 6.12.63+d │ │ │ │ -00000d30: 6562 3133 2d61 6d64 3634 2023 3120 534d eb13-amd64 #1 SM │ │ │ │ -00000d40: 5020 5052 4545 4d50 545f 4459 4e41 4d49 P PREEMPT_DYNAMI │ │ │ │ -00000d50: 4320 4465 6269 616e 2036 2e31 322e 3633 C Debian 6.12.63 │ │ │ │ -00000d60: 2d31 2020 7c0a 7c2d 2d20 414d 4420 4550 -1 |.|-- AMD EP │ │ │ │ -00000d70: 5943 2037 3730 3250 2036 342d 436f 7265 YC 7702P 64-Core │ │ │ │ -00000d80: 2050 726f 6365 7373 6f72 2020 4175 7468 Processor Auth │ │ │ │ -00000d90: 656e 7469 6341 4d44 2020 6370 7520 4d48 enticAMD cpu MH │ │ │ │ -00000da0: 7a20 3139 3936 2e32 3439 2020 2020 2020 z 1996.249 │ │ │ │ +00000d30: 6562 3133 2d63 6c6f 7564 2d61 6d64 3634 eb13-cloud-amd64 │ │ │ │ +00000d40: 2023 3120 534d 5020 5052 4545 4d50 545f #1 SMP PREEMPT_ │ │ │ │ +00000d50: 4459 4e41 4d49 4320 4465 6269 616e 2020 DYNAMIC Debian │ │ │ │ +00000d60: 2020 2020 7c0a 7c2d 2d20 496e 7465 6c20 |.|-- Intel │ │ │ │ +00000d70: 5865 6f6e 2050 726f 6365 7373 6f72 2028 Xeon Processor ( │ │ │ │ +00000d80: 536b 796c 616b 652c 2049 4252 5329 2020 Skylake, IBRS) │ │ │ │ +00000d90: 4765 6e75 696e 6549 6e74 656c 2020 6370 GenuineIntel cp │ │ │ │ +00000da0: 7520 4d48 7a20 3230 3939 2e39 3938 2020 u MHz 2099.998 │ │ │ │ 00000db0: 2020 2020 7c0a 7c2d 2d20 4d61 6361 756c |.|-- Macaul │ │ │ │ 00000dc0: 6179 3220 312e 3235 2e31 312c 2063 6f6d ay2 1.25.11, com │ │ │ │ 00000dd0: 7069 6c65 6420 7769 7468 2067 6363 2031 piled with gcc 1 │ │ │ │ 00000de0: 352e 322e 3020 2020 2020 2020 2020 2020 5.2.0 │ │ │ │ 00000df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000e00: 2020 2020 7c0a 7c2d 2d20 7265 7333 393a |.|-- res39: │ │ │ │ 00000e10: 2072 6573 206f 6620 6120 6765 6e65 7269 res of a generi │ │ │ │ 00000e20: 6320 3320 6279 2039 206d 6174 7269 7820 c 3 by 9 matrix │ │ │ │ -00000e30: 6f76 6572 205a 5a2f 3130 313a 202e 3132 over ZZ/101: .12 │ │ │ │ -00000e40: 3136 3620 7365 636f 6e64 7320 2020 2020 166 seconds │ │ │ │ +00000e30: 6f76 6572 205a 5a2f 3130 313a 202e 3136 over ZZ/101: .16 │ │ │ │ +00000e40: 3534 3939 2073 6563 6f6e 6473 2020 2020 5499 seconds │ │ │ │ 00000e50: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d |.|--------- │ │ │ │ 00000e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000ea0: 2d2d 2d2d 7c0a 7c28 3230 3235 2d31 322d ----|.|(2025-12- │ │ │ │ -00000eb0: 3330 2920 7838 365f 3634 2047 4e55 2f4c 30) x86_64 GNU/L │ │ │ │ -00000ec0: 696e 7578 2020 2020 2020 2020 2020 2020 inux │ │ │ │ +00000ea0: 2d2d 2d2d 7c0a 7c36 2e31 322e 3633 2d31 ----|.|6.12.63-1 │ │ │ │ +00000eb0: 2028 3230 3235 2d31 322d 3330 2920 7838 (2025-12-30) x8 │ │ │ │ +00000ec0: 365f 3634 2047 4e55 2f4c 696e 7578 2020 6_64 GNU/Linux │ │ │ │ 00000ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00000ef0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00000f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 2b0a 0a46 6f72 2074 6865 2070 ----+..For the p │ │ │ │ -00000f50: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -00000f60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -00000f70: 6520 6f62 6a65 6374 202a 6e6f 7465 2072 e object *note r │ │ │ │ -00000f80: 756e 4265 6e63 686d 6172 6b73 3a20 7275 unBenchmarks: ru │ │ │ │ -00000f90: 6e42 656e 6368 6d61 726b 732c 2069 7320 nBenchmarks, is │ │ │ │ -00000fa0: 6120 2a6e 6f74 6520 636f 6d6d 616e 643a a *note command: │ │ │ │ -00000fb0: 0a28 4d61 6361 756c 6179 3244 6f63 2943 .(Macaulay2Doc)C │ │ │ │ -00000fc0: 6f6d 6d61 6e64 2c2e 0a0a 2d2d 2d2d 2d2d ommand,...------ │ │ │ │ -00000fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00001000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00001010: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -00001020: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -00001030: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -00001040: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -00001050: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -00001060: 312e 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+ds/M2/Ma │ │ │ │ -00001070: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -00001080: 2f42 656e 6368 6d61 726b 2e0a 6d32 3a32 /Benchmark..m2:2 │ │ │ │ -00001090: 3937 3a30 2e0a 1f0a 5461 6720 5461 626c 97:0....Tag Tabl │ │ │ │ -000010a0: 653a 0a4e 6f64 653a 2054 6f70 7f32 3334 e:.Node: Top.234 │ │ │ │ -000010b0: 0a4e 6f64 653a 2072 756e 4265 6e63 686d .Node: runBenchm │ │ │ │ -000010c0: 6172 6b73 7f32 3033 350a 1f0a 456e 6420 arks.2035...End │ │ │ │ -000010d0: 5461 6720 5461 626c 650a Tag Table. │ │ │ │ +00000ef0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00000f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000f40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00000f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00000f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00000f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00000f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00000f90: 2d2d 2d2d 2b0a 0a46 6f72 2074 6865 2070 ----+..For the p │ │ │ │ +00000fa0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00000fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00000fc0: 6520 6f62 6a65 6374 202a 6e6f 7465 2072 e object *note r │ │ │ │ +00000fd0: 756e 4265 6e63 686d 6172 6b73 3a20 7275 unBenchmarks: ru │ │ │ │ +00000fe0: 6e42 656e 6368 6d61 726b 732c 2069 7320 nBenchmarks, is │ │ │ │ +00000ff0: 6120 2a6e 6f74 6520 636f 6d6d 616e 643a a *note command: │ │ │ │ +00001000: 0a28 4d61 6361 756c 6179 3244 6f63 2943 .(Macaulay2Doc)C │ │ │ │ +00001010: 6f6d 6d61 6e64 2c2e 0a0a 2d2d 2d2d 2d2d ommand,...------ │ │ │ │ +00001020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00001060: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00001070: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00001080: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00001090: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +000010a0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +000010b0: 312e 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+ds/M2/Ma │ │ │ │ +000010c0: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +000010d0: 2f42 656e 6368 6d61 726b 2e0a 6d32 3a32 /Benchmark..m2:2 │ │ │ │ +000010e0: 3937 3a30 2e0a 1f0a 5461 6720 5461 626c 97:0....Tag Tabl │ │ │ │ +000010f0: 653a 0a4e 6f64 653a 2054 6f70 7f32 3334 e:.Node: Top.234 │ │ │ │ +00001100: 0a4e 6f64 653a 2072 756e 4265 6e63 686d .Node: runBenchm │ │ │ │ +00001110: 6172 6b73 7f32 3033 350a 1f0a 456e 6420 arks.2035...End │ │ │ │ +00001120: 5461 6720 5461 626c 650a Tag Table. │ │ ├── ./usr/share/info/Bertini.info.gz │ │ │ ├── Bertini.info │ │ │ │ @@ -2253,15 +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 3238 3638 "/tmp/M2-2868 │ │ │ │ +00008d30: 2020 2022 2f74 6d70 2f4d 322d 3430 3931 "/tmp/M2-4091 │ │ │ │ 00008d40: 302d 302f 3022 2c20 4f70 7469 6f6e 2074 0-0/0", 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 3836 3830 2d30 2f30 222c p/M2-28680-0/0", │ │ │ │ +00013710: 702f 4d32 2d34 3039 3130 2d30 2f30 222c p/M2-40910-0/0", │ │ │ │ 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 3836 "/tmp/M2-286 │ │ │ │ -00015670: 3830 2d30 2f30 222c 204f 7074 696f 6e20 80-0/0", Option │ │ │ │ +00015660: 2020 2020 222f 746d 702f 4d32 2d34 3039 "/tmp/M2-409 │ │ │ │ +00015670: 3130 2d30 2f30 222c 204f 7074 696f 6e20 10-0/0", Option │ │ │ │ 00015680: 746f 2063 6861 6e67 6520 6469 7265 6374 to change direct │ │ │ │ 00015690: 6f72 7920 666f 7220 6669 6c65 2073 746f ory for file sto │ │ │ │ 000156a0: 7261 6765 2e0a 2020 2020 2020 2a20 5573 rage.. * Us │ │ │ │ 000156b0: 6552 6567 656e 6572 6174 696f 6e20 286d eRegeneration (m │ │ │ │ 000156c0: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ 000156d0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ 000156e0: 6661 756c 7420 7661 6c75 6520 2d31 2c20 fault value -1, │ │ ├── ./usr/share/info/BettiCharacters.info.gz │ │ │ ├── BettiCharacters.info │ │ │ │ @@ -12972,15 +12972,15 @@ │ │ │ │ 00032ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00032ad0: 0a7c 6939 203a 2065 6c61 7073 6564 5469 .|i9 : elapsedTi │ │ │ │ 00032ae0: 6d65 2063 203d 2063 6861 7261 6374 6572 me c = character │ │ │ │ 00032af0: 2041 2020 2020 2020 2020 2020 2020 2020 A │ │ │ │ 00032b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00032b20: 0a7c 202d 2d20 2e34 3337 3332 3973 2065 .| -- .437329s e │ │ │ │ +00032b20: 0a7c 202d 2d20 2e33 3735 3536 3673 2065 .| -- .375566s e │ │ │ │ 00032b30: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00032b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00032b70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00032b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -14183,15 +14183,15 @@ │ │ │ │ 00037660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037680: 2b0a 7c69 3720 3a20 656c 6170 7365 6454 +.|i7 : elapsedT │ │ │ │ 00037690: 696d 6520 633d 6368 6172 6163 7465 7220 ime c=character │ │ │ │ 000376a0: 4120 2020 2020 2020 2020 2020 2020 2020 A │ │ │ │ 000376b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000376c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000376d0: 7c0a 7c20 2d2d 202e 3436 3133 3635 7320 |.| -- .461365s │ │ │ │ +000376d0: 7c0a 7c20 2d2d 202e 3434 3339 3032 7320 |.| -- .443902s │ │ │ │ 000376e0: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 000376f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037720: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00037730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -15614,16 +15614,16 @@ │ │ │ │ 0003cfd0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0003cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d010: 2d2b 0a7c 6932 3020 3a20 656c 6170 7365 -+.|i20 : elapse │ │ │ │ 0003d020: 6454 696d 6520 6131 203d 2063 6861 7261 dTime a1 = chara │ │ │ │ 0003d030: 6374 6572 2041 3120 2020 2020 2020 2020 cter A1 │ │ │ │ -0003d040: 2020 2020 2020 7c0a 7c20 2d2d 202e 3936 |.| -- .96 │ │ │ │ -0003d050: 3632 3631 7320 656c 6170 7365 6420 2020 6261s elapsed │ │ │ │ +0003d040: 2020 2020 2020 7c0a 7c20 2d2d 202e 3733 |.| -- .73 │ │ │ │ +0003d050: 3230 3938 7320 656c 6170 7365 6420 2020 2098s elapsed │ │ │ │ 0003d060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d070: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0003d080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d0b0: 7c0a 7c6f 3230 203d 2043 6861 7261 6374 |.|o20 = Charact │ │ │ │ 0003d0c0: 6572 206f 7665 7220 5220 2020 2020 2020 er over R │ │ │ │ @@ -15654,16 +15654,16 @@ │ │ │ │ 0003d250: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 0003d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0003d290: 6932 3120 3a20 656c 6170 7365 6454 696d i21 : elapsedTim │ │ │ │ 0003d2a0: 6520 6132 203d 2063 6861 7261 6374 6572 e a2 = character │ │ │ │ 0003d2b0: 2041 3220 2020 2020 2020 2020 2020 2020 A2 │ │ │ │ -0003d2c0: 2020 7c0a 7c20 2d2d 2033 332e 3230 3337 |.| -- 33.2037 │ │ │ │ -0003d2d0: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ +0003d2c0: 2020 7c0a 7c20 2d2d 2032 362e 3938 3473 |.| -- 26.984s │ │ │ │ +0003d2d0: 2065 6c61 7073 6564 2020 2020 2020 2020 elapsed │ │ │ │ 0003d2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d2f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0003d300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d320: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0003d330: 3231 203d 2043 6861 7261 6374 6572 206f 21 = Character o │ │ │ │ 0003d340: 7665 7220 5220 2020 2020 2020 2020 2020 ver R │ │ │ │ @@ -16112,16 +16112,16 @@ │ │ │ │ 0003eef0: 6f33 3120 3a20 4163 7469 6f6e 4f6e 4772 o31 : ActionOnGr │ │ │ │ 0003ef00: 6164 6564 4d6f 6475 6c65 2020 2020 2020 adedModule │ │ │ │ 0003ef10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0003ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0003ef40: 6933 3220 3a20 656c 6170 7365 6454 696d i32 : elapsedTim │ │ │ │ 0003ef50: 6520 6220 3d20 6368 6172 6163 7465 7228 e b = character( │ │ │ │ -0003ef60: 422c 3231 297c 0a7c 202d 2d20 3135 2e37 B,21)|.| -- 15.7 │ │ │ │ -0003ef70: 3833 3273 2065 6c61 7073 6564 2020 2020 832s elapsed │ │ │ │ +0003ef60: 422c 3231 297c 0a7c 202d 2d20 3132 2e32 B,21)|.| -- 12.2 │ │ │ │ +0003ef70: 3333 3573 2065 6c61 7073 6564 2020 2020 335s elapsed │ │ │ │ 0003ef80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0003ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003efb0: 2020 2020 207c 0a7c 6f33 3220 3d20 4368 |.|o32 = Ch │ │ │ │ 0003efc0: 6172 6163 7465 7220 6f76 6572 2052 2020 aracter over R │ │ │ │ 0003efd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0003efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/Bruns.info.gz │ │ │ ├── Bruns.info │ │ │ │ @@ -1095,17 +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 2e32 3235 3033 -- used 0.22503 │ │ │ │ -000044e0: 3173 2028 6370 7529 3b20 302e 3137 3237 1s (cpu); 0.1727 │ │ │ │ -000044f0: 3032 7320 2874 6872 6561 6429 3b20 3073 02s (thread); 0s │ │ │ │ +000044d0: 202d 2d20 7573 6564 2030 2e33 3132 3038 -- used 0.31208 │ │ │ │ +000044e0: 3873 2028 6370 7529 3b20 302e 3232 3436 8s (cpu); 0.2246 │ │ │ │ +000044f0: 3635 7320 2874 6872 6561 6429 3b20 3073 65s (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/CellularResolutions.info.gz │ │ │ ├── CellularResolutions.info │ │ │ │ @@ -984,26 +984,26 @@ │ │ │ │ 00003d70: 2843 656c 6c20 6f66 2064 696d 656e 7369 (Cell of dimensi │ │ │ │ 00003d80: 6f6e 2031 2077 6974 6820 6c61 6265 6c7c on 1 with label| │ │ │ │ 00003d90: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ 00003da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -00003de0: 0a7c 2020 2020 2020 312c 2031 292c 2028 .| 1, 1), ( │ │ │ │ -00003df0: 4365 6c6c 206f 6620 6469 6d65 6e73 696f Cell of dimensio │ │ │ │ -00003e00: 6e20 3120 7769 7468 206c 6162 656c 2031 n 1 with label 1 │ │ │ │ -00003e10: 2c20 2d31 292c 2028 4365 6c6c 206f 6620 , -1), (Cell of │ │ │ │ -00003e20: 6469 6d65 6e73 696f 6e20 3120 2020 207c dimension 1 | │ │ │ │ +00003de0: 0a7c 2020 2020 2020 312c 202d 3129 2c20 .| 1, -1), │ │ │ │ +00003df0: 2843 656c 6c20 6f66 2064 696d 656e 7369 (Cell of dimensi │ │ │ │ +00003e00: 6f6e 2031 2077 6974 6820 6c61 6265 6c20 on 1 with label │ │ │ │ +00003e10: 312c 202d 3129 2c20 2843 656c 6c20 6f66 1, -1), (Cell of │ │ │ │ +00003e20: 2064 696d 656e 7369 6f6e 2031 2020 207c dimension 1 | │ │ │ │ 00003e30: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ 00003e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ 00003e80: 0a7c 2020 2020 2020 7769 7468 206c 6162 .| with lab │ │ │ │ -00003e90: 656c 2031 2c20 2d31 297d 2020 2020 2020 el 1, -1)} │ │ │ │ +00003e90: 656c 2031 2c20 3129 7d20 2020 2020 2020 el 1, 1)} │ │ │ │ 00003ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003ec0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00003ed0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00003ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2297,63 +2297,63 @@ │ │ │ │ 00008f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008f90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00008fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008fd0: 207c 0a7c 6f38 203d 2052 656c 6174 696f |.|o8 = Relatio │ │ │ │ 00008fe0: 6e20 4d61 7472 6978 3a20 7c20 3120 3020 n Matrix: | 1 0 │ │ │ │ -00008ff0: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ -00009000: 3120 3120 3120 3120 3120 3120 3120 7c7c 1 1 1 1 1 1 1 || │ │ │ │ +00008ff0: 3020 3020 3020 3120 3020 3120 3120 3120 0 0 0 1 0 1 1 1 │ │ │ │ +00009000: 3020 3020 3020 3120 3120 3120 3120 7c7c 0 0 0 1 1 1 1 || │ │ │ │ 00009010: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00009020: 2020 2020 2020 2020 7c20 3020 3120 3020 | 0 1 0 │ │ │ │ -00009030: 3020 3020 3120 3120 3020 3120 3020 3020 0 0 1 1 0 1 0 0 │ │ │ │ -00009040: 3020 3020 3120 3120 3020 3020 7c7c 0a7c 0 0 1 1 0 0 ||.| │ │ │ │ +00009030: 3020 3020 3120 3020 3020 3020 3020 3120 0 0 1 0 0 0 0 1 │ │ │ │ +00009040: 3120 3020 3120 3120 3020 3020 7c7c 0a7c 1 0 1 1 0 0 ||.| │ │ │ │ 00009050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009060: 2020 2020 2020 7c20 3020 3020 3120 3020 | 0 0 1 0 │ │ │ │ -00009070: 3020 3120 3020 3120 3020 3020 3120 3020 0 1 0 1 0 0 1 0 │ │ │ │ -00009080: 3020 3120 3020 3120 3020 7c7c 0a7c 2020 0 1 0 1 0 ||.| │ │ │ │ +00009070: 3020 3020 3020 3120 3020 3020 3120 3020 0 0 0 1 0 0 1 0 │ │ │ │ +00009080: 3120 3120 3020 3120 3020 7c7c 0a7c 2020 1 1 0 1 0 ||.| │ │ │ │ 00009090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000090a0: 2020 2020 7c20 3020 3020 3020 3120 3020 | 0 0 0 1 0 │ │ │ │ -000090b0: 3020 3120 3020 3020 3120 3020 3120 3020 0 1 0 0 1 0 1 0 │ │ │ │ +000090b0: 3020 3120 3020 3120 3020 3020 3120 3020 0 1 0 1 0 0 1 0 │ │ │ │ 000090c0: 3020 3120 3020 3120 7c7c 0a7c 2020 2020 0 1 0 1 ||.| │ │ │ │ 000090d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000090e0: 2020 7c20 3020 3020 3020 3020 3120 3020 | 0 0 0 0 1 0 │ │ │ │ -000090f0: 3020 3120 3020 3120 3020 3020 3120 3020 0 1 0 1 0 0 1 0 │ │ │ │ +000090f0: 3120 3020 3020 3120 3020 3020 3120 3020 1 0 0 1 0 0 1 0 │ │ │ │ 00009100: 3020 3120 3120 7c7c 0a7c 2020 2020 2020 0 1 1 ||.| │ │ │ │ 00009110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009120: 7c20 3020 3020 3020 3020 3020 3120 3020 | 0 0 0 0 0 1 0 │ │ │ │ -00009130: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ +00009130: 3020 3020 3020 3020 3020 3020 3120 3120 0 0 0 0 0 0 1 1 │ │ │ │ 00009140: 3020 3020 7c7c 0a7c 2020 2020 2020 2020 0 0 ||.| │ │ │ │ 00009150: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ 00009160: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ -00009170: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ -00009180: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ +00009170: 3020 3020 3020 3020 3020 3020 3020 3020 0 0 0 0 0 0 0 0 │ │ │ │ +00009180: 3120 7c7c 0a7c 2020 2020 2020 2020 2020 1 ||.| │ │ │ │ 00009190: 2020 2020 2020 2020 2020 2020 7c20 3020 | 0 │ │ │ │ 000091a0: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ -000091b0: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ +000091b0: 3020 3020 3020 3020 3120 3020 3120 3020 0 0 0 0 1 0 1 0 │ │ │ │ 000091c0: 7c7c 0a7c 2020 2020 2020 2020 2020 2020 ||.| │ │ │ │ 000091d0: 2020 2020 2020 2020 2020 7c20 3020 3020 | 0 0 │ │ │ │ 000091e0: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ -000091f0: 3020 3020 3020 3120 3120 3020 3020 7c7c 0 0 0 1 1 0 0 || │ │ │ │ +000091f0: 3020 3020 3020 3020 3120 3020 3120 7c7c 0 0 0 0 1 0 1 || │ │ │ │ 00009200: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00009210: 2020 2020 2020 2020 7c20 3020 3020 3020 | 0 0 0 │ │ │ │ 00009220: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ -00009230: 3020 3020 3020 3020 3020 3120 7c7c 0a7c 0 0 0 0 0 1 ||.| │ │ │ │ +00009230: 3020 3020 3020 3020 3120 3120 7c7c 0a7c 0 0 0 0 1 1 ||.| │ │ │ │ 00009240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009250: 2020 2020 2020 7c20 3020 3020 3020 3020 | 0 0 0 0 │ │ │ │ 00009260: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ -00009270: 3020 3120 3020 3120 3020 7c7c 0a7c 2020 0 1 0 1 0 ||.| │ │ │ │ +00009270: 3020 3120 3020 3020 3020 7c7c 0a7c 2020 0 1 0 0 0 ||.| │ │ │ │ 00009280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009290: 2020 2020 7c20 3020 3020 3020 3020 3020 | 0 0 0 0 0 │ │ │ │ 000092a0: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ -000092b0: 3020 3120 3020 3120 7c7c 0a7c 2020 2020 0 1 0 1 ||.| │ │ │ │ +000092b0: 3020 3120 3020 3020 7c7c 0a7c 2020 2020 0 1 0 0 ||.| │ │ │ │ 000092c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000092d0: 2020 7c20 3020 3020 3020 3020 3020 3020 | 0 0 0 0 0 0 │ │ │ │ 000092e0: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ -000092f0: 3020 3120 3120 7c7c 0a7c 2020 2020 2020 0 1 1 ||.| │ │ │ │ +000092f0: 3020 3120 3020 7c7c 0a7c 2020 2020 2020 0 1 0 ||.| │ │ │ │ 00009300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009310: 7c20 3020 3020 3020 3020 3020 3020 3020 | 0 0 0 0 0 0 0 │ │ │ │ 00009320: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ 00009330: 3020 3020 7c7c 0a7c 2020 2020 2020 2020 0 0 ||.| │ │ │ │ 00009340: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ 00009350: 3020 3020 3020 3020 3020 3020 3020 3020 0 0 0 0 0 0 0 0 │ │ │ │ 00009360: 3020 3020 3020 3020 3020 3020 3120 3020 0 0 0 0 0 0 1 0 │ │ │ │ @@ -2973,23 +2973,23 @@ │ │ │ │ 0000b9c0: 2020 2020 2020 2020 207c 0a7c 6f38 203d |.|o8 = │ │ │ │ 0000b9d0: 2048 6173 6854 6162 6c65 7b30 203d 3e20 HashTable{0 => │ │ │ │ 0000b9e0: 7b78 202c 2078 2079 2c20 7820 7920 2c20 {x , x y, x y , │ │ │ │ 0000b9f0: 7820 7920 2c20 782a 7920 2c20 7820 7d20 x y , x*y , x } │ │ │ │ 0000ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ba10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0000ba20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ba30: 2020 3520 2020 2033 2033 2020 2035 2032 5 3 3 5 2 │ │ │ │ -0000ba40: 2020 2032 2034 2020 2035 2033 2020 2035 2 4 5 3 5 │ │ │ │ -0000ba50: 2034 2020 2035 2020 2020 3520 3220 2020 4 5 5 2 │ │ │ │ -0000ba60: 3520 3320 2020 2020 207c 0a7c 2020 2020 5 3 |.| │ │ │ │ +0000ba30: 2020 3220 3420 2020 3520 3320 2020 3520 2 4 5 3 5 │ │ │ │ +0000ba40: 3420 2020 3520 2020 2035 2032 2020 2035 4 5 5 2 5 │ │ │ │ +0000ba50: 2033 2020 2035 2034 2020 2034 2032 2020 3 5 4 4 2 │ │ │ │ +0000ba60: 2034 2034 2020 2020 207c 0a7c 2020 2020 4 4 |.| │ │ │ │ 0000ba70: 2020 2020 2020 2020 2020 2031 203d 3e20 1 => │ │ │ │ -0000ba80: 7b78 2079 2c20 7820 7920 2c20 7820 7920 {x y, x y , x y │ │ │ │ -0000ba90: 2c20 7820 7920 2c20 7820 7920 2c20 7820 , x y , x y , x │ │ │ │ -0000baa0: 7920 2c20 7820 792c 2078 2079 202c 2078 y , x y, x y , x │ │ │ │ -0000bab0: 2079 202c 2020 2020 207c 0a7c 2020 2020 y , |.| │ │ │ │ +0000ba80: 7b78 2079 202c 2078 2079 202c 2078 2079 {x y , x y , x y │ │ │ │ +0000ba90: 202c 2078 2079 2c20 7820 7920 2c20 7820 , x y, x y , x │ │ │ │ +0000baa0: 7920 2c20 7820 7920 2c20 7820 7920 2c20 y , x y , x y , │ │ │ │ +0000bab0: 7820 7920 2c20 2020 207c 0a7c 2020 2020 x y , |.| │ │ │ │ 0000bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bad0: 2020 3520 3220 2020 3520 3420 2020 3520 5 2 5 4 5 │ │ │ │ 0000bae0: 3320 2020 3520 3420 2020 3520 3220 2020 3 5 4 5 2 │ │ │ │ 0000baf0: 3520 3420 2020 3520 3320 2020 3520 3420 5 4 5 3 5 4 │ │ │ │ 0000bb00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0000bb10: 2020 2020 2020 2020 2020 2032 203d 3e20 2 => │ │ │ │ 0000bb20: 7b78 2079 202c 2078 2079 202c 2078 2079 {x y , x y , x y │ │ │ │ @@ -3007,25 +3007,25 @@ │ │ │ │ 0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbf0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d |.|---- │ │ │ │ 0000bc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bc40: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 ---------|.| │ │ │ │ -0000bc50: 2020 2020 2020 2020 2020 2020 207d 2020 } │ │ │ │ +0000bc50: 2020 2020 2020 2020 2020 2020 7d20 2020 } │ │ │ │ 0000bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000bc90: 2020 2020 2020 2020 207c 0a7c 2035 2034 |.| 5 4 │ │ │ │ -0000bca0: 2020 2034 2032 2020 2034 2034 2020 2020 4 2 4 4 │ │ │ │ +0000bc90: 2020 2020 2020 2020 207c 0a7c 2035 2020 |.| 5 │ │ │ │ +0000bca0: 2020 3320 3320 2020 3520 3220 2020 2020 3 3 5 2 │ │ │ │ 0000bcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000bce0: 2020 2020 2020 2020 207c 0a7c 7820 7920 |.|x y │ │ │ │ -0000bcf0: 2c20 7820 7920 2c20 7820 7920 7d20 2020 , x y , x y } │ │ │ │ +0000bce0: 2020 2020 2020 2020 207c 0a7c 7820 792c |.|x y, │ │ │ │ +0000bcf0: 2078 2079 202c 2078 2079 207d 2020 2020 x y , x y } │ │ │ │ 0000bd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 0000bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -3989,15 +3989,15 @@ │ │ │ │ 0000f940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f970: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000f980: 6f37 203d 2048 6173 6854 6162 6c65 7b30 o7 = HashTable{0 │ │ │ │ 0000f990: 203d 3e20 7b43 656c 6c20 6f66 2064 696d => {Cell of dim │ │ │ │ 0000f9a0: 656e 7369 6f6e 2030 2077 6974 6820 6c61 ension 0 with la │ │ │ │ -0000f9b0: 6265 6c20 792c 2043 656c 6c20 6f66 2064 bel y, Cell of d │ │ │ │ +0000f9b0: 6265 6c20 782c 2043 656c 6c20 6f66 2064 bel x, Cell of d │ │ │ │ 0000f9c0: 696d 656e 7369 6f6e 2030 2020 207c 0a7c imension 0 |.| │ │ │ │ 0000f9d0: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ 0000f9e0: 203d 3e20 7b43 656c 6c20 6f66 2064 696d => {Cell of dim │ │ │ │ 0000f9f0: 656e 7369 6f6e 2031 2077 6974 6820 6c61 ension 1 with la │ │ │ │ 0000fa00: 6265 6c20 782a 797d 2020 2020 2020 2020 bel x*y} │ │ │ │ 0000fa10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000fa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4013,15 +4013,15 @@ │ │ │ │ 0000fac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ 0000fb10: 7769 7468 206c 6162 656c 207a 2c20 4365 with label z, Ce │ │ │ │ 0000fb20: 6c6c 206f 6620 6469 6d65 6e73 696f 6e20 ll of dimension │ │ │ │ -0000fb30: 3020 7769 7468 206c 6162 656c 2078 7d7d 0 with label x}} │ │ │ │ +0000fb30: 3020 7769 7468 206c 6162 656c 2079 7d7d 0 with label y}} │ │ │ │ 0000fb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb50: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 0000fb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2b -------------+.+ │ │ │ │ @@ -4290,25 +4290,25 @@ │ │ │ │ 00010c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010c50: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ 00010c60: 3d20 7b43 656c 6c20 6f66 2064 696d 656e = {Cell of dimen │ │ │ │ 00010c70: 7369 6f6e 2030 2077 6974 6820 6c61 6265 sion 0 with labe │ │ │ │ -00010c80: 6c20 792c 2043 656c 6c20 6f66 2064 696d l y, Cell of dim │ │ │ │ +00010c80: 6c20 7a2c 2043 656c 6c20 6f66 2064 696d l z, Cell of dim │ │ │ │ 00010c90: 656e 7369 6f6e 2030 2077 6974 6820 6c61 ension 0 with la │ │ │ │ 00010ca0: 6265 6c20 782c 2020 2020 7c0a 7c20 2020 bel x, |.| │ │ │ │ 00010cb0: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ 00010cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ 00010d00: 2020 4365 6c6c 206f 6620 6469 6d65 6e73 Cell of dimens │ │ │ │ 00010d10: 696f 6e20 3020 7769 7468 206c 6162 656c ion 0 with label │ │ │ │ -00010d20: 207a 7d20 2020 2020 2020 2020 2020 2020 z} │ │ │ │ +00010d20: 2079 7d20 2020 2020 2020 2020 2020 2020 y} │ │ │ │ 00010d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00010d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d90: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ @@ -8275,22 +8275,22 @@ │ │ │ │ 00020520: 6c4c 6162 656c 2863 2920 2020 2020 2020 lLabel(c) │ │ │ │ 00020530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020540: 2020 7c0a 7c20 2020 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 2020 │ │ │ │ -00020590: 2020 7c0a 7c20 2020 2020 2020 2032 2020 |.| 2 │ │ │ │ -000205a0: 2032 2020 2032 2032 2020 2032 2032 2020 2 2 2 2 2 │ │ │ │ -000205b0: 2020 2020 2032 2020 2020 2032 2020 2020 2 2 │ │ │ │ +00020590: 2020 7c0a 7c20 2020 2020 2020 2032 2032 |.| 2 2 │ │ │ │ +000205a0: 2020 2032 2020 2032 2020 2020 2032 2020 2 2 2 │ │ │ │ +000205b0: 2020 3220 3220 2020 2020 2020 3220 2020 2 2 2 │ │ │ │ 000205c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000205d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000205e0: 2020 7c0a 7c6f 3134 203d 207b 6120 622a |.|o14 = {a b* │ │ │ │ -000205f0: 6320 2c20 6120 6220 2c20 6220 6320 2c20 c , a b , b c , │ │ │ │ -00020600: 612a 622a 6320 2c20 612a 6220 637d 2020 a*b*c , a*b c} │ │ │ │ +000205e0: 2020 7c0a 7c6f 3134 203d 207b 6220 6320 |.|o14 = {b c │ │ │ │ +000205f0: 2c20 6120 622a 6320 2c20 612a 6220 632c , a b*c , a*b c, │ │ │ │ +00020600: 2061 2062 202c 2061 2a62 2a63 207d 2020 a b , a*b*c } │ │ │ │ 00020610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020630: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00020640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 │ │ ├── ./usr/share/info/ChainComplexExtras.info.gz │ │ │ ├── ChainComplexExtras.info │ │ │ │ @@ -4819,16 +4819,16 @@ │ │ │ │ 00012d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00012d50: 3133 203a 2074 696d 6520 6d20 3d20 6d69 13 : time m = mi │ │ │ │ 00012d60: 6e69 6d69 7a65 2028 455b 315d 293b 2020 nimize (E[1]); │ │ │ │ 00012d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012d80: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00012d90: 302e 3234 3338 3533 7320 2863 7075 293b 0.243853s (cpu); │ │ │ │ -00012da0: 2030 2e31 3939 3631 3573 2028 7468 7265 0.199615s (thre │ │ │ │ +00012d90: 302e 3438 3539 3639 7320 2863 7075 293b 0.485969s (cpu); │ │ │ │ +00012da0: 2030 2e33 3738 3330 3573 2028 7468 7265 0.378305s (thre │ │ │ │ 00012db0: 6164 293b 2030 7320 2867 6329 7c0a 2b2d ad); 0s (gc)|.+- │ │ │ │ 00012dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012df0: 2d2d 2d2d 2b0a 7c69 3134 203a 2069 7351 ----+.|i14 : isQ │ │ │ │ 00012e00: 7561 7369 4973 6f6d 6f72 7068 6973 6d20 uasiIsomorphism │ │ │ │ 00012e10: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ @@ -6579,33 +6579,33 @@ │ │ │ │ 00019b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00019b40: 3820 3a20 7469 6d65 206d 203d 2072 6573 8 : time m = res │ │ │ │ 00019b50: 6f6c 7574 696f 6e4f 6643 6861 696e 436f olutionOfChainCo │ │ │ │ 00019b60: 6d70 6c65 7820 433b 2020 2020 2020 2020 mplex C; │ │ │ │ 00019b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00019b90: 2d2d 2075 7365 6420 302e 3039 3730 3831 -- used 0.097081 │ │ │ │ -00019ba0: 3873 2028 6370 7529 3b20 302e 3039 3730 8s (cpu); 0.0970 │ │ │ │ -00019bb0: 3831 3173 2028 7468 7265 6164 293b 2030 811s (thread); 0 │ │ │ │ -00019bc0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ +00019b90: 2d2d 2075 7365 6420 302e 3135 3733 3539 -- used 0.157359 │ │ │ │ +00019ba0: 7320 2863 7075 293b 2030 2e31 3537 3335 s (cpu); 0.15735 │ │ │ │ +00019bb0: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ +00019bc0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00019bd0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00019be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00019c30: 3920 3a20 7469 6d65 206e 203d 2063 6172 9 : time n = car │ │ │ │ 00019c40: 7461 6e45 696c 656e 6265 7267 5265 736f tanEilenbergReso │ │ │ │ 00019c50: 6c75 7469 6f6e 2043 3b20 2020 2020 2020 lution C; │ │ │ │ 00019c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00019c80: 2d2d 2075 7365 6420 302e 3231 3135 3631 -- used 0.211561 │ │ │ │ -00019c90: 7320 2863 7075 293b 2030 2e31 3437 3731 s (cpu); 0.14771 │ │ │ │ -00019ca0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -00019cb0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +00019c80: 2d2d 2075 7365 6420 302e 3238 3632 3239 -- used 0.286229 │ │ │ │ +00019c90: 7320 2863 7075 293b 2030 2e31 3930 3536 s (cpu); 0.19056 │ │ │ │ +00019ca0: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ +00019cb0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00019cc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00019cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00019d20: 3130 203a 2062 6574 7469 2073 6f75 7263 10 : betti sourc │ │ ├── ./usr/share/info/CharacteristicClasses.info.gz │ │ │ ├── CharacteristicClasses.info │ │ │ │ @@ -1215,16 +1215,16 @@ │ │ │ │ 00004be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004bf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ 00004c00: 2074 696d 6520 4353 4d20 5520 2020 2020 time CSM U │ │ │ │ 00004c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004c40: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00004c50: 7573 6564 2030 2e33 3032 3430 3173 2028 used 0.302401s ( │ │ │ │ -00004c60: 6370 7529 3b20 302e 3232 3233 3838 7320 cpu); 0.222388s │ │ │ │ +00004c50: 7573 6564 2030 2e33 3233 3838 3573 2028 used 0.323885s ( │ │ │ │ +00004c60: 6370 7529 3b20 302e 3230 3735 3339 7320 cpu); 0.207539s │ │ │ │ 00004c70: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00004c80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00004c90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1300,16 +1300,16 @@ │ │ │ │ 00005130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005140: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ 00005150: 2074 696d 6520 4353 4d28 552c 4368 6563 time CSM(U,Chec │ │ │ │ 00005160: 6b53 6d6f 6f74 683d 3e66 616c 7365 2920 kSmooth=>false) │ │ │ │ 00005170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005190: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000051a0: 7573 6564 2030 2e33 3539 3830 3373 2028 used 0.359803s ( │ │ │ │ -000051b0: 6370 7529 3b20 302e 3238 3632 3032 7320 cpu); 0.286202s │ │ │ │ +000051a0: 7573 6564 2030 2e35 3337 3836 3273 2028 used 0.537862s ( │ │ │ │ +000051b0: 6370 7529 3b20 302e 3431 3530 3734 7320 cpu); 0.415074s │ │ │ │ 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 │ │ │ │ @@ -4341,16 +4341,16 @@ │ │ │ │ 00010f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ 00010f70: 2074 696d 6520 4353 4d28 492c 436f 6d70 time CSM(I,Comp │ │ │ │ 00010f80: 4d65 7468 6f64 3d3e 5072 6f6a 6563 7469 Method=>Projecti │ │ │ │ 00010f90: 7665 4465 6772 6565 2920 2020 2020 2020 veDegree) │ │ │ │ 00010fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00010fb0: 2d2d 2075 7365 6420 302e 3336 3730 3435 -- used 0.367045 │ │ │ │ -00010fc0: 7320 2863 7075 293b 2030 2e32 3739 3335 s (cpu); 0.27935 │ │ │ │ +00010fb0: 2d2d 2075 7365 6420 302e 3835 3236 3239 -- used 0.852629 │ │ │ │ +00010fc0: 7320 2863 7075 293b 2030 2e34 3239 3033 s (cpu); 0.42903 │ │ │ │ 00010fd0: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ 00010fe0: 2867 6329 2020 2020 2020 2020 2020 207c (gc) | │ │ │ │ 00010ff0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00011000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011030: 2020 7c0a 7c20 2020 2020 2020 3520 2020 |.| 5 │ │ │ │ @@ -4400,16 +4400,16 @@ │ │ │ │ 000112f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011310: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520 ---+.|i6 : time │ │ │ │ 00011320: 4353 4d28 492c 436f 6d70 4d65 7468 6f64 CSM(I,CompMethod │ │ │ │ 00011330: 3d3e 506e 5265 7369 6475 616c 2920 2020 =>PnResidual) │ │ │ │ 00011340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011350: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00011360: 6420 322e 3334 3536 3373 2028 6370 7529 d 2.34563s (cpu) │ │ │ │ -00011370: 3b20 312e 3938 3731 3973 2028 7468 7265 ; 1.98719s (thre │ │ │ │ +00011360: 6420 322e 3535 3233 3473 2028 6370 7529 d 2.55234s (cpu) │ │ │ │ +00011370: 3b20 322e 3138 3837 3373 2028 7468 7265 ; 2.18873s (thre │ │ │ │ 00011380: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00011390: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000113a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000113e0: 2020 2020 2020 3520 2020 2020 2034 2020 5 4 │ │ │ │ @@ -4488,16 +4488,16 @@ │ │ │ │ 00011870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011890: 2d2d 2b0a 7c69 3130 203a 2074 696d 6520 --+.|i10 : time │ │ │ │ 000118a0: 4353 4d28 4b2c 436f 6d70 4d65 7468 6f64 CSM(K,CompMethod │ │ │ │ 000118b0: 3d3e 5072 6f6a 6563 7469 7665 4465 6772 =>ProjectiveDegr │ │ │ │ 000118c0: 6565 2920 2020 2020 2020 2020 2020 2020 ee) │ │ │ │ 000118d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000118e0: 2030 2e32 3738 3639 3473 2028 6370 7529 0.278694s (cpu) │ │ │ │ -000118f0: 3b20 302e 3232 3034 3832 7320 2874 6872 ; 0.220482s (thr │ │ │ │ +000118e0: 2030 2e33 3632 3231 3473 2028 6370 7529 0.362214s (cpu) │ │ │ │ +000118f0: 3b20 302e 3234 3233 3434 7320 2874 6872 ; 0.242344s (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 3835 .| -- used 0.085 │ │ │ │ -00011c90: 3737 3234 7320 2863 7075 293b 2030 2e30 7724s (cpu); 0.0 │ │ │ │ -00011ca0: 3835 3737 3939 7320 2874 6872 6561 6429 857799s (thread) │ │ │ │ -00011cb0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +00011c80: 0a7c 202d 2d20 7573 6564 2030 2e31 3136 .| -- used 0.116 │ │ │ │ +00011c90: 3839 3673 2028 6370 7529 3b20 302e 3131 896s (cpu); 0.11 │ │ │ │ +00011ca0: 3639 3037 7320 2874 6872 6561 6429 3b20 6907s (thread); │ │ │ │ +00011cb0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00011cc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00011d10: 3320 2020 2020 3220 2020 2020 2020 2020 3 2 │ │ │ │ 00011d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5446,6791 +5446,6785 @@ │ │ │ │ 00015450: 2072 6574 7572 6e65 6420 696e 2074 6865 returned in the │ │ │ │ 00015460: 2073 616d 6520 7269 6e67 2e20 5765 206d same ring. We m │ │ │ │ 00015470: 6179 2061 6c73 6f20 7265 7475 726e 2061 ay also return a │ │ │ │ 00015480: 0a4d 7574 6162 6c65 4861 7368 5461 626c .MutableHashTabl │ │ │ │ 00015490: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d e...+----------- │ │ │ │ 000154a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000154b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000154c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 ----------+.|i11 │ │ │ │ -000154d0: 203a 2052 3d4d 756c 7469 5072 6f6a 436f : R=MultiProjCo │ │ │ │ -000154e0: 6f72 6452 696e 6728 7b32 2c32 7d29 2020 ordRing({2,2}) │ │ │ │ -000154f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015500: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000154c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a --------+.|i11 : │ │ │ │ +000154d0: 2052 3d4d 756c 7469 5072 6f6a 436f 6f72 R=MultiProjCoor │ │ │ │ +000154e0: 6452 696e 6728 7b32 2c32 7d29 2020 2020 dRing({2,2}) │ │ │ │ +000154f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00015500: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00015510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015530: 2020 2020 2020 2020 2020 7c0a 7c6f 3131 |.|o11 │ │ │ │ -00015540: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +00015530: 2020 2020 7c0a 7c6f 3131 203d 2052 2020 |.|o11 = R │ │ │ │ +00015540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015570: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015560: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00015570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000155a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3131 |.|o11 │ │ │ │ -000155b0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -000155c0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000155e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000155a0: 7c0a 7c6f 3131 203a 2050 6f6c 796e 6f6d |.|o11 : Polynom │ │ │ │ +000155b0: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +000155c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000155d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +000155e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000155f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015610: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 ----------+.|i12 │ │ │ │ -00015620: 203a 2041 3d43 686f 7752 696e 6728 5229 : A=ChowRing(R) │ │ │ │ +00015600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00015610: 3132 203a 2041 3d43 686f 7752 696e 6728 12 : A=ChowRing( │ │ │ │ +00015620: 5229 2020 2020 2020 2020 2020 2020 2020 R) │ │ │ │ 00015630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015650: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015640: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015680: 2020 2020 2020 2020 2020 7c0a 7c6f 3132 |.|o12 │ │ │ │ -00015690: 203d 2041 2020 2020 2020 2020 2020 2020 = A │ │ │ │ -000156a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000156b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000156c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015670: 2020 2020 2020 2020 7c0a 7c6f 3132 203d |.|o12 = │ │ │ │ +00015680: 2041 2020 2020 2020 2020 2020 2020 2020 A │ │ │ │ +00015690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000156a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000156b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000156c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000156d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000156e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000156f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3132 |.|o12 │ │ │ │ -00015700: 203a 2051 756f 7469 656e 7452 696e 6720 : QuotientRing │ │ │ │ -00015710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015730: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000156e0: 2020 2020 7c0a 7c6f 3132 203a 2051 756f |.|o12 : Quo │ │ │ │ +000156f0: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ +00015700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015710: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00015720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015760: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 ----------+.|i13 │ │ │ │ -00015770: 203a 2072 3d67 656e 7320 5220 2020 2020 : r=gens R │ │ │ │ -00015780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015750: 2b0a 7c69 3133 203a 2072 3d67 656e 7320 +.|i13 : r=gens │ │ │ │ +00015760: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00015770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015780: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00015790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000157a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -000157b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000157c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000157d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3133 |.|o13 │ │ │ │ -000157e0: 203d 207b 7820 2c20 7820 2c20 7820 2c20 = {x , x , x , │ │ │ │ -000157f0: 7820 2c20 7820 2c20 7820 7d20 2020 2020 x , x , x } │ │ │ │ -00015800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015810: 2020 7c0a 7c20 2020 2020 2020 2030 2020 |.| 0 │ │ │ │ -00015820: 2031 2020 2032 2020 2033 2020 2034 2020 1 2 3 4 │ │ │ │ -00015830: 2035 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ -00015840: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00015850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000157a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000157b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000157c0: 3133 203d 207b 7820 2c20 7820 2c20 7820 13 = {x , x , x │ │ │ │ +000157d0: 2c20 7820 2c20 7820 2c20 7820 7d20 2020 , x , x , x } │ │ │ │ +000157e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000157f0: 2020 7c0a 7c20 2020 2020 2020 2030 2020 |.| 0 │ │ │ │ +00015800: 2031 2020 2032 2020 2033 2020 2034 2020 1 2 3 4 │ │ │ │ +00015810: 2035 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ +00015820: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00015830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015850: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00015860: 7c6f 3133 203a 204c 6973 7420 2020 2020 |o13 : List │ │ │ │ 00015870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015880: 2020 7c0a 7c6f 3133 203a 204c 6973 7420 |.|o13 : List │ │ │ │ -00015890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000158a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000158b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -000158c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000158d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000158e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000158f0: 2d2d 2b0a 7c69 3134 203a 204b 3d69 6465 --+.|i14 : K=ide │ │ │ │ -00015900: 616c 2872 5f30 5e32 2a72 5f33 2d72 5f34 al(r_0^2*r_3-r_4 │ │ │ │ -00015910: 2a72 5f31 2a72 5f32 2c72 5f32 5e32 2a72 *r_1*r_2,r_2^2*r │ │ │ │ -00015920: 5f35 2920 2020 2020 2020 7c0a 7c20 2020 _5) |.| │ │ │ │ -00015930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015960: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00015970: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00015980: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00015990: 2020 2020 2020 2020 2020 7c0a 7c6f 3134 |.|o14 │ │ │ │ -000159a0: 203d 2069 6465 616c 2028 7820 7820 202d = ideal (x x - │ │ │ │ -000159b0: 2078 2078 2078 202c 2078 2078 2029 2020 x x x , x x ) │ │ │ │ -000159c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000159d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -000159e0: 2020 2030 2033 2020 2020 3120 3220 3420 0 3 1 2 4 │ │ │ │ -000159f0: 2020 3220 3520 2020 2020 2020 2020 2020 2 5 │ │ │ │ -00015a00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00015a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015890: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000158a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000158b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000158c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 ----------+.|i14 │ │ │ │ +000158d0: 203a 204b 3d69 6465 616c 2872 5f30 5e32 : K=ideal(r_0^2 │ │ │ │ +000158e0: 2a72 5f33 2d72 5f34 2a72 5f31 2a72 5f32 *r_3-r_4*r_1*r_2 │ │ │ │ +000158f0: 2c72 5f32 5e32 2a72 5f35 2920 2020 2020 ,r_2^2*r_5) │ │ │ │ +00015900: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015930: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00015940: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00015950: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00015960: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00015970: 3134 203d 2069 6465 616c 2028 7820 7820 14 = ideal (x x │ │ │ │ +00015980: 202d 2078 2078 2078 202c 2078 2078 2029 - x x x , x x ) │ │ │ │ +00015990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000159a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000159b0: 2020 2030 2033 2020 2020 3120 3220 3420 0 3 1 2 4 │ │ │ │ +000159c0: 2020 3220 3520 2020 2020 2020 2020 2020 2 5 │ │ │ │ +000159d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000159e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000159f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015a00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00015a10: 7c6f 3134 203a 2049 6465 616c 206f 6620 |o14 : Ideal of │ │ │ │ +00015a20: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00015a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015a40: 2020 7c0a 7c6f 3134 203a 2049 6465 616c |.|o14 : Ideal │ │ │ │ -00015a50: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ -00015a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015a70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -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 3530 3133 3731 7320 used 0.501371s │ │ │ │ -00015b00: 2863 7075 293b 2030 2e33 3332 3732 3273 (cpu); 0.332722s │ │ │ │ -00015b10: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -00015b20: 6329 7c0a 7c20 2020 2020 2020 2020 2020 c)|.| │ │ │ │ -00015b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015b50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00015b60: 2020 2020 2032 2032 2020 2020 2032 2020 2 2 2 │ │ │ │ -00015b70: 2020 2020 2020 2032 2020 2020 3220 2020 2 2 │ │ │ │ -00015b80: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00015b90: 2020 7c0a 7c6f 3135 203d 2037 6820 6820 |.|o15 = 7h h │ │ │ │ -00015ba0: 202b 2035 6820 6820 202b 2034 6820 6820 + 5h h + 4h h │ │ │ │ -00015bb0: 202b 2068 2020 2b20 3368 2068 2020 2b20 + h + 3h h + │ │ │ │ -00015bc0: 6820 2020 2020 2020 2020 7c0a 7c20 2020 h |.| │ │ │ │ -00015bd0: 2020 2020 2031 2032 2020 2020 2031 2032 1 2 1 2 │ │ │ │ -00015be0: 2020 2020 2031 2032 2020 2020 3120 2020 1 2 1 │ │ │ │ -00015bf0: 2020 3120 3220 2020 2032 2020 2020 2020 1 2 2 │ │ │ │ -00015c00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015a40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00015a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 ----------+.|i15 │ │ │ │ +00015a80: 203a 2074 696d 6520 6373 6d4b 3d43 534d : time csmK=CSM │ │ │ │ +00015a90: 2841 2c4b 2920 2020 2020 2020 2020 2020 (A,K) │ │ │ │ +00015aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015ab0: 7c0a 7c20 2d2d 2075 7365 6420 312e 3331 |.| -- used 1.31 │ │ │ │ +00015ac0: 3438 7320 2863 7075 293b 2030 2e34 3233 48s (cpu); 0.423 │ │ │ │ +00015ad0: 3832 3473 2028 7468 7265 6164 293b 2030 824s (thread); 0 │ │ │ │ +00015ae0: 7320 2867 6329 7c0a 7c20 2020 2020 2020 s (gc)|.| │ │ │ │ +00015af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015b10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00015b20: 2020 2020 2020 2032 2032 2020 2020 2032 2 2 2 │ │ │ │ +00015b30: 2020 2020 2020 2020 2032 2020 2020 3220 2 2 │ │ │ │ +00015b40: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +00015b50: 2020 7c0a 7c6f 3135 203d 2037 6820 6820 |.|o15 = 7h h │ │ │ │ +00015b60: 202b 2035 6820 6820 202b 2034 6820 6820 + 5h h + 4h h │ │ │ │ +00015b70: 202b 2068 2020 2b20 3368 2068 2020 2b20 + h + 3h h + │ │ │ │ +00015b80: 6820 2020 2020 2020 7c0a 7c20 2020 2020 h |.| │ │ │ │ +00015b90: 2020 2031 2032 2020 2020 2031 2032 2020 1 2 1 2 │ │ │ │ +00015ba0: 2020 2031 2032 2020 2020 3120 2020 2020 1 2 1 │ │ │ │ +00015bb0: 3120 3220 2020 2032 2020 2020 2020 7c0a 1 2 2 |. │ │ │ │ +00015bc0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00015bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015bf0: 2020 2020 7c0a 7c6f 3135 203a 2041 2020 |.|o15 : A │ │ │ │ +00015c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015c30: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ -00015c40: 203a 2041 2020 2020 2020 2020 2020 2020 : A │ │ │ │ -00015c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015c70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00015c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136 ----------+.|i16 │ │ │ │ -00015cb0: 203a 2063 736d 4b48 6173 683d 2043 534d : csmKHash= CSM │ │ │ │ -00015cc0: 2841 2c4b 2c4f 7574 7075 743d 3e48 6173 (A,K,Output=>Has │ │ │ │ -00015cd0: 6846 6f72 6d29 2020 2020 2020 2020 2020 hForm) │ │ │ │ -00015ce0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015c20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00015c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015c60: 2b0a 7c69 3136 203a 2063 736d 4b48 6173 +.|i16 : csmKHas │ │ │ │ +00015c70: 683d 2043 534d 2841 2c4b 2c4f 7574 7075 h= CSM(A,K,Outpu │ │ │ │ +00015c80: 743d 3e48 6173 6846 6f72 6d29 2020 2020 t=>HashForm) │ │ │ │ +00015c90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00015ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00015cd0: 3136 203d 204d 7574 6162 6c65 4861 7368 16 = MutableHash │ │ │ │ +00015ce0: 5461 626c 657b 2e2e 2e34 2e2e 2e7d 2020 Table{...4...} │ │ │ │ 00015cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d10: 2020 2020 2020 2020 2020 7c0a 7c6f 3136 |.|o16 │ │ │ │ -00015d20: 203d 204d 7574 6162 6c65 4861 7368 5461 = MutableHashTa │ │ │ │ -00015d30: 626c 657b 2e2e 2e34 2e2e 2e7d 2020 2020 ble{...4...} │ │ │ │ -00015d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00015d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d80: 2020 2020 2020 2020 2020 7c0a 7c6f 3136 |.|o16 │ │ │ │ -00015d90: 203a 204d 7574 6162 6c65 4861 7368 5461 : MutableHashTa │ │ │ │ -00015da0: 626c 6520 2020 2020 2020 2020 2020 2020 ble │ │ │ │ -00015db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015dc0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00015dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 ----------+.|i17 │ │ │ │ -00015e00: 203a 2063 736d 4b3d 3d63 736d 4b48 6173 : csmK==csmKHas │ │ │ │ -00015e10: 6823 2243 534d 2220 2020 2020 2020 2020 h#"CSM" │ │ │ │ +00015d00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015d30: 2020 2020 2020 2020 7c0a 7c6f 3136 203a |.|o16 : │ │ │ │ +00015d40: 204d 7574 6162 6c65 4861 7368 5461 626c MutableHashTabl │ │ │ │ +00015d50: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00015d60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00015d70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00015d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015da0: 2d2d 2d2d 2b0a 7c69 3137 203a 2063 736d ----+.|i17 : csm │ │ │ │ +00015db0: 4b3d 3d63 736d 4b48 6173 6823 2243 534d K==csmKHash#"CSM │ │ │ │ +00015dc0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ +00015dd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00015de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015e10: 7c0a 7c6f 3137 203d 2074 7275 6520 2020 |.|o17 = true │ │ │ │ 00015e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015e30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00015e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015e60: 2020 2020 2020 2020 2020 7c0a 7c6f 3137 |.|o17 │ │ │ │ -00015e70: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ -00015e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015ea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00015eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 ----------+.|i18 │ │ │ │ -00015ee0: 203a 2043 534d 2841 2c69 6465 616c 284b : CSM(A,ideal(K │ │ │ │ -00015ef0: 5f30 2929 3d3d 6373 6d4b 4861 7368 237b _0))==csmKHash#{ │ │ │ │ -00015f00: 307d 2020 2020 2020 2020 2020 2020 2020 0} │ │ │ │ -00015f10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00015f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015f40: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ -00015f50: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ -00015f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015f80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00015f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 7570 ----------+..Sup │ │ │ │ -00015fc0: 706f 7365 2077 6520 6861 7665 2061 6c72 pose we have alr │ │ │ │ -00015fd0: 6561 6479 2063 6f6d 7075 7465 6420 736f eady computed so │ │ │ │ -00015fe0: 6d65 206f 6620 4353 4d20 636c 6173 7365 me of CSM classe │ │ │ │ -00015ff0: 7320 6f66 2068 7970 6572 7375 7266 6163 s of hypersurfac │ │ │ │ -00016000: 6573 2069 6e76 6f6c 7665 640a 696e 2074 es involved.in t │ │ │ │ -00016010: 6865 2069 6e63 6c75 7369 6f6e 2d65 7863 he inclusion-exc │ │ │ │ -00016020: 6c75 7369 6f6e 2070 726f 6365 6475 7265 lusion procedure │ │ │ │ -00016030: 2c20 7468 656e 2077 6520 6d61 7920 696e , then we may in │ │ │ │ -00016040: 7075 7420 7468 6573 6520 746f 2062 6520 put these to be │ │ │ │ -00016050: 7573 6564 2062 7920 7468 650a 4353 4d20 used by the.CSM │ │ │ │ -00016060: 6675 6e63 7469 6f6e 2e20 496e 2074 6865 function. In the │ │ │ │ -00016070: 2065 7861 6d70 6c65 2062 656c 6f77 2077 example below w │ │ │ │ -00016080: 6520 696e 7075 7420 7468 6520 4353 4d20 e input the CSM │ │ │ │ -00016090: 636c 6173 7320 6f66 2056 284b 5f30 2920 class of V(K_0) │ │ │ │ -000160a0: 2874 6861 7420 6973 206f 660a 7468 6520 (that is of.the │ │ │ │ -000160b0: 6879 7065 7273 7572 6661 6365 2064 6566 hypersurface def │ │ │ │ -000160c0: 696e 6564 2062 7920 7468 6520 6669 7273 ined by the firs │ │ │ │ -000160d0: 7420 706f 6c79 6e6f 6d69 616c 2067 656e t polynomial gen │ │ │ │ -000160e0: 6572 6174 696e 6720 4b29 2061 6e64 2074 erating K) and t │ │ │ │ -000160f0: 6865 2043 534d 0a63 6c61 7373 206f 6620 he CSM.class of │ │ │ │ -00016100: 7468 6520 6879 7065 7273 7572 6661 6365 the hypersurface │ │ │ │ -00016110: 2064 6566 696e 6564 2062 7920 7468 6520 defined by the │ │ │ │ -00016120: 7072 6f64 7563 7420 6f66 2074 6865 2067 product of the g │ │ │ │ -00016130: 656e 6572 6174 6f72 7320 6f66 204b 2e0a enerators of K.. │ │ │ │ -00016140: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00016150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015e40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00015e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00015e80: 3138 203a 2043 534d 2841 2c69 6465 616c 18 : CSM(A,ideal │ │ │ │ +00015e90: 284b 5f30 2929 3d3d 6373 6d4b 4861 7368 (K_0))==csmKHash │ │ │ │ +00015ea0: 237b 307d 2020 2020 2020 2020 2020 2020 #{0} │ │ │ │ +00015eb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015ee0: 2020 2020 2020 2020 7c0a 7c6f 3138 203d |.|o18 = │ │ │ │ +00015ef0: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ +00015f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015f10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00015f20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00015f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015f50: 2d2d 2d2d 2b0a 0a53 7570 706f 7365 2077 ----+..Suppose w │ │ │ │ +00015f60: 6520 6861 7665 2061 6c72 6561 6479 2063 e have already c │ │ │ │ +00015f70: 6f6d 7075 7465 6420 736f 6d65 206f 6620 omputed some of │ │ │ │ +00015f80: 4353 4d20 636c 6173 7365 7320 6f66 2068 CSM classes of h │ │ │ │ +00015f90: 7970 6572 7375 7266 6163 6573 2069 6e76 ypersurfaces inv │ │ │ │ +00015fa0: 6f6c 7665 640a 696e 2074 6865 2069 6e63 olved.in the inc │ │ │ │ +00015fb0: 6c75 7369 6f6e 2d65 7863 6c75 7369 6f6e lusion-exclusion │ │ │ │ +00015fc0: 2070 726f 6365 6475 7265 2c20 7468 656e procedure, then │ │ │ │ +00015fd0: 2077 6520 6d61 7920 696e 7075 7420 7468 we may input th │ │ │ │ +00015fe0: 6573 6520 746f 2062 6520 7573 6564 2062 ese to be used b │ │ │ │ +00015ff0: 7920 7468 650a 4353 4d20 6675 6e63 7469 y the.CSM functi │ │ │ │ +00016000: 6f6e 2e20 496e 2074 6865 2065 7861 6d70 on. In the examp │ │ │ │ +00016010: 6c65 2062 656c 6f77 2077 6520 696e 7075 le below we inpu │ │ │ │ +00016020: 7420 7468 6520 4353 4d20 636c 6173 7320 t the CSM class │ │ │ │ +00016030: 6f66 2056 284b 5f30 2920 2874 6861 7420 of V(K_0) (that │ │ │ │ +00016040: 6973 206f 660a 7468 6520 6879 7065 7273 is of.the hypers │ │ │ │ +00016050: 7572 6661 6365 2064 6566 696e 6564 2062 urface defined b │ │ │ │ +00016060: 7920 7468 6520 6669 7273 7420 706f 6c79 y the first poly │ │ │ │ +00016070: 6e6f 6d69 616c 2067 656e 6572 6174 696e nomial generatin │ │ │ │ +00016080: 6720 4b29 2061 6e64 2074 6865 2043 534d g K) and the CSM │ │ │ │ +00016090: 0a63 6c61 7373 206f 6620 7468 6520 6879 .class of the hy │ │ │ │ +000160a0: 7065 7273 7572 6661 6365 2064 6566 696e persurface defin │ │ │ │ +000160b0: 6564 2062 7920 7468 6520 7072 6f64 7563 ed by the produc │ │ │ │ +000160c0: 7420 6f66 2074 6865 2067 656e 6572 6174 t of the generat │ │ │ │ +000160d0: 6f72 7320 6f66 204b 2e0a 0a2b 2d2d 2d2d ors of K...+---- │ │ │ │ +000160e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000160f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016110: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 -------+.|i19 : │ │ │ │ +00016120: 6d3d 6e65 7720 4d75 7461 626c 6548 6173 m=new MutableHas │ │ │ │ +00016130: 6854 6162 6c65 3b20 2020 2020 2020 2020 hTable; │ │ │ │ +00016140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016150: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00016160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00016180: 6931 3920 3a20 6d3d 6e65 7720 4d75 7461 i19 : m=new Muta │ │ │ │ -00016190: 626c 6548 6173 6854 6162 6c65 3b20 2020 bleHashTable; │ │ │ │ -000161a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000161b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000161d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000161e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000161f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020 ---------+.|i20 │ │ │ │ -00016200: 3a20 6d23 7b30 7d3d 6373 6d4b 4861 7368 : m#{0}=csmKHash │ │ │ │ -00016210: 237b 307d 2020 2020 2020 2020 2020 2020 #{0} │ │ │ │ -00016220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016230: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00016240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016270: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00016280: 3220 3220 2020 2020 3220 2020 2020 2020 2 2 2 │ │ │ │ -00016290: 2020 3220 2020 2020 3220 2020 2020 2020 2 2 │ │ │ │ -000162a0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -000162b0: 2020 207c 0a7c 6f32 3020 3d20 3868 2068 |.|o20 = 8h h │ │ │ │ -000162c0: 2020 2b20 3768 2068 2020 2b20 3668 2068 + 7h h + 6h h │ │ │ │ -000162d0: 2020 2b20 3268 2020 2b20 3568 2068 2020 + 2h + 5h h │ │ │ │ -000162e0: 2b20 3268 2020 2b20 3268 2020 2b20 6820 + 2h + 2h + h │ │ │ │ -000162f0: 207c 0a7c 2020 2020 2020 2020 3120 3220 |.| 1 2 │ │ │ │ -00016300: 2020 2020 3120 3220 2020 2020 3120 3220 1 2 1 2 │ │ │ │ -00016310: 2020 2020 3120 2020 2020 3120 3220 2020 1 1 2 │ │ │ │ -00016320: 2020 3220 2020 2020 3120 2020 2032 207c 2 1 2 | │ │ │ │ -00016330: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00016340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016360: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00016370: 6f32 3020 3a20 4120 2020 2020 2020 2020 o20 : A │ │ │ │ -00016380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000163a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -000163b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000163c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000163d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000163e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 ---------+.|i21 │ │ │ │ -000163f0: 3a20 6d23 7b30 2c31 7d3d 6373 6d4b 4861 : m#{0,1}=csmKHa │ │ │ │ -00016400: 7368 237b 302c 317d 2020 2020 2020 2020 sh#{0,1} │ │ │ │ -00016410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016420: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00016430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016460: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00016470: 3220 3220 2020 2020 3220 2020 2020 2020 2 2 2 │ │ │ │ -00016480: 2020 3220 2020 2020 3220 2020 2020 2020 2 2 │ │ │ │ -00016490: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -000164a0: 2020 207c 0a7c 6f32 3120 3d20 3968 2068 |.|o21 = 9h h │ │ │ │ -000164b0: 2020 2b20 3968 2068 2020 2b20 3968 2068 + 9h h + 9h h │ │ │ │ -000164c0: 2020 2b20 3368 2020 2b20 3768 2068 2020 + 3h + 7h h │ │ │ │ -000164d0: 2b20 3368 2020 2b20 3368 2020 2b20 3268 + 3h + 3h + 2h │ │ │ │ -000164e0: 207c 0a7c 2020 2020 2020 2020 3120 3220 |.| 1 2 │ │ │ │ -000164f0: 2020 2020 3120 3220 2020 2020 3120 3220 1 2 1 2 │ │ │ │ -00016500: 2020 2020 3120 2020 2020 3120 3220 2020 1 1 2 │ │ │ │ -00016510: 2020 3220 2020 2020 3120 2020 2020 327c 2 1 2| │ │ │ │ -00016520: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00016530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016550: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00016560: 6f32 3120 3a20 4120 2020 2020 2020 2020 o21 : A │ │ │ │ -00016570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016590: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -000165a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000165b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000165c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000165d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220 ---------+.|i22 │ │ │ │ -000165e0: 3a20 7469 6d65 2043 534d 2841 2c4b 2c6d : time CSM(A,K,m │ │ │ │ -000165f0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00016170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016190: 2d2d 2d2b 0a7c 6932 3020 3a20 6d23 7b30 ---+.|i20 : m#{0 │ │ │ │ +000161a0: 7d3d 6373 6d4b 4861 7368 237b 307d 2020 }=csmKHash#{0} │ │ │ │ +000161b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000161c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000161d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000161e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000161f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016200: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00016210: 0a7c 2020 2020 2020 2020 3220 3220 2020 .| 2 2 │ │ │ │ +00016220: 2020 3220 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ +00016230: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00016240: 3220 2020 2020 2020 2020 2020 207c 0a7c 2 |.| │ │ │ │ +00016250: 6f32 3020 3d20 3868 2068 2020 2b20 3768 o20 = 8h h + 7h │ │ │ │ +00016260: 2068 2020 2b20 3668 2068 2020 2b20 3268 h + 6h h + 2h │ │ │ │ +00016270: 2020 2b20 3568 2068 2020 2b20 3268 2020 + 5h h + 2h │ │ │ │ +00016280: 2b20 3268 2020 2b20 6820 207c 0a7c 2020 + 2h + h |.| │ │ │ │ +00016290: 2020 2020 2020 3120 3220 2020 2020 3120 1 2 1 │ │ │ │ +000162a0: 3220 2020 2020 3120 3220 2020 2020 3120 2 1 2 1 │ │ │ │ +000162b0: 2020 2020 3120 3220 2020 2020 3220 2020 1 2 2 │ │ │ │ +000162c0: 2020 3120 2020 2032 207c 0a7c 2020 2020 1 2 |.| │ │ │ │ +000162d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000162e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000162f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016300: 2020 2020 2020 207c 0a7c 6f32 3020 3a20 |.|o20 : │ │ │ │ +00016310: 4120 2020 2020 2020 2020 2020 2020 2020 A │ │ │ │ +00016320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016340: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00016350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016380: 2d2d 2d2b 0a7c 6932 3120 3a20 6d23 7b30 ---+.|i21 : m#{0 │ │ │ │ +00016390: 2c31 7d3d 6373 6d4b 4861 7368 237b 302c ,1}=csmKHash#{0, │ │ │ │ +000163a0: 317d 2020 2020 2020 2020 2020 2020 2020 1} │ │ │ │ +000163b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000163c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000163d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000163e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000163f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00016400: 0a7c 2020 2020 2020 2020 3220 3220 2020 .| 2 2 │ │ │ │ +00016410: 2020 3220 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ +00016420: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00016430: 3220 2020 2020 2020 2020 2020 207c 0a7c 2 |.| │ │ │ │ +00016440: 6f32 3120 3d20 3968 2068 2020 2b20 3968 o21 = 9h h + 9h │ │ │ │ +00016450: 2068 2020 2b20 3968 2068 2020 2b20 3368 h + 9h h + 3h │ │ │ │ +00016460: 2020 2b20 3768 2068 2020 2b20 3368 2020 + 7h h + 3h │ │ │ │ +00016470: 2b20 3368 2020 2b20 3268 207c 0a7c 2020 + 3h + 2h |.| │ │ │ │ +00016480: 2020 2020 2020 3120 3220 2020 2020 3120 1 2 1 │ │ │ │ +00016490: 3220 2020 2020 3120 3220 2020 2020 3120 2 1 2 1 │ │ │ │ +000164a0: 2020 2020 3120 3220 2020 2020 3220 2020 1 2 2 │ │ │ │ +000164b0: 2020 3120 2020 2020 327c 0a7c 2020 2020 1 2|.| │ │ │ │ +000164c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000164d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000164e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000164f0: 2020 2020 2020 207c 0a7c 6f32 3120 3a20 |.|o21 : │ │ │ │ +00016500: 4120 2020 2020 2020 2020 2020 2020 2020 A │ │ │ │ +00016510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016530: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00016540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016570: 2d2d 2d2b 0a7c 6932 3220 3a20 7469 6d65 ---+.|i22 : time │ │ │ │ +00016580: 2043 534d 2841 2c4b 2c6d 2920 2020 2020 CSM(A,K,m) │ │ │ │ +00016590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000165a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000165b0: 207c 0a7c 202d 2d20 7573 6564 2030 2e31 |.| -- used 0.1 │ │ │ │ +000165c0: 3130 3739 3273 2028 6370 7529 3b20 302e 10792s (cpu); 0. │ │ │ │ +000165d0: 3037 3837 3239 3573 2028 7468 7265 6164 0787295s (thread │ │ │ │ +000165e0: 293b 2030 7320 2867 6329 2020 2020 207c ); 0s (gc) | │ │ │ │ +000165f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00016600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016610: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00016620: 6564 2030 2e30 3537 3738 3638 7320 2863 ed 0.0577868s (c │ │ │ │ -00016630: 7075 293b 2030 2e30 3536 3832 3773 2028 pu); 0.056827s ( │ │ │ │ -00016640: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -00016650: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00016660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016690: 2020 207c 0a7c 2020 2020 2020 2020 3220 |.| 2 │ │ │ │ -000166a0: 3220 2020 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ -000166b0: 3220 2020 2032 2020 2020 2020 2020 2020 2 2 │ │ │ │ -000166c0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000166d0: 207c 0a7c 6f32 3220 3d20 3768 2068 2020 |.|o22 = 7h h │ │ │ │ -000166e0: 2b20 3568 2068 2020 2b20 3468 2068 2020 + 5h h + 4h h │ │ │ │ -000166f0: 2b20 6820 202b 2033 6820 6820 202b 2068 + h + 3h h + h │ │ │ │ -00016700: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016710: 0a7c 2020 2020 2020 2020 3120 3220 2020 .| 1 2 │ │ │ │ -00016720: 2020 3120 3220 2020 2020 3120 3220 2020 1 2 1 2 │ │ │ │ -00016730: 2031 2020 2020 2031 2032 2020 2020 3220 1 1 2 2 │ │ │ │ -00016740: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00016610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016620: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00016630: 2020 2020 2020 2020 3220 3220 2020 2020 2 2 │ │ │ │ +00016640: 3220 2020 2020 2020 2020 3220 2020 2032 2 2 2 │ │ │ │ +00016650: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +00016660: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +00016670: 3220 3d20 3768 2068 2020 2b20 3568 2068 2 = 7h h + 5h h │ │ │ │ +00016680: 2020 2b20 3468 2068 2020 2b20 6820 202b + 4h h + h + │ │ │ │ +00016690: 2033 6820 6820 202b 2068 2020 2020 2020 3h h + h │ │ │ │ +000166a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000166b0: 2020 2020 3120 3220 2020 2020 3120 3220 1 2 1 2 │ │ │ │ +000166c0: 2020 2020 3120 3220 2020 2031 2020 2020 1 2 1 │ │ │ │ +000166d0: 2031 2032 2020 2020 3220 2020 2020 2020 1 2 2 │ │ │ │ +000166e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000166f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016720: 2020 2020 207c 0a7c 6f32 3220 3a20 4120 |.|o22 : A │ │ │ │ +00016730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016780: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00016790: 3220 3a20 4120 2020 2020 2020 2020 2020 2 : A │ │ │ │ -000167a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000167b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000167c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000167d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000167e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000167f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016800: 2d2d 2d2d 2d2d 2d2b 0a0a 496e 2074 6865 -------+..In the │ │ │ │ -00016810: 2063 6173 6520 7768 6572 6520 7468 6520 case where the │ │ │ │ -00016820: 616d 6269 656e 7420 7370 6163 6520 6973 ambient space is │ │ │ │ -00016830: 2061 2074 6f72 6963 2076 6172 6965 7479 a toric variety │ │ │ │ -00016840: 2077 6869 6368 2069 7320 6e6f 7420 6120 which is not a │ │ │ │ -00016850: 7072 6f64 7563 740a 6f66 2070 726f 6a65 product.of proje │ │ │ │ -00016860: 6374 6976 6520 7370 6163 6573 2077 6520 ctive spaces we │ │ │ │ -00016870: 6d75 7374 206c 6f61 6420 7468 6520 4e6f must load the No │ │ │ │ -00016880: 726d 616c 546f 7269 6356 6172 6965 7469 rmalToricVarieti │ │ │ │ -00016890: 6573 2070 6163 6b61 6765 2061 6e64 206d es package and m │ │ │ │ -000168a0: 7573 740a 616c 736f 2069 6e70 7574 2074 ust.also input t │ │ │ │ -000168b0: 6865 2074 6f72 6963 2076 6172 6965 7479 he toric variety │ │ │ │ -000168c0: 2e20 4966 2074 6865 2074 6f72 6963 2076 . If the toric v │ │ │ │ -000168d0: 6172 6965 7479 2069 7320 6120 7072 6f64 ariety is a prod │ │ │ │ -000168e0: 7563 7420 6f66 2070 726f 6a65 6374 6976 uct of projectiv │ │ │ │ -000168f0: 650a 7370 6163 6520 6974 2069 7320 7265 e.space it is re │ │ │ │ -00016900: 636f 6d6d 656e 6420 746f 2075 7365 2074 commend to use t │ │ │ │ -00016910: 6865 2066 6f72 6d20 6162 6f76 6520 7261 he form above ra │ │ │ │ -00016920: 7468 6572 2074 6861 6e20 696e 7075 7474 ther than inputt │ │ │ │ -00016930: 696e 6720 7468 6520 746f 7269 630a 7661 ing the toric.va │ │ │ │ -00016940: 7269 6574 7920 666f 7220 6566 6669 6369 riety for effici │ │ │ │ -00016950: 656e 6379 2072 6561 736f 6e73 2e0a 0a2b ency reasons...+ │ │ │ │ -00016960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000169a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a --------+.|i23 : │ │ │ │ -000169b0: 206e 6565 6473 5061 636b 6167 6520 224e needsPackage "N │ │ │ │ -000169c0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574 ormalToricVariet │ │ │ │ -000169d0: 6965 7322 2020 2020 2020 2020 2020 2020 ies" │ │ │ │ -000169e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000169f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00016760: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00016770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000167a0: 2d2b 0a0a 496e 2074 6865 2063 6173 6520 -+..In the case │ │ │ │ +000167b0: 7768 6572 6520 7468 6520 616d 6269 656e where the ambien │ │ │ │ +000167c0: 7420 7370 6163 6520 6973 2061 2074 6f72 t space is a tor │ │ │ │ +000167d0: 6963 2076 6172 6965 7479 2077 6869 6368 ic variety which │ │ │ │ +000167e0: 2069 7320 6e6f 7420 6120 7072 6f64 7563 is not a produc │ │ │ │ +000167f0: 740a 6f66 2070 726f 6a65 6374 6976 6520 t.of projective │ │ │ │ +00016800: 7370 6163 6573 2077 6520 6d75 7374 206c spaces we must l │ │ │ │ +00016810: 6f61 6420 7468 6520 4e6f 726d 616c 546f oad the NormalTo │ │ │ │ +00016820: 7269 6356 6172 6965 7469 6573 2070 6163 ricVarieties pac │ │ │ │ +00016830: 6b61 6765 2061 6e64 206d 7573 740a 616c kage and must.al │ │ │ │ +00016840: 736f 2069 6e70 7574 2074 6865 2074 6f72 so input the tor │ │ │ │ +00016850: 6963 2076 6172 6965 7479 2e20 4966 2074 ic variety. If t │ │ │ │ +00016860: 6865 2074 6f72 6963 2076 6172 6965 7479 he toric variety │ │ │ │ +00016870: 2069 7320 6120 7072 6f64 7563 7420 6f66 is a product of │ │ │ │ +00016880: 2070 726f 6a65 6374 6976 650a 7370 6163 projective.spac │ │ │ │ +00016890: 6520 6974 2069 7320 7265 636f 6d6d 656e e it is recommen │ │ │ │ +000168a0: 6420 746f 2075 7365 2074 6865 2066 6f72 d to use the for │ │ │ │ +000168b0: 6d20 6162 6f76 6520 7261 7468 6572 2074 m above rather t │ │ │ │ +000168c0: 6861 6e20 696e 7075 7474 696e 6720 7468 han inputting th │ │ │ │ +000168d0: 6520 746f 7269 630a 7661 7269 6574 7920 e toric.variety │ │ │ │ +000168e0: 666f 7220 6566 6669 6369 656e 6379 2072 for efficiency r │ │ │ │ +000168f0: 6561 736f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d easons...+------ │ │ │ │ +00016900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016940: 2d2d 2b0a 7c69 3233 203a 206e 6565 6473 --+.|i23 : needs │ │ │ │ +00016950: 5061 636b 6167 6520 224e 6f72 6d61 6c54 Package "NormalT │ │ │ │ +00016960: 6f72 6963 5661 7269 6574 6965 7322 2020 oricVarieties" │ │ │ │ +00016970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00016990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000169a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000169b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000169c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000169d0: 2020 2020 2020 2020 7c0a 7c6f 3233 203d |.|o23 = │ │ │ │ +000169e0: 204e 6f72 6d61 6c54 6f72 6963 5661 7269 NormalToricVari │ │ │ │ +000169f0: 6574 6965 7320 2020 2020 2020 2020 2020 eties │ │ │ │ 00016a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016a30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00016a40: 7c6f 3233 203d 204e 6f72 6d61 6c54 6f72 |o23 = NormalTor │ │ │ │ -00016a50: 6963 5661 7269 6574 6965 7320 2020 2020 icVarieties │ │ │ │ -00016a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016a80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00016a20: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00016a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016a60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00016a70: 7c6f 3233 203a 2050 6163 6b61 6765 2020 |o23 : Package │ │ │ │ +00016a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ad0: 2020 2020 7c0a 7c6f 3233 203a 2050 6163 |.|o23 : Pac │ │ │ │ -00016ae0: 6b61 6765 2020 2020 2020 2020 2020 2020 kage │ │ │ │ -00016af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016b10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016b20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00016b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234 ----------+.|i24 │ │ │ │ -00016b70: 203a 2052 686f 203d 207b 7b31 2c30 2c30 : Rho = {{1,0,0 │ │ │ │ -00016b80: 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30 2c31 },{0,1,0},{0,0,1 │ │ │ │ -00016b90: 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30 2c30 },{-1,-1,0},{0,0 │ │ │ │ -00016ba0: 2c2d 317d 7d20 2020 2020 2020 2020 2020 ,-1}} │ │ │ │ -00016bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00016bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016ab0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00016ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016b00: 2d2d 2d2d 2b0a 7c69 3234 203a 2052 686f ----+.|i24 : Rho │ │ │ │ +00016b10: 203d 207b 7b31 2c30 2c30 7d2c 7b30 2c31 = {{1,0,0},{0,1 │ │ │ │ +00016b20: 2c30 7d2c 7b30 2c30 2c31 7d2c 7b2d 312c ,0},{0,0,1},{-1, │ │ │ │ +00016b30: 2d31 2c30 7d2c 7b30 2c30 2c2d 317d 7d20 -1,0},{0,0,-1}} │ │ │ │ +00016b40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00016b50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016b90: 2020 2020 2020 2020 2020 7c0a 7c6f 3234 |.|o24 │ │ │ │ +00016ba0: 203d 207b 7b31 2c20 302c 2030 7d2c 207b = {{1, 0, 0}, { │ │ │ │ +00016bb0: 302c 2031 2c20 307d 2c20 7b30 2c20 302c 0, 1, 0}, {0, 0, │ │ │ │ +00016bc0: 2031 7d2c 207b 2d31 2c20 2d31 2c20 307d 1}, {-1, -1, 0} │ │ │ │ +00016bd0: 2c20 7b30 2c20 302c 202d 317d 7d20 2020 , {0, 0, -1}} │ │ │ │ +00016be0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00016bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016c00: 7c0a 7c6f 3234 203d 207b 7b31 2c20 302c |.|o24 = {{1, 0, │ │ │ │ -00016c10: 2030 7d2c 207b 302c 2031 2c20 307d 2c20 0}, {0, 1, 0}, │ │ │ │ -00016c20: 7b30 2c20 302c 2031 7d2c 207b 2d31 2c20 {0, 0, 1}, {-1, │ │ │ │ -00016c30: 2d31 2c20 307d 2c20 7b30 2c20 302c 202d -1, 0}, {0, 0, - │ │ │ │ -00016c40: 317d 7d20 2020 2020 2020 207c 0a7c 2020 1}} |.| │ │ │ │ +00016c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016c30: 7c0a 7c6f 3234 203a 204c 6973 7420 2020 |.|o24 : List │ │ │ │ +00016c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016c90: 2020 2020 2020 7c0a 7c6f 3234 203a 204c |.|o24 : L │ │ │ │ -00016ca0: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ -00016cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ce0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00016cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00016d30: 3235 203a 2053 6967 6d61 203d 207b 7b30 25 : Sigma = {{0 │ │ │ │ -00016d40: 2c31 2c32 7d2c 7b31 2c32 2c33 7d2c 7b30 ,1,2},{1,2,3},{0 │ │ │ │ -00016d50: 2c32 2c33 7d2c 7b30 2c31 2c34 7d2c 7b31 ,2,3},{0,1,4},{1 │ │ │ │ -00016d60: 2c33 2c34 7d2c 7b30 2c33 2c34 7d7d 2020 ,3,4},{0,3,4}} │ │ │ │ -00016d70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00016d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016c70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 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207b 7b30 2c20 312c 2032 7d2c 25 = {{0, 1, 2}, │ │ │ │ +00016d70: 207b 312c 2032 2c20 337d 2c20 7b30 2c20 {1, 2, 3}, {0, │ │ │ │ +00016d80: 322c 2033 7d2c 207b 302c 2031 2c20 347d 2, 3}, {0, 1, 4} │ │ │ │ +00016d90: 2c20 7b31 2c20 332c 2034 7d2c 207b 302c , {1, 3, 4}, {0, │ │ │ │ +00016da0: 2033 2c20 347d 7d7c 0a7c 2020 2020 2020 3, 4}}|.| │ │ │ │ 00016db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016dc0: 2020 7c0a 7c6f 3235 203d 207b 7b30 2c20 |.|o25 = {{0, │ │ │ │ -00016dd0: 312c 2032 7d2c 207b 312c 2032 2c20 337d 1, 2}, {1, 2, 3} │ │ │ │ -00016de0: 2c20 7b30 2c20 322c 2033 7d2c 207b 302c , {0, 2, 3}, {0, │ │ │ │ -00016df0: 2031 2c20 347d 2c20 7b31 2c20 332c 2034 1, 4}, {1, 3, 4 │ │ │ │ -00016e00: 7d2c 207b 302c 2033 2c20 347d 7d7c 0a7c }, {0, 3, 4}}|.| │ │ │ │ +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 2020 │ │ │ │ +00016df0: 2020 7c0a 7c6f 3235 203a 204c 6973 7420 |.|o25 : List │ │ │ │ +00016e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e50: 2020 2020 2020 2020 7c0a 7c6f 3235 203a |.|o25 : │ │ │ │ -00016e60: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ -00016e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ea0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00016eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00016ef0: 7c69 3236 203a 2058 203d 206e 6f72 6d61 |i26 : X = norma │ │ │ │ -00016f00: 6c54 6f72 6963 5661 7269 6574 7928 5268 lToricVariety(Rh │ │ │ │ -00016f10: 6f2c 5369 676d 612c 436f 6566 6669 6369 o,Sigma,Coeffici │ │ │ │ -00016f20: 656e 7452 696e 6720 3d3e 5a5a 2f33 3237 entRing =>ZZ/327 │ │ │ │ -00016f30: 3439 2920 2020 2020 207c 0a7c 2020 2020 49) |.| │ │ │ │ +00016e30: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00016e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016e80: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3236 203a --------+.|i26 : │ │ │ │ +00016e90: 2058 203d 206e 6f72 6d61 6c54 6f72 6963 X = normalToric │ │ │ │ +00016ea0: 5661 7269 6574 7928 5268 6f2c 5369 676d Variety(Rho,Sigm │ │ │ │ +00016eb0: 612c 436f 6566 6669 6369 656e 7452 696e a,CoefficientRin │ │ │ │ +00016ec0: 6720 3d3e 5a5a 2f33 3237 3439 2920 2020 g =>ZZ/32749) │ │ │ │ +00016ed0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00016ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 7c0a |. │ │ │ │ +00016f20: 7c6f 3236 203d 2058 2020 2020 2020 2020 |o26 = X │ │ │ │ +00016f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016f60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00016f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016f80: 2020 2020 7c0a 7c6f 3236 203d 2058 2020 |.|o26 = X │ │ │ │ +00016f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 2020 2020 2020 │ │ │ │ -00017760: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017750: 2020 7c0a 7c6f 3330 203d 2052 2020 2020 |.|o30 = R │ │ │ │ +00017760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017790: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000177a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000177b0: 2020 2020 2020 2020 7c0a 7c6f 3330 203d |.|o30 = │ │ │ │ -000177c0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +000177b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000177c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000177d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000177e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000177f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017800: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000177e0: 2020 2020 2020 2020 7c0a 7c6f 3330 203a |.|o30 : │ │ │ │ +000177f0: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ +00017800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017840: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00017850: 7c6f 3330 203a 2050 6f6c 796e 6f6d 6961 |o30 : Polynomia │ │ │ │ -00017860: 6c52 696e 6720 2020 2020 2020 2020 2020 lRing │ │ │ │ -00017870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017890: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000178a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000178b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000178c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000178d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000178e0: 2d2d 2d2d 2b0a 7c69 3331 203a 2049 3d69 ----+.|i31 : I=i │ │ │ │ -000178f0: 6465 616c 2852 5f30 5e34 2a52 5f31 2c52 deal(R_0^4*R_1,R │ │ │ │ -00017900: 5f30 2a52 5f33 2a52 5f34 2a52 5f32 2d52 _0*R_3*R_4*R_2-R │ │ │ │ -00017910: 5f32 5e32 2a52 5f30 5e32 2920 2020 2020 _2^2*R_0^2) │ │ │ │ -00017920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017930: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00017830: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00017840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00017880: 7c69 3331 203a 2049 3d69 6465 616c 2852 |i31 : I=ideal(R │ │ │ │ +00017890: 5f30 5e34 2a52 5f31 2c52 5f30 2a52 5f33 _0^4*R_1,R_0*R_3 │ │ │ │ +000178a0: 2a52 5f34 2a52 5f32 2d52 5f32 5e32 2a52 *R_4*R_2-R_2^2*R │ │ │ │ +000178b0: 5f30 5e32 2920 2020 2020 2020 2020 2020 _0^2) │ │ │ │ +000178c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000178d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000178e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000178f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017910: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00017920: 2020 2020 2034 2020 2020 2020 2032 2032 4 2 2 │ │ │ │ +00017930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017970: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00017980: 2020 2020 2020 2020 2020 2034 2020 2020 4 │ │ │ │ -00017990: 2020 2032 2032 2020 2020 2020 2020 2020 2 2 │ │ │ │ -000179a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000179b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000179c0: 2020 2020 207c 0a7c 6f33 3120 3d20 6964 |.|o31 = id │ │ │ │ -000179d0: 6561 6c20 2878 2078 202c 202d 2078 2078 eal (x x , - x x │ │ │ │ -000179e0: 2020 2b20 7820 7820 7820 7820 2920 2020 + x x x x ) │ │ │ │ -000179f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017950: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00017960: 0a7c 6f33 3120 3d20 6964 6561 6c20 2878 .|o31 = ideal (x │ │ │ │ +00017970: 2078 202c 202d 2078 2078 2020 2b20 7820 x , - x x + x │ │ │ │ +00017980: 7820 7820 7820 2920 2020 2020 2020 2020 x x x ) │ │ │ │ +00017990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000179a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000179b0: 2020 2020 2020 2020 2020 2030 2031 2020 0 1 │ │ │ │ +000179c0: 2020 2030 2032 2020 2020 3020 3220 3320 0 2 0 2 3 │ │ │ │ +000179d0: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +000179e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000179f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00017a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017a10: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00017a20: 2030 2031 2020 2020 2030 2032 2020 2020 0 1 0 2 │ │ │ │ -00017a30: 3020 3220 3320 3420 2020 2020 2020 2020 0 2 3 4 │ │ │ │ -00017a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017a50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00017a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017a40: 7c0a 7c6f 3331 203a 2049 6465 616c 206f |.|o31 : Ideal o │ │ │ │ +00017a50: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ 00017a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017aa0: 2020 2020 2020 7c0a 7c6f 3331 203a 2049 |.|o31 : I │ │ │ │ -00017ab0: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ -00017ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017af0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00017b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00017b40: 3332 203a 2043 534d 2858 2c49 2920 2020 32 : CSM(X,I) │ │ │ │ +00017a80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00017a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017ad0: 2d2d 2d2d 2d2d 2b0a 7c69 3332 203a 2043 ------+.|i32 : C │ │ │ │ +00017ae0: 534d 2858 2c49 2920 2020 2020 2020 2020 SM(X,I) │ │ │ │ +00017af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017b20: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00017b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017b80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00017b60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00017b70: 2020 2020 2020 2032 2020 2020 2020 2032 2 2 │ │ │ │ +00017b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017bd0: 2020 7c0a 7c20 2020 2020 2020 2032 2020 |.| 2 │ │ │ │ -00017be0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00017bb0: 2020 2020 2020 207c 0a7c 6f33 3220 3d20 |.|o32 = │ │ │ │ +00017bc0: 3578 2078 2020 2b20 3378 2020 2b20 3478 5x x + 3x + 4x │ │ │ │ +00017bd0: 2078 2020 2b20 7820 2020 2020 2020 2020 x + x │ │ │ │ +00017be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017c10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00017c20: 6f33 3220 3d20 3578 2078 2020 2b20 3378 o32 = 5x x + 3x │ │ │ │ -00017c30: 2020 2b20 3478 2078 2020 2b20 7820 2020 + 4x x + x │ │ │ │ -00017c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017c00: 2020 7c0a 7c20 2020 2020 2020 2033 2034 |.| 3 4 │ │ │ │ +00017c10: 2020 2020 2033 2020 2020 2033 2034 2020 3 3 4 │ │ │ │ +00017c20: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00017c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017c40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00017c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017c60: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00017c70: 2020 2033 2034 2020 2020 2033 2020 2020 3 4 3 │ │ │ │ -00017c80: 2033 2034 2020 2020 3320 2020 2020 2020 3 4 3 │ │ │ │ -00017c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017c90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00017ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00017cb0: 205a 5a5b 7820 2e2e 7820 5d20 2020 2020 ZZ[x ..x ] │ │ │ │ 00017cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017cf0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00017d00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00017d10: 2020 2020 2020 205a 5a5b 7820 2e2e 7820 ZZ[x ..x │ │ │ │ -00017d20: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -00017d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017d40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00017d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017d60: 2020 2020 2020 3020 2020 3420 2020 2020 0 4 │ │ │ │ -00017d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017d90: 2020 2020 7c0a 7c6f 3332 203a 202d 2d2d |.|o32 : --- │ │ │ │ -00017da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017dc0: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020 ------ │ │ │ │ -00017dd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017de0: 0a7c 2020 2020 2020 2878 2078 202c 2078 .| (x x , x │ │ │ │ -00017df0: 2078 2078 202c 2078 2020 2d20 7820 2c20 x x , x - x , │ │ │ │ -00017e00: 7820 202d 2078 202c 2078 2020 2d20 7820 x - x , x - x │ │ │ │ -00017e10: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -00017e20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00017e30: 2020 2020 2032 2034 2020 2030 2031 2033 2 4 0 1 3 │ │ │ │ -00017e40: 2020 2030 2020 2020 3320 2020 3120 2020 0 3 1 │ │ │ │ -00017e50: 2033 2020 2032 2020 2020 3420 2020 2020 3 2 4 │ │ │ │ -00017e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017e70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017ec0: 2b0a 7c69 3333 203a 2043 534d 2843 682c +.|i33 : CSM(Ch, │ │ │ │ -00017ed0: 582c 4929 2020 2020 2020 2020 2020 2020 X,I) │ │ │ │ +00017ce0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00017cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017d00: 3020 2020 3420 2020 2020 2020 2020 2020 0 4 │ │ │ │ +00017d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017d20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00017d30: 7c6f 3332 203a 202d 2d2d 2d2d 2d2d 2d2d |o32 : --------- │ │ │ │ +00017d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017d70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00017d80: 2020 2878 2078 202c 2078 2078 2078 202c (x x , x x x , │ │ │ │ +00017d90: 2078 2020 2d20 7820 2c20 7820 202d 2078 x - x , x - x │ │ │ │ +00017da0: 202c 2078 2020 2d20 7820 2920 2020 2020 , x - x ) │ │ │ │ +00017db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017dc0: 2020 2020 7c0a 7c20 2020 2020 2020 2032 |.| 2 │ │ │ │ +00017dd0: 2034 2020 2030 2031 2033 2020 2030 2020 4 0 1 3 0 │ │ │ │ +00017de0: 2020 3320 2020 3120 2020 2033 2020 2032 3 1 3 2 │ │ │ │ +00017df0: 2020 2020 3420 2020 2020 2020 2020 2020 4 │ │ │ │ +00017e00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00017e10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00017e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3333 ----------+.|i33 │ │ │ │ +00017e60: 203a 2043 534d 2843 682c 582c 4929 2020 : CSM(Ch,X,I) │ │ │ │ +00017e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017ea0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00017eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017ed0: 2020 2020 2020 2020 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 207c 0a7c 2020 |.| │ │ │ │ +00017ef0: 7c0a 7c20 2020 2020 2020 2032 2020 2020 |.| 2 │ │ │ │ +00017f00: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 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 2020 2020 2020 2020 │ │ │ │ -00017f50: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00017f60: 2032 2020 2020 2020 2032 2020 2020 2020 2 2 │ │ │ │ +00017f30: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +00017f40: 3320 3d20 3878 2078 2020 2b20 3378 2020 3 = 8x x + 3x │ │ │ │ +00017f50: 2b20 3578 2078 2020 2b20 7820 2020 2020 + 5x x + x │ │ │ │ +00017f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017fa0: 207c 0a7c 6f33 3320 3d20 3878 2078 2020 |.|o33 = 8x x │ │ │ │ -00017fb0: 2b20 3378 2020 2b20 3578 2078 2020 2b20 + 3x + 5x x + │ │ │ │ -00017fc0: 7820 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ -00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00017ff0: 2020 2020 2020 2033 2034 2020 2020 2033 3 4 3 │ │ │ │ -00018000: 2020 2020 2033 2034 2020 2020 3320 2020 3 4 3 │ │ │ │ -00018010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018030: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00017f80: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00017f90: 2033 2034 2020 2020 2033 2020 2020 2033 3 4 3 3 │ │ │ │ +00017fa0: 2034 2020 2020 3320 2020 2020 2020 2020 4 3 │ │ │ │ +00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017fd0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018010: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00018020: 3333 203a 2043 6820 2020 2020 2020 2020 33 : Ch │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018080: 2020 7c0a 7c6f 3333 203a 2043 6820 2020 |.|o33 : Ch │ │ │ │ -00018090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -000180d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000180e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000180f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018110: 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6869 7320 --------+..This │ │ │ │ -00018120: 6675 6e63 7469 6f6e 206d 6179 2061 6c73 function may als │ │ │ │ -00018130: 6f20 636f 6d70 7574 6520 7468 6520 4353 o compute the CS │ │ │ │ -00018140: 4d20 636c 6173 7320 6f66 2061 206e 6f72 M class of a nor │ │ │ │ -00018150: 6d61 6c20 746f 7269 6320 7661 7269 6574 mal toric variet │ │ │ │ -00018160: 7920 6465 6669 6e65 640a 6279 2061 2066 y defined.by a f │ │ │ │ -00018170: 616e 2e20 496e 2074 6869 7320 6361 7365 an. In this case │ │ │ │ -00018180: 2061 2063 6f6d 6269 6e61 746f 7269 616c a combinatorial │ │ │ │ -00018190: 206d 6574 686f 6420 6973 2075 7365 642e method is used. │ │ │ │ -000181a0: 2054 6869 7320 6d65 7468 6f64 2069 7320 This method is │ │ │ │ -000181b0: 6163 6365 7373 6564 0a77 6974 6820 7468 accessed.with th │ │ │ │ -000181c0: 6520 7573 7561 6c20 4353 4d20 636f 6d6d e usual CSM comm │ │ │ │ -000181d0: 616e 6420 7769 7468 2065 6974 6865 7220 and with either │ │ │ │ -000181e0: 6f6e 6c79 2061 2074 6f72 6963 2076 6172 only a toric var │ │ │ │ -000181f0: 6965 7479 206f 7220 6120 746f 7269 6320 iety or a toric │ │ │ │ -00018200: 7661 7269 6574 790a 616e 6420 6120 4368 variety.and a Ch │ │ │ │ -00018210: 6f77 2072 696e 6720 6173 2069 6e70 7574 ow ring as input │ │ │ │ -00018220: 2e20 496e 2074 6869 7320 6361 7365 2077 . In this case w │ │ │ │ -00018230: 6520 6f6e 6c79 2072 6571 7569 7265 2074 e only require t │ │ │ │ -00018240: 6861 7420 7468 6520 696e 7075 7420 746f hat the input to │ │ │ │ -00018250: 7269 630a 7661 7269 6574 7920 6973 2063 ric.variety is c │ │ │ │ -00018260: 6f6d 706c 6574 6520 616e 6420 7369 6d70 omplete and simp │ │ │ │ -00018270: 6c69 6369 616c 2028 696e 2070 6172 7469 licial (in parti │ │ │ │ -00018280: 6375 6c61 7220 7765 2064 6f20 6e6f 7420 cular we do not │ │ │ │ -00018290: 6e65 6564 2069 7420 746f 2062 650a 736d need it to be.sm │ │ │ │ -000182a0: 6f6f 7468 292e 0a0a 2b2d 2d2d 2d2d 2d2d ooth)...+------- │ │ │ │ -000182b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000182c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000182d0: 2d2d 2d2b 0a7c 6933 3420 3a20 6e65 6564 ---+.|i34 : need │ │ │ │ -000182e0: 7350 6163 6b61 6765 2022 4e6f 726d 616c sPackage "Normal │ │ │ │ -000182f0: 546f 7269 6356 6172 6965 7469 6573 2220 ToricVarieties" │ │ │ │ -00018300: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018060: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00018070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000180a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000180b0: 2d2d 2b0a 0a54 6869 7320 6675 6e63 7469 --+..This functi │ │ │ │ +000180c0: 6f6e 206d 6179 2061 6c73 6f20 636f 6d70 on may also comp │ │ │ │ +000180d0: 7574 6520 7468 6520 4353 4d20 636c 6173 ute the CSM clas │ │ │ │ +000180e0: 7320 6f66 2061 206e 6f72 6d61 6c20 746f s of a normal to │ │ │ │ +000180f0: 7269 6320 7661 7269 6574 7920 6465 6669 ric variety defi │ │ │ │ +00018100: 6e65 640a 6279 2061 2066 616e 2e20 496e ned.by a fan. In │ │ │ │ +00018110: 2074 6869 7320 6361 7365 2061 2063 6f6d this case a com │ │ │ │ +00018120: 6269 6e61 746f 7269 616c 206d 6574 686f binatorial metho │ │ │ │ +00018130: 6420 6973 2075 7365 642e 2054 6869 7320 d is used. This │ │ │ │ +00018140: 6d65 7468 6f64 2069 7320 6163 6365 7373 method is access │ │ │ │ +00018150: 6564 0a77 6974 6820 7468 6520 7573 7561 ed.with the usua │ │ │ │ +00018160: 6c20 4353 4d20 636f 6d6d 616e 6420 7769 l CSM command wi │ │ │ │ +00018170: 7468 2065 6974 6865 7220 6f6e 6c79 2061 th either only a │ │ │ │ +00018180: 2074 6f72 6963 2076 6172 6965 7479 206f toric variety o │ │ │ │ +00018190: 7220 6120 746f 7269 6320 7661 7269 6574 r a toric variet │ │ │ │ +000181a0: 790a 616e 6420 6120 4368 6f77 2072 696e y.and a Chow rin │ │ │ │ +000181b0: 6720 6173 2069 6e70 7574 2e20 496e 2074 g as input. In t │ │ │ │ +000181c0: 6869 7320 6361 7365 2077 6520 6f6e 6c79 his case we only │ │ │ │ +000181d0: 2072 6571 7569 7265 2074 6861 7420 7468 require that th │ │ │ │ +000181e0: 6520 696e 7075 7420 746f 7269 630a 7661 e input toric.va │ │ │ │ +000181f0: 7269 6574 7920 6973 2063 6f6d 706c 6574 riety is complet │ │ │ │ +00018200: 6520 616e 6420 7369 6d70 6c69 6369 616c e and simplicial │ │ │ │ +00018210: 2028 696e 2070 6172 7469 6375 6c61 7220 (in particular │ │ │ │ +00018220: 7765 2064 6f20 6e6f 7420 6e65 6564 2069 we do not need i │ │ │ │ +00018230: 7420 746f 2062 650a 736d 6f6f 7468 292e t to be.smooth). │ │ │ │ +00018240: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +00018250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00018270: 6933 3420 3a20 6e65 6564 7350 6163 6b61 i34 : needsPacka │ │ │ │ +00018280: 6765 2022 4e6f 726d 616c 546f 7269 6356 ge "NormalToricV │ │ │ │ +00018290: 6172 6965 7469 6573 2220 7c0a 7c20 2020 arieties" |.| │ │ │ │ +000182a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000182b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000182c0: 2020 2020 2020 207c 0a7c 6f33 3420 3d20 |.|o34 = │ │ │ │ +000182d0: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ +000182e0: 7469 6573 2020 2020 2020 2020 2020 2020 ties │ │ │ │ +000182f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00018300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018320: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00018330: 6f33 3420 3d20 4e6f 726d 616c 546f 7269 o34 = NormalTori │ │ │ │ -00018340: 6356 6172 6965 7469 6573 2020 2020 2020 cVarieties │ │ │ │ -00018350: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00018360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018380: 2020 2020 2020 207c 0a7c 6f33 3420 3a20 |.|o34 : │ │ │ │ -00018390: 5061 636b 6167 6520 2020 2020 2020 2020 Package │ │ │ │ -000183a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000183b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -000183c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000183d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000183e0: 2d2b 0a7c 6933 3520 3a20 5520 3d20 6869 -+.|i35 : U = hi │ │ │ │ -000183f0: 727a 6562 7275 6368 5375 7266 6163 6520 rzebruchSurface │ │ │ │ -00018400: 3720 2020 2020 2020 2020 2020 2020 7c0a 7 |. │ │ │ │ -00018410: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00018420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018430: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -00018440: 3520 3d20 5520 2020 2020 2020 2020 2020 5 = U │ │ │ │ -00018450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018460: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00018470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018490: 2020 2020 207c 0a7c 6f33 3520 3a20 4e6f |.|o35 : No │ │ │ │ -000184a0: 726d 616c 546f 7269 6356 6172 6965 7479 rmalToricVariety │ │ │ │ -000184b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000184c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -000184d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000184e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000184f0: 0a7c 6933 3620 3a20 4368 3d54 6f72 6963 .|i36 : Ch=Toric │ │ │ │ -00018500: 4368 6f77 5269 6e67 2855 2920 2020 2020 ChowRing(U) │ │ │ │ -00018510: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00018320: 207c 0a7c 6f33 3420 3a20 5061 636b 6167 |.|o34 : Packag │ │ │ │ +00018330: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00018350: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00018360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ +00018380: 3520 3a20 5520 3d20 6869 727a 6562 7275 5 : U = hirzebru │ │ │ │ +00018390: 6368 5375 7266 6163 6520 3720 2020 2020 chSurface 7 │ │ │ │ +000183a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000183b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000183c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000183d0: 2020 2020 207c 0a7c 6f33 3520 3d20 5520 |.|o35 = U │ │ │ │ +000183e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000183f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018400: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018420: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00018430: 0a7c 6f33 3520 3a20 4e6f 726d 616c 546f .|o35 : NormalTo │ │ │ │ +00018440: 7269 6356 6172 6965 7479 2020 2020 2020 ricVariety │ │ │ │ +00018450: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00018460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018480: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3620 ---------+.|i36 │ │ │ │ +00018490: 3a20 4368 3d54 6f72 6963 4368 6f77 5269 : Ch=ToricChowRi │ │ │ │ +000184a0: 6e67 2855 2920 2020 2020 2020 2020 2020 ng(U) │ │ │ │ +000184b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000184c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000184d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000184e0: 2020 207c 0a7c 6f33 3620 3d20 4368 2020 |.|o36 = Ch │ │ │ │ +000184f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018510: 7c0a 7c20 2020 2020 2020 2020 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 207c 0a7c 6f33 3620 |.|o36 │ │ │ │ -00018550: 3d20 4368 2020 2020 2020 2020 2020 2020 = Ch │ │ │ │ -00018560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018570: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00018580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000185a0: 2020 207c 0a7c 6f33 3620 3a20 5175 6f74 |.|o36 : Quot │ │ │ │ -000185b0: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ -000185c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000185d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -000185e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000185f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00018600: 6933 3720 3a20 4353 4d20 5520 2020 2020 i37 : CSM U │ │ │ │ -00018610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018620: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00018630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018650: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00018660: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00018670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018680: 2020 2020 7c0a 7c6f 3337 203d 202d 2033 |.|o37 = - 3 │ │ │ │ -00018690: 7820 7820 202b 2078 2020 2d20 3578 2020 x x + x - 5x │ │ │ │ -000186a0: 2b20 3278 2020 2b20 3120 2020 2020 2020 + 2x + 1 │ │ │ │ -000186b0: 207c 0a7c 2020 2020 2020 2020 2020 3220 |.| 2 │ │ │ │ -000186c0: 3320 2020 2033 2020 2020 2032 2020 2020 3 3 2 │ │ │ │ -000186d0: 2033 2020 2020 2020 2020 2020 2020 7c0a 3 |. │ │ │ │ -000186e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000186f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018700: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00018710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018720: 205a 5a5b 7820 2e2e 7820 5d20 2020 2020 ZZ[x ..x ] │ │ │ │ -00018730: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00018740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018750: 2020 3020 2020 3320 2020 2020 2020 2020 0 3 │ │ │ │ -00018760: 2020 2020 207c 0a7c 6f33 3720 3a20 2d2d |.|o37 : -- │ │ │ │ -00018770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018790: 2d2d 7c0a 7c20 2020 2020 2028 7820 7820 --|.| (x x │ │ │ │ -000187a0: 2c20 7820 7820 2c20 7820 202d 2078 202c , x x , x - x , │ │ │ │ -000187b0: 2078 2020 2b20 3778 2020 2d20 7820 297c x + 7x - x )| │ │ │ │ -000187c0: 0a7c 2020 2020 2020 2020 3020 3220 2020 .| 0 2 │ │ │ │ -000187d0: 3120 3320 2020 3020 2020 2032 2020 2031 1 3 0 2 1 │ │ │ │ -000187e0: 2020 2020 2032 2020 2020 3320 7c0a 2b2d 2 3 |.+- │ │ │ │ -000187f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018810: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3820 ---------+.|i38 │ │ │ │ -00018820: 3a20 6373 6d31 3d43 534d 2843 682c 5529 : csm1=CSM(Ch,U) │ │ │ │ -00018830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018840: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00018850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018870: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00018880: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00018890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000188a0: 7c0a 7c6f 3338 203d 202d 2033 7820 7820 |.|o38 = - 3x x │ │ │ │ -000188b0: 202b 2078 2020 2d20 3578 2020 2b20 3278 + x - 5x + 2x │ │ │ │ -000188c0: 2020 2b20 3120 2020 2020 2020 207c 0a7c + 1 |.| │ │ │ │ -000188d0: 2020 2020 2020 2020 2020 3220 3320 2020 2 3 │ │ │ │ -000188e0: 2033 2020 2020 2032 2020 2020 2033 2020 3 2 3 │ │ │ │ -000188f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00018900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018920: 2020 2020 2020 207c 0a7c 6f33 3820 3a20 |.|o38 : │ │ │ │ -00018930: 4368 2020 2020 2020 2020 2020 2020 2020 Ch │ │ │ │ -00018940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018950: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00018960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018980: 2d2b 0a0a 416c 6c20 7468 6520 6578 616d -+..All the exam │ │ │ │ -00018990: 706c 6573 2077 6572 6520 646f 6e65 2075 ples were done u │ │ │ │ -000189a0: 7369 6e67 2073 796d 626f 6c69 6320 636f sing symbolic co │ │ │ │ -000189b0: 6d70 7574 6174 696f 6e73 2077 6974 6820 mputations with │ │ │ │ -000189c0: 4772 5c22 6f62 6e65 7220 6261 7365 732e Gr\"obner bases. │ │ │ │ -000189d0: 0a43 6861 6e67 696e 6720 7468 6520 6f70 .Changing the op │ │ │ │ -000189e0: 7469 6f6e 202a 6e6f 7465 2043 6f6d 704d tion *note CompM │ │ │ │ -000189f0: 6574 686f 643a 2043 6f6d 704d 6574 686f ethod: CompMetho │ │ │ │ -00018a00: 642c 2074 6f20 6265 7274 696e 6920 7769 d, to bertini wi │ │ │ │ -00018a10: 6c6c 2064 6f20 7468 6520 6d61 696e 0a63 ll do the main.c │ │ │ │ -00018a20: 6f6d 7075 7461 7469 6f6e 7320 6e75 6d65 omputations nume │ │ │ │ -00018a30: 7269 6361 6c6c 792c 2070 726f 7669 6465 rically, provide │ │ │ │ -00018a40: 6420 4265 7274 696e 6920 6973 202a 6e6f d Bertini is *no │ │ │ │ -00018a50: 7465 2069 6e73 7461 6c6c 6564 2061 6e64 te installed and │ │ │ │ -00018a60: 2063 6f6e 6669 6775 7265 643a 0a63 6f6e configured:.con │ │ │ │ -00018a70: 6669 6775 7269 6e67 2042 6572 7469 6e69 figuring Bertini │ │ │ │ -00018a80: 2c2e 204e 6f74 6520 7468 6174 2074 6865 ,. Note that the │ │ │ │ -00018a90: 2062 6572 7469 6e69 2061 6e64 2050 6e52 bertini and PnR │ │ │ │ -00018aa0: 6573 6964 7561 6c20 6f70 7469 6f6e 7320 esidual options │ │ │ │ -00018ab0: 6d61 7920 6f6e 6c79 2062 650a 7573 6564 may only be.used │ │ │ │ -00018ac0: 2066 6f72 2073 7562 7363 6865 6d65 7320 for subschemes │ │ │ │ -00018ad0: 6f66 205c 5050 5e6e 2e0a 0a4f 6273 6572 of \PP^n...Obser │ │ │ │ -00018ae0: 7665 2074 6861 7420 7468 6520 616c 676f ve that the algo │ │ │ │ -00018af0: 7269 7468 6d20 6973 2061 2070 726f 6261 rithm is a proba │ │ │ │ -00018b00: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ -00018b10: 686d 2061 6e64 206d 6179 2067 6976 6520 hm and may give │ │ │ │ -00018b20: 6120 7772 6f6e 670a 616e 7377 6572 2077 a wrong.answer w │ │ │ │ -00018b30: 6974 6820 6120 736d 616c 6c20 6275 7420 ith a small but │ │ │ │ -00018b40: 6e6f 6e7a 6572 6f20 7072 6f62 6162 696c nonzero probabil │ │ │ │ -00018b50: 6974 792e 2052 6561 6420 6d6f 7265 2075 ity. Read more u │ │ │ │ -00018b60: 6e64 6572 202a 6e6f 7465 0a70 726f 6261 nder *note.proba │ │ │ │ -00018b70: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ -00018b80: 686d 3a20 7072 6f62 6162 696c 6973 7469 hm: probabilisti │ │ │ │ -00018b90: 6320 616c 676f 7269 7468 6d2c 2e0a 0a0a c algorithm,.... │ │ │ │ -00018ba0: 0a57 6179 7320 746f 2075 7365 2043 534d .Ways to use CSM │ │ │ │ -00018bb0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -00018bc0: 3d3d 0a0a 2020 2a20 2243 534d 2849 6465 ==.. * "CSM(Ide │ │ │ │ -00018bd0: 616c 2922 0a20 202a 2022 4353 4d28 4964 al)". * "CSM(Id │ │ │ │ -00018be0: 6561 6c2c 5379 6d62 6f6c 2922 0a20 202a eal,Symbol)". * │ │ │ │ -00018bf0: 2022 4353 4d28 5175 6f74 6965 6e74 5269 "CSM(QuotientRi │ │ │ │ -00018c00: 6e67 2c49 6465 616c 2922 0a20 202a 2022 ng,Ideal)". * " │ │ │ │ -00018c10: 4353 4d28 5175 6f74 6965 6e74 5269 6e67 CSM(QuotientRing │ │ │ │ -00018c20: 2c49 6465 616c 2c4d 7574 6162 6c65 4861 ,Ideal,MutableHa │ │ │ │ -00018c30: 7368 5461 626c 6529 220a 0a46 6f72 2074 shTable)"..For t │ │ │ │ -00018c40: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -00018c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00018c60: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00018c70: 7465 2043 534d 3a20 4353 4d2c 2069 7320 te CSM: CSM, is │ │ │ │ -00018c80: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ -00018c90: 756e 6374 696f 6e20 7769 7468 206f 7074 unction with opt │ │ │ │ -00018ca0: 696f 6e73 3a0a 284d 6163 6175 6c61 7932 ions:.(Macaulay2 │ │ │ │ -00018cb0: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -00018cc0: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a onWithOptions,.. │ │ │ │ -00018cd0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ -00018ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018d20: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00018d30: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -00018d40: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -00018d50: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -00018d60: 6175 6c61 7932 2d31 2e32 352e 3131 2b64 aulay2-1.25.11+d │ │ │ │ -00018d70: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -00018d80: 6163 6b61 6765 732f 0a43 6861 7261 6374 ackages/.Charact │ │ │ │ -00018d90: 6572 6973 7469 6343 6c61 7373 6573 2e6d eristicClasses.m │ │ │ │ -00018da0: 323a 3232 3231 3a30 2e0a 1f0a 4669 6c65 2:2221:0....File │ │ │ │ -00018db0: 3a20 4368 6172 6163 7465 7269 7374 6963 : Characteristic │ │ │ │ -00018dc0: 436c 6173 7365 732e 696e 666f 2c20 4e6f Classes.info, No │ │ │ │ -00018dd0: 6465 3a20 4575 6c65 722c 204e 6578 743a de: Euler, Next: │ │ │ │ -00018de0: 2045 756c 6572 4166 6669 6e65 2c20 5072 EulerAffine, Pr │ │ │ │ -00018df0: 6576 3a20 4353 4d2c 2055 703a 2054 6f70 ev: CSM, Up: Top │ │ │ │ -00018e00: 0a0a 4575 6c65 7220 2d2d 2054 6865 2045 ..Euler -- The E │ │ │ │ -00018e10: 756c 6572 2043 6861 7261 6374 6572 6973 uler Characteris │ │ │ │ -00018e20: 7469 630a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a tic.************ │ │ │ │ -00018e30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00018e40: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -00018e50: 3a20 0a20 2020 2020 2020 2045 756c 6572 : . Euler │ │ │ │ -00018e60: 2049 0a20 2020 2020 2020 2045 756c 6572 I. Euler │ │ │ │ -00018e70: 2858 2c4a 290a 2020 2020 2020 2020 4575 (X,J). Eu │ │ │ │ -00018e80: 6c65 7220 6373 6d0a 2020 2a20 496e 7075 ler csm. * Inpu │ │ │ │ -00018e90: 7473 3a0a 2020 2020 2020 2a20 492c 2061 ts:. * I, a │ │ │ │ -00018ea0: 6e20 2a6e 6f74 6520 6964 6561 6c3a 2028 n *note ideal: ( │ │ │ │ -00018eb0: 4d61 6361 756c 6179 3244 6f63 2949 6465 Macaulay2Doc)Ide │ │ │ │ -00018ec0: 616c 2c2c 2061 206d 756c 7469 2d68 6f6d al,, a multi-hom │ │ │ │ -00018ed0: 6f67 656e 656f 7573 2069 6465 616c 2069 ogeneous ideal i │ │ │ │ -00018ee0: 6e20 610a 2020 2020 2020 2020 6772 6164 n a. grad │ │ │ │ -00018ef0: 6564 2070 6f6c 796e 6f6d 6961 6c20 7269 ed polynomial ri │ │ │ │ -00018f00: 6e67 206f 7665 7220 6120 6669 656c 6420 ng over a field │ │ │ │ -00018f10: 6465 6669 6e69 6e67 2061 2063 6c6f 7365 defining a close │ │ │ │ -00018f20: 6420 7375 6273 6368 656d 6520 5620 6f66 d subscheme V of │ │ │ │ -00018f30: 0a20 2020 2020 2020 205c 5050 5e7b 6e5f . \PP^{n_ │ │ │ │ -00018f40: 317d 782e 2e2e 785c 5050 5e7b 6e5f 6d7d 1}x...x\PP^{n_m} │ │ │ │ -00018f50: 0a20 2020 2020 202a 204a 2c20 616e 202a . * J, an * │ │ │ │ -00018f60: 6e6f 7465 2069 6465 616c 3a20 284d 6163 note ideal: (Mac │ │ │ │ -00018f70: 6175 6c61 7932 446f 6329 4964 6561 6c2c aulay2Doc)Ideal, │ │ │ │ -00018f80: 2c20 616e 2069 6465 616c 2069 6e20 7468 , an ideal in th │ │ │ │ -00018f90: 6520 6772 6164 6564 0a20 2020 2020 2020 e graded. │ │ │ │ -00018fa0: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ -00018fb0: 2077 6869 6368 2069 7320 636f 6f72 6469 which is coordi │ │ │ │ -00018fc0: 6e61 7465 2072 696e 6720 6f66 2074 6865 nate ring of the │ │ │ │ -00018fd0: 204e 6f72 6d61 6c20 546f 7269 6320 5661 Normal Toric Va │ │ │ │ -00018fe0: 7269 6574 7920 580a 2020 2020 2020 2a20 riety X. * │ │ │ │ -00018ff0: 582c 2061 202a 6e6f 7465 206e 6f72 6d61 X, a *note norma │ │ │ │ -00019000: 6c20 746f 7269 6320 7661 7269 6574 793a l toric variety: │ │ │ │ -00019010: 0a20 2020 2020 2020 2028 4e6f 726d 616c . (Normal │ │ │ │ -00019020: 546f 7269 6356 6172 6965 7469 6573 294e ToricVarieties)N │ │ │ │ -00019030: 6f72 6d61 6c54 6f72 6963 5661 7269 6574 ormalToricVariet │ │ │ │ -00019040: 792c 2c20 6120 6e6f 726d 616c 2074 6f72 y,, a normal tor │ │ │ │ -00019050: 6963 2076 6172 6965 7479 2077 6869 6368 ic variety which │ │ │ │ -00019060: 0a20 2020 2020 2020 2069 7320 7468 6520 . is the │ │ │ │ -00019070: 616d 6269 656e 7420 7370 6163 6520 7468 ambient space th │ │ │ │ -00019080: 6174 2077 6520 6172 6520 776f 726b 696e at we are workin │ │ │ │ -00019090: 6720 696e 0a20 2020 2020 202a 2063 736d g in. * csm │ │ │ │ -000190a0: 2c20 6120 2a6e 6f74 6520 7269 6e67 2065 , a *note ring e │ │ │ │ -000190b0: 6c65 6d65 6e74 3a20 284d 6163 6175 6c61 lement: (Macaula │ │ │ │ -000190c0: 7932 446f 6329 5269 6e67 456c 656d 656e y2Doc)RingElemen │ │ │ │ -000190d0: 742c 2c20 7468 6520 4353 4d20 636c 6173 t,, the CSM clas │ │ │ │ -000190e0: 7320 6f66 0a20 2020 2020 2020 2073 6f6d s of. som │ │ │ │ -000190f0: 6520 7661 7269 6574 7920 560a 2020 2a20 e variety V. * │ │ │ │ -00019100: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069 *note Optional i │ │ │ │ -00019110: 6e70 7574 733a 2028 4d61 6361 756c 6179 nputs: (Macaulay │ │ │ │ -00019120: 3244 6f63 2975 7369 6e67 2066 756e 6374 2Doc)using funct │ │ │ │ -00019130: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ -00019140: 616c 2069 6e70 7574 732c 3a0a 2020 2020 al inputs,:. │ │ │ │ -00019150: 2020 2a20 436f 6d70 4d65 7468 6f64 2028 * CompMethod ( │ │ │ │ -00019160: 6d69 7373 696e 6720 646f 6375 6d65 6e74 missing document │ │ │ │ -00019170: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064 ation) => ..., d │ │ │ │ -00019180: 6566 6175 6c74 2076 616c 7565 0a20 2020 efault value. │ │ │ │ -00019190: 2020 2020 2050 726f 6a65 6374 6976 6544 ProjectiveD │ │ │ │ -000191a0: 6567 7265 652c 2050 726f 6a65 6374 6976 egree, Projectiv │ │ │ │ -000191b0: 6544 6567 7265 652c 2061 7070 6c69 6361 eDegree, applica │ │ │ │ -000191c0: 626c 6520 666f 7220 616c 6c20 6361 7365 ble for all case │ │ │ │ -000191d0: 7320 7768 6572 6520 7468 650a 2020 2020 s where the. │ │ │ │ -000191e0: 2020 2020 6d65 7468 6f64 7320 696e 2074 methods in t │ │ │ │ -000191f0: 6865 2070 6163 6b61 6765 206d 6179 2062 he package may b │ │ │ │ -00019200: 6520 7573 6564 0a20 2020 2020 202a 2043 e used. * C │ │ │ │ -00019210: 6f6d 704d 6574 686f 6420 286d 6973 7369 ompMethod (missi │ │ │ │ -00019220: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ -00019230: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ -00019240: 7420 7661 6c75 650a 2020 2020 2020 2020 t value. │ │ │ │ -00019250: 5072 6f6a 6563 7469 7665 4465 6772 6565 ProjectiveDegree │ │ │ │ -00019260: 2c20 506e 5265 7369 6475 616c 2c20 7468 , PnResidual, th │ │ │ │ -00019270: 6973 2061 6c67 6f72 6974 686d 206d 6179 is algorithm may │ │ │ │ -00019280: 2062 6520 7573 6564 2066 6f72 2073 7562 be used for sub │ │ │ │ -00019290: 7363 6865 6d65 730a 2020 2020 2020 2020 schemes. │ │ │ │ -000192a0: 6f66 205c 5050 5e6e 206f 6e6c 790a 2020 of \PP^n only. │ │ │ │ -000192b0: 2020 2020 2a20 4d65 7468 6f64 2028 6d69 * Method (mi │ │ │ │ -000192c0: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ -000192d0: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ -000192e0: 6175 6c74 2076 616c 7565 0a20 2020 2020 ault value. │ │ │ │ -000192f0: 2020 2049 6e63 6c75 7369 6f6e 4578 636c InclusionExcl │ │ │ │ -00019300: 7573 696f 6e2c 2049 6e63 6c75 7369 6f6e usion, Inclusion │ │ │ │ -00019310: 4578 636c 7573 696f 6e2c 2061 7070 6c69 Exclusion, appli │ │ │ │ -00019320: 6361 626c 6520 666f 7220 616c 6c20 696e cable for all in │ │ │ │ -00019330: 7075 7473 0a20 2020 2020 202a 204d 6574 puts. * Met │ │ │ │ -00019340: 686f 6420 286d 6973 7369 6e67 2064 6f63 hod (missing doc │ │ │ │ -00019350: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ -00019360: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ -00019370: 650a 2020 2020 2020 2020 496e 636c 7573 e. Inclus │ │ │ │ -00019380: 696f 6e45 7863 6c75 7369 6f6e 2c20 4469 ionExclusion, Di │ │ │ │ -00019390: 7265 6374 436f 6d70 6c65 7465 496e 742c rectCompleteInt, │ │ │ │ -000193a0: 2074 6869 7320 6d65 7468 6f64 206d 6179 this method may │ │ │ │ -000193b0: 2070 726f 7669 6465 2061 0a20 2020 2020 provide a. │ │ │ │ -000193c0: 2020 2070 6572 666f 726d 616e 6365 2069 performance i │ │ │ │ -000193d0: 6d70 726f 7665 6d65 6e74 2077 6865 6e20 mprovement when │ │ │ │ -000193e0: 7468 6520 696e 7075 7420 6973 2061 2063 the input is a c │ │ │ │ -000193f0: 6f6d 706c 6574 6520 696e 7465 7273 6563 omplete intersec │ │ │ │ -00019400: 7469 6f6e 2c20 6966 0a20 2020 2020 2020 tion, if. │ │ │ │ -00019410: 2074 6865 2069 6e70 7574 2069 7320 6e6f the input is no │ │ │ │ -00019420: 7420 6120 636f 6d70 6c65 7465 2069 6e74 t a complete int │ │ │ │ -00019430: 6572 7365 6374 696f 6e20 696e 636c 7573 ersection inclus │ │ │ │ -00019440: 696f 6e2f 6578 636c 7573 696f 6e20 6974 ion/exclusion it │ │ │ │ -00019450: 2077 696c 6c0a 2020 2020 2020 2020 7265 will. re │ │ │ │ -00019460: 7475 726e 2061 6e20 6572 726f 720a 2020 turn an error. │ │ │ │ -00019470: 2020 2020 2a20 496e 7075 7449 7353 6d6f * InputIsSmo │ │ │ │ -00019480: 6f74 6820 286d 6973 7369 6e67 2064 6f63 oth (missing doc │ │ │ │ -00019490: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ -000194a0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ -000194b0: 6520 6661 6c73 652c 2074 6869 730a 2020 e false, this. │ │ │ │ -000194c0: 2020 2020 2020 6f70 7469 6f6e 2068 6173 option has │ │ │ │ -000194d0: 2076 616c 7565 7320 7472 7565 2f66 616c values true/fal │ │ │ │ -000194e0: 7365 2061 6e64 2074 656c 6c73 2074 6865 se and tells the │ │ │ │ -000194f0: 206d 6574 686f 6420 7768 6574 6865 7220 method whether │ │ │ │ -00019500: 746f 2061 7373 756d 6520 7468 650a 2020 to assume the. │ │ │ │ -00019510: 2020 2020 2020 696e 7075 7420 6964 6561 input idea │ │ │ │ -00019520: 6c20 6465 6669 6e65 7320 6120 736d 6f6f l defines a smoo │ │ │ │ -00019530: 7468 2073 6368 656d 652c 2061 6e64 2068 th scheme, and h │ │ │ │ -00019540: 656e 6365 2074 6f20 6361 6c6c 2074 6865 ence to call the │ │ │ │ -00019550: 206d 6574 686f 6420 4368 6572 6e0a 2020 method Chern. │ │ │ │ -00019560: 2020 2020 2020 696e 7374 6561 6420 666f instead fo │ │ │ │ -00019570: 7220 7265 6475 6365 6420 7275 6e20 7469 r reduced run ti │ │ │ │ -00019580: 6d65 2c20 616c 7465 726e 6174 6976 656c me, alternativel │ │ │ │ -00019590: 7920 7468 6520 4368 6572 6e20 6675 6e63 y the Chern func │ │ │ │ -000195a0: 7469 6f6e 2063 616e 2062 650a 2020 2020 tion can be. │ │ │ │ -000195b0: 2020 2020 7573 6564 2064 6972 6563 746c used directl │ │ │ │ -000195c0: 790a 2020 2020 2020 2a20 4f75 7470 7574 y. * Output │ │ │ │ -000195d0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ -000195e0: 2076 616c 7565 2043 686f 7752 696e 6745 value ChowRingE │ │ │ │ -000195f0: 6c65 6d65 6e74 2c20 7468 6520 7479 7065 lement, the type │ │ │ │ -00019600: 206f 6620 6f75 7470 7574 2074 6f0a 2020 of output to. │ │ │ │ -00019610: 2020 2020 2020 7265 7475 726e 2074 6865 return the │ │ │ │ -00019620: 2064 6566 6175 6c74 206f 7574 7075 7420 default output │ │ │ │ -00019630: 6973 2061 6e20 696e 7465 6765 720a 2020 is an integer. │ │ │ │ -00019640: 2020 2020 2a20 4f75 7470 7574 203d 3e20 * Output => │ │ │ │ -00019650: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -00019660: 7565 2043 686f 7752 696e 6745 6c65 6d65 ue ChowRingEleme │ │ │ │ -00019670: 6e74 2c20 4861 7368 466f 726d 2c20 7468 nt, HashForm, th │ │ │ │ -00019680: 6520 7479 7065 206f 660a 2020 2020 2020 e type of. │ │ │ │ -00019690: 2020 6f75 7470 7574 2074 6f20 7265 7475 output to retu │ │ │ │ -000196a0: 726e 2c20 4861 7368 466f 726d 2072 6574 rn, HashForm ret │ │ │ │ -000196b0: 7572 6e73 2061 204d 7574 6162 6c65 4861 urns a MutableHa │ │ │ │ -000196c0: 7368 5461 626c 6520 636f 6e74 6169 6e69 shTable containi │ │ │ │ -000196d0: 6e67 2074 6865 0a20 2020 2020 2020 206b ng the. k │ │ │ │ -000196e0: 6579 2022 4353 4d22 2028 7468 6520 4353 ey "CSM" (the CS │ │ │ │ -000196f0: 4d20 636c 6173 7329 2c20 616e 6420 6b65 M class), and ke │ │ │ │ -00019700: 7973 206f 6620 7468 6520 666f 726d 0a20 ys of the form. │ │ │ │ -00019710: 2020 2020 2020 205c 7b30 5c7d 2c5c 7b31 \{0\},\{1 │ │ │ │ -00019720: 5c7d 2c5c 7b32 5c7d 2c2e 2e2e 2c5c 7b30 \},\{2\},...,\{0 │ │ │ │ -00019730: 2c31 5c7d 2c5c 7b30 2c32 5c7d 202e 2e2e ,1\},\{0,2\} ... │ │ │ │ -00019740: 2e5c 7b30 2c31 2c32 5c7d 2e2e 2e20 616e .\{0,1,2\}... an │ │ │ │ -00019750: 6420 736f 206f 6e20 7768 6963 680a 2020 d so on which. │ │ │ │ -00019760: 2020 2020 2020 636f 7272 6573 706f 6e64 correspond │ │ │ │ -00019770: 2074 6f20 7468 6520 696e 6469 6365 7320 to the indices │ │ │ │ -00019780: 6f66 2074 6865 2070 6f73 7369 626c 6520 of the possible │ │ │ │ -00019790: 7375 6273 6574 7320 6f66 2074 6865 2067 subsets of the g │ │ │ │ -000197a0: 656e 6572 6174 6f72 7320 6f66 0a20 2020 enerators of. │ │ │ │ -000197b0: 2020 2020 2074 6865 2069 6e70 7574 2069 the input i │ │ │ │ -000197c0: 6465 616c 2c20 666f 7220 6561 6368 2073 deal, for each s │ │ │ │ -000197d0: 6574 206f 6620 696e 6469 6365 7320 7468 et of indices th │ │ │ │ -000197e0: 6520 4353 4d20 636c 6173 7320 6f66 2074 e CSM class of t │ │ │ │ -000197f0: 6865 0a20 2020 2020 2020 2068 7970 6572 he. hyper │ │ │ │ -00019800: 7375 7266 6163 6520 6769 7665 6e20 6279 surface given by │ │ │ │ -00019810: 2074 6865 2070 726f 6475 6374 206f 6620 the product of │ │ │ │ -00019820: 616c 6c20 706f 6c79 6e6f 6d69 616c 7320 all polynomials │ │ │ │ -00019830: 696e 2074 6865 0a20 2020 2020 2020 2063 in the. c │ │ │ │ -00019840: 6f72 7265 7370 6f6e 6469 6e67 2073 6574 orresponding set │ │ │ │ -00019850: 206f 6620 6765 6e65 7261 746f 7273 2069 of generators i │ │ │ │ -00019860: 7320 7374 6f72 6564 2c20 7468 6572 6520 s stored, there │ │ │ │ -00019870: 6973 206e 6f20 6578 7472 6120 636f 7374 is no extra cost │ │ │ │ -00019880: 2074 6f0a 2020 2020 2020 2020 7573 696e to. usin │ │ │ │ -00019890: 6720 7468 6973 206f 7074 696f 6e0a 2020 g this option. │ │ │ │ -000198a0: 2020 2020 2a20 496e 6473 4f66 536d 6f6f * IndsOfSmoo │ │ │ │ -000198b0: 7468 2028 6d69 7373 696e 6720 646f 6375 th (missing docu │ │ │ │ -000198c0: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e mentation) => .. │ │ │ │ -000198d0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -000198e0: 207b 7d2c 2074 6869 730a 2020 2020 2020 {}, this. │ │ │ │ -000198f0: 2020 6f70 7469 6f6e 206d 6179 2073 7065 option may spe │ │ │ │ -00019900: 6564 2075 7020 7468 6520 7275 6e20 7469 ed up the run ti │ │ │ │ -00019910: 6d65 2077 6865 6e20 7573 696e 6720 7468 me when using th │ │ │ │ -00019920: 6520 4469 7265 6374 436f 6d70 6c65 7465 e DirectComplete │ │ │ │ -00019930: 496e 740a 2020 2020 2020 2020 4d65 7468 Int. Meth │ │ │ │ -00019940: 6f64 2069 6620 7468 6520 7573 6572 206b od if the user k │ │ │ │ -00019950: 6e6f 7773 2061 6464 6974 696f 6e61 6c20 nows additional │ │ │ │ -00019960: 696e 666f 726d 6174 696f 6e20 6162 6f75 information abou │ │ │ │ -00019970: 7420 7468 6520 696e 7075 7420 6964 6561 t the input idea │ │ │ │ -00019980: 6c2c 0a20 2020 2020 2020 2073 6565 202a l,. see * │ │ │ │ -00019990: 6e6f 7465 2049 6e64 734f 6653 6d6f 6f74 note IndsOfSmoot │ │ │ │ -000199a0: 683a 2049 6e64 734f 6653 6d6f 6f74 682c h: IndsOfSmooth, │ │ │ │ -000199b0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -000199c0: 2020 2020 2a20 6120 2a6e 6f74 6520 7269 * a *note ri │ │ │ │ -000199d0: 6e67 2065 6c65 6d65 6e74 3a20 284d 6163 ng element: (Mac │ │ │ │ -000199e0: 6175 6c61 7932 446f 6329 5269 6e67 456c aulay2Doc)RingEl │ │ │ │ -000199f0: 656d 656e 742c 2c20 7468 6520 4575 6c65 ement,, the Eule │ │ │ │ -00019a00: 720a 2020 2020 2020 2020 6368 6172 6163 r. charac │ │ │ │ -00019a10: 7465 7269 7374 6963 0a0a 4465 7363 7269 teristic..Descri │ │ │ │ -00019a20: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00019a30: 3d0a 0a46 6f72 2061 2073 7562 7363 6865 =..For a subsche │ │ │ │ -00019a40: 6d65 2056 206f 6620 616e 2061 7070 6c69 me V of an appli │ │ │ │ -00019a50: 6361 626c 6520 746f 7269 6320 7661 7269 cable toric vari │ │ │ │ -00019a60: 6574 7920 582c 2074 6869 7320 636f 6d6d ety X, this comm │ │ │ │ -00019a70: 616e 6420 636f 6d70 7574 6573 2074 6865 and computes the │ │ │ │ -00019a80: 0a45 756c 6572 2063 6861 7261 6374 6572 .Euler character │ │ │ │ -00019a90: 6973 7469 630a 0a2b 2d2d 2d2d 2d2d 2d2d istic..+-------- │ │ │ │ -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 2d2b 0a7c 6931 203a 206b 6b3d -----+.|i1 : kk= │ │ │ │ -00019af0: 5a5a 2f33 3237 3439 3b20 2020 2020 2020 ZZ/32749; │ │ │ │ -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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00019b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019b80: 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 3d6b -----+.|i2 : R=k │ │ │ │ -00019b90: 6b5b 785f 302e 2e78 5f34 5d20 2020 2020 k[x_0..x_4] │ │ │ │ +00018530: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018540: 6f33 3620 3a20 5175 6f74 6965 6e74 5269 o36 : QuotientRi │ │ │ │ +00018550: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ +00018560: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00018570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018590: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3720 3a20 -------+.|i37 : │ │ │ │ +000185a0: 4353 4d20 5520 2020 2020 2020 2020 2020 CSM U │ │ │ │ +000185b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000185c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000185d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000185e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000185f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018600: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00018610: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00018620: 7c6f 3337 203d 202d 2033 7820 7820 202b |o37 = - 3x x + │ │ │ │ +00018630: 2078 2020 2d20 3578 2020 2b20 3278 2020 x - 5x + 2x │ │ │ │ +00018640: 2b20 3120 2020 2020 2020 207c 0a7c 2020 + 1 |.| │ │ │ │ +00018650: 2020 2020 2020 2020 3220 3320 2020 2033 2 3 3 │ │ │ │ +00018660: 2020 2020 2032 2020 2020 2033 2020 2020 2 3 │ │ │ │ +00018670: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00018680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000186a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000186b0: 2020 2020 2020 2020 2020 205a 5a5b 7820 ZZ[x │ │ │ │ +000186c0: 2e2e 7820 5d20 2020 2020 2020 2020 2020 ..x ] │ │ │ │ +000186d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000186e0: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +000186f0: 3320 2020 2020 2020 2020 2020 2020 207c 3 | │ │ │ │ +00018700: 0a7c 6f33 3720 3a20 2d2d 2d2d 2d2d 2d2d .|o37 : -------- │ │ │ │ +00018710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00018730: 2020 2020 2028 7820 7820 2c20 7820 7820 (x x , x x │ │ │ │ +00018740: 2c20 7820 202d 2078 202c 2078 2020 2b20 , x - x , x + │ │ │ │ +00018750: 3778 2020 2d20 7820 297c 0a7c 2020 2020 7x - x )|.| │ │ │ │ +00018760: 2020 2020 3020 3220 2020 3120 3320 2020 0 2 1 3 │ │ │ │ +00018770: 3020 2020 2032 2020 2031 2020 2020 2032 0 2 1 2 │ │ │ │ +00018780: 2020 2020 3320 7c0a 2b2d 2d2d 2d2d 2d2d 3 |.+------- │ │ │ │ +00018790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000187a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000187b0: 2d2d 2d2b 0a7c 6933 3820 3a20 6373 6d31 ---+.|i38 : csm1 │ │ │ │ +000187c0: 3d43 534d 2843 682c 5529 2020 2020 2020 =CSM(Ch,U) │ │ │ │ +000187d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000187e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000187f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018800: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018820: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00018830: 2020 2020 2020 2020 2020 7c0a 7c6f 3338 |.|o38 │ │ │ │ +00018840: 203d 202d 2033 7820 7820 202b 2078 2020 = - 3x x + x │ │ │ │ +00018850: 2d20 3578 2020 2b20 3278 2020 2b20 3120 - 5x + 2x + 1 │ │ │ │ +00018860: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00018870: 2020 2020 3220 3320 2020 2033 2020 2020 2 3 3 │ │ │ │ +00018880: 2032 2020 2020 2033 2020 2020 2020 2020 2 3 │ │ │ │ +00018890: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000188a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000188b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000188c0: 207c 0a7c 6f33 3820 3a20 4368 2020 2020 |.|o38 : Ch │ │ │ │ +000188d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000188e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000188f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00018900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 416c -----------+..Al │ │ │ │ +00018920: 6c20 7468 6520 6578 616d 706c 6573 2077 l the examples w │ │ │ │ +00018930: 6572 6520 646f 6e65 2075 7369 6e67 2073 ere done using s │ │ │ │ +00018940: 796d 626f 6c69 6320 636f 6d70 7574 6174 ymbolic computat │ │ │ │ +00018950: 696f 6e73 2077 6974 6820 4772 5c22 6f62 ions with Gr\"ob │ │ │ │ +00018960: 6e65 7220 6261 7365 732e 0a43 6861 6e67 ner bases..Chang │ │ │ │ +00018970: 696e 6720 7468 6520 6f70 7469 6f6e 202a ing the option * │ │ │ │ +00018980: 6e6f 7465 2043 6f6d 704d 6574 686f 643a note CompMethod: │ │ │ │ +00018990: 2043 6f6d 704d 6574 686f 642c 2074 6f20 CompMethod, to │ │ │ │ +000189a0: 6265 7274 696e 6920 7769 6c6c 2064 6f20 bertini will do │ │ │ │ +000189b0: 7468 6520 6d61 696e 0a63 6f6d 7075 7461 the main.computa │ │ │ │ +000189c0: 7469 6f6e 7320 6e75 6d65 7269 6361 6c6c tions numericall │ │ │ │ +000189d0: 792c 2070 726f 7669 6465 6420 4265 7274 y, provided Bert │ │ │ │ +000189e0: 696e 6920 6973 202a 6e6f 7465 2069 6e73 ini is *note ins │ │ │ │ +000189f0: 7461 6c6c 6564 2061 6e64 2063 6f6e 6669 talled and confi │ │ │ │ +00018a00: 6775 7265 643a 0a63 6f6e 6669 6775 7269 gured:.configuri │ │ │ │ +00018a10: 6e67 2042 6572 7469 6e69 2c2e 204e 6f74 ng Bertini,. Not │ │ │ │ +00018a20: 6520 7468 6174 2074 6865 2062 6572 7469 e that the berti │ │ │ │ +00018a30: 6e69 2061 6e64 2050 6e52 6573 6964 7561 ni and PnResidua │ │ │ │ +00018a40: 6c20 6f70 7469 6f6e 7320 6d61 7920 6f6e l options may on │ │ │ │ +00018a50: 6c79 2062 650a 7573 6564 2066 6f72 2073 ly be.used for s │ │ │ │ +00018a60: 7562 7363 6865 6d65 7320 6f66 205c 5050 ubschemes of \PP │ │ │ │ +00018a70: 5e6e 2e0a 0a4f 6273 6572 7665 2074 6861 ^n...Observe tha │ │ │ │ +00018a80: 7420 7468 6520 616c 676f 7269 7468 6d20 t the algorithm │ │ │ │ +00018a90: 6973 2061 2070 726f 6261 6269 6c69 7374 is a probabilist │ │ │ │ +00018aa0: 6963 2061 6c67 6f72 6974 686d 2061 6e64 ic algorithm and │ │ │ │ +00018ab0: 206d 6179 2067 6976 6520 6120 7772 6f6e may give a wron │ │ │ │ +00018ac0: 670a 616e 7377 6572 2077 6974 6820 6120 g.answer with a │ │ │ │ +00018ad0: 736d 616c 6c20 6275 7420 6e6f 6e7a 6572 small but nonzer │ │ │ │ +00018ae0: 6f20 7072 6f62 6162 696c 6974 792e 2052 o probability. R │ │ │ │ +00018af0: 6561 6420 6d6f 7265 2075 6e64 6572 202a ead more under * │ │ │ │ +00018b00: 6e6f 7465 0a70 726f 6261 6269 6c69 7374 note.probabilist │ │ │ │ +00018b10: 6963 2061 6c67 6f72 6974 686d 3a20 7072 ic algorithm: pr │ │ │ │ +00018b20: 6f62 6162 696c 6973 7469 6320 616c 676f obabilistic algo │ │ │ │ +00018b30: 7269 7468 6d2c 2e0a 0a0a 0a57 6179 7320 rithm,.....Ways │ │ │ │ +00018b40: 746f 2075 7365 2043 534d 3a0a 3d3d 3d3d to use CSM:.==== │ │ │ │ +00018b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +00018b60: 2a20 2243 534d 2849 6465 616c 2922 0a20 * "CSM(Ideal)". │ │ │ │ +00018b70: 202a 2022 4353 4d28 4964 6561 6c2c 5379 * "CSM(Ideal,Sy │ │ │ │ +00018b80: 6d62 6f6c 2922 0a20 202a 2022 4353 4d28 mbol)". * "CSM( │ │ │ │ +00018b90: 5175 6f74 6965 6e74 5269 6e67 2c49 6465 QuotientRing,Ide │ │ │ │ +00018ba0: 616c 2922 0a20 202a 2022 4353 4d28 5175 al)". * "CSM(Qu │ │ │ │ +00018bb0: 6f74 6965 6e74 5269 6e67 2c49 6465 616c otientRing,Ideal │ │ │ │ +00018bc0: 2c4d 7574 6162 6c65 4861 7368 5461 626c ,MutableHashTabl │ │ │ │ +00018bd0: 6529 220a 0a46 6f72 2074 6865 2070 726f e)"..For the pro │ │ │ │ +00018be0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00018bf0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00018c00: 6f62 6a65 6374 202a 6e6f 7465 2043 534d object *note CSM │ │ │ │ +00018c10: 3a20 4353 4d2c 2069 7320 6120 2a6e 6f74 : CSM, is a *not │ │ │ │ +00018c20: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +00018c30: 6e20 7769 7468 206f 7074 696f 6e73 3a0a n with options:. │ │ │ │ +00018c40: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +00018c50: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ +00018c60: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +00018cc0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +00018cd0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +00018ce0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00018cf0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +00018d00: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d -1.25.11+ds/M2/M │ │ │ │ +00018d10: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +00018d20: 732f 0a43 6861 7261 6374 6572 6973 7469 s/.Characteristi │ │ │ │ +00018d30: 6343 6c61 7373 6573 2e6d 323a 3232 3231 cClasses.m2:2221 │ │ │ │ +00018d40: 3a30 2e0a 1f0a 4669 6c65 3a20 4368 6172 :0....File: Char │ │ │ │ +00018d50: 6163 7465 7269 7374 6963 436c 6173 7365 acteristicClasse │ │ │ │ +00018d60: 732e 696e 666f 2c20 4e6f 6465 3a20 4575 s.info, Node: Eu │ │ │ │ +00018d70: 6c65 722c 204e 6578 743a 2045 756c 6572 ler, Next: Euler │ │ │ │ +00018d80: 4166 6669 6e65 2c20 5072 6576 3a20 4353 Affine, Prev: CS │ │ │ │ +00018d90: 4d2c 2055 703a 2054 6f70 0a0a 4575 6c65 M, Up: Top..Eule │ │ │ │ +00018da0: 7220 2d2d 2054 6865 2045 756c 6572 2043 r -- The Euler C │ │ │ │ +00018db0: 6861 7261 6374 6572 6973 7469 630a 2a2a haracteristic.** │ │ │ │ +00018dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00018dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +00018de0: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +00018df0: 2020 2020 2045 756c 6572 2049 0a20 2020 Euler I. │ │ │ │ +00018e00: 2020 2020 2045 756c 6572 2858 2c4a 290a Euler(X,J). │ │ │ │ +00018e10: 2020 2020 2020 2020 4575 6c65 7220 6373 Euler cs │ │ │ │ +00018e20: 6d0a 2020 2a20 496e 7075 7473 3a0a 2020 m. * Inputs:. │ │ │ │ +00018e30: 2020 2020 2a20 492c 2061 6e20 2a6e 6f74 * I, an *not │ │ │ │ +00018e40: 6520 6964 6561 6c3a 2028 4d61 6361 756c e ideal: (Macaul │ │ │ │ +00018e50: 6179 3244 6f63 2949 6465 616c 2c2c 2061 ay2Doc)Ideal,, a │ │ │ │ +00018e60: 206d 756c 7469 2d68 6f6d 6f67 656e 656f multi-homogeneo │ │ │ │ +00018e70: 7573 2069 6465 616c 2069 6e20 610a 2020 us ideal in a. │ │ │ │ +00018e80: 2020 2020 2020 6772 6164 6564 2070 6f6c graded pol │ │ │ │ +00018e90: 796e 6f6d 6961 6c20 7269 6e67 206f 7665 ynomial ring ove │ │ │ │ +00018ea0: 7220 6120 6669 656c 6420 6465 6669 6e69 r a field defini │ │ │ │ +00018eb0: 6e67 2061 2063 6c6f 7365 6420 7375 6273 ng a closed subs │ │ │ │ +00018ec0: 6368 656d 6520 5620 6f66 0a20 2020 2020 cheme V of. │ │ │ │ +00018ed0: 2020 205c 5050 5e7b 6e5f 317d 782e 2e2e \PP^{n_1}x... │ │ │ │ +00018ee0: 785c 5050 5e7b 6e5f 6d7d 0a20 2020 2020 x\PP^{n_m}. │ │ │ │ +00018ef0: 202a 204a 2c20 616e 202a 6e6f 7465 2069 * J, an *note i │ │ │ │ +00018f00: 6465 616c 3a20 284d 6163 6175 6c61 7932 deal: (Macaulay2 │ │ │ │ +00018f10: 446f 6329 4964 6561 6c2c 2c20 616e 2069 Doc)Ideal,, an i │ │ │ │ +00018f20: 6465 616c 2069 6e20 7468 6520 6772 6164 deal in the grad │ │ │ │ +00018f30: 6564 0a20 2020 2020 2020 2070 6f6c 796e ed. polyn │ │ │ │ +00018f40: 6f6d 6961 6c20 7269 6e67 2077 6869 6368 omial ring which │ │ │ │ +00018f50: 2069 7320 636f 6f72 6469 6e61 7465 2072 is coordinate r │ │ │ │ +00018f60: 696e 6720 6f66 2074 6865 204e 6f72 6d61 ing of the Norma │ │ │ │ +00018f70: 6c20 546f 7269 6320 5661 7269 6574 7920 l Toric Variety │ │ │ │ +00018f80: 580a 2020 2020 2020 2a20 582c 2061 202a X. * X, a * │ │ │ │ +00018f90: 6e6f 7465 206e 6f72 6d61 6c20 746f 7269 note normal tori │ │ │ │ +00018fa0: 6320 7661 7269 6574 793a 0a20 2020 2020 c variety:. │ │ │ │ +00018fb0: 2020 2028 4e6f 726d 616c 546f 7269 6356 (NormalToricV │ │ │ │ +00018fc0: 6172 6965 7469 6573 294e 6f72 6d61 6c54 arieties)NormalT │ │ │ │ +00018fd0: 6f72 6963 5661 7269 6574 792c 2c20 6120 oricVariety,, a │ │ │ │ +00018fe0: 6e6f 726d 616c 2074 6f72 6963 2076 6172 normal toric var │ │ │ │ +00018ff0: 6965 7479 2077 6869 6368 0a20 2020 2020 iety which. │ │ │ │ +00019000: 2020 2069 7320 7468 6520 616d 6269 656e is the ambien │ │ │ │ +00019010: 7420 7370 6163 6520 7468 6174 2077 6520 t space that we │ │ │ │ +00019020: 6172 6520 776f 726b 696e 6720 696e 0a20 are working in. │ │ │ │ +00019030: 2020 2020 202a 2063 736d 2c20 6120 2a6e * csm, a *n │ │ │ │ +00019040: 6f74 6520 7269 6e67 2065 6c65 6d65 6e74 ote ring element │ │ │ │ +00019050: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00019060: 5269 6e67 456c 656d 656e 742c 2c20 7468 RingElement,, th │ │ │ │ +00019070: 6520 4353 4d20 636c 6173 7320 6f66 0a20 e CSM class of. │ │ │ │ +00019080: 2020 2020 2020 2073 6f6d 6520 7661 7269 some vari │ │ │ │ +00019090: 6574 7920 560a 2020 2a20 2a6e 6f74 6520 ety V. * *note │ │ │ │ +000190a0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a Optional inputs: │ │ │ │ +000190b0: 2028 4d61 6361 756c 6179 3244 6f63 2975 (Macaulay2Doc)u │ │ │ │ +000190c0: 7369 6e67 2066 756e 6374 696f 6e73 2077 sing functions w │ │ │ │ +000190d0: 6974 6820 6f70 7469 6f6e 616c 2069 6e70 ith optional inp │ │ │ │ +000190e0: 7574 732c 3a0a 2020 2020 2020 2a20 436f uts,:. * Co │ │ │ │ +000190f0: 6d70 4d65 7468 6f64 2028 6d69 7373 696e mpMethod (missin │ │ │ │ +00019100: 6720 646f 6375 6d65 6e74 6174 696f 6e29 g documentation) │ │ │ │ +00019110: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +00019120: 2076 616c 7565 0a20 2020 2020 2020 2050 value. P │ │ │ │ +00019130: 726f 6a65 6374 6976 6544 6567 7265 652c rojectiveDegree, │ │ │ │ +00019140: 2050 726f 6a65 6374 6976 6544 6567 7265 ProjectiveDegre │ │ │ │ +00019150: 652c 2061 7070 6c69 6361 626c 6520 666f e, applicable fo │ │ │ │ +00019160: 7220 616c 6c20 6361 7365 7320 7768 6572 r all cases wher │ │ │ │ +00019170: 6520 7468 650a 2020 2020 2020 2020 6d65 e the. me │ │ │ │ +00019180: 7468 6f64 7320 696e 2074 6865 2070 6163 thods in the pac │ │ │ │ +00019190: 6b61 6765 206d 6179 2062 6520 7573 6564 kage may be used │ │ │ │ +000191a0: 0a20 2020 2020 202a 2043 6f6d 704d 6574 . * CompMet │ │ │ │ +000191b0: 686f 6420 286d 6973 7369 6e67 2064 6f63 hod (missing doc │ │ │ │ +000191c0: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ +000191d0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ +000191e0: 650a 2020 2020 2020 2020 5072 6f6a 6563 e. Projec │ │ │ │ +000191f0: 7469 7665 4465 6772 6565 2c20 506e 5265 tiveDegree, PnRe │ │ │ │ +00019200: 7369 6475 616c 2c20 7468 6973 2061 6c67 sidual, this alg │ │ │ │ +00019210: 6f72 6974 686d 206d 6179 2062 6520 7573 orithm may be us │ │ │ │ +00019220: 6564 2066 6f72 2073 7562 7363 6865 6d65 ed for subscheme │ │ │ │ +00019230: 730a 2020 2020 2020 2020 6f66 205c 5050 s. of \PP │ │ │ │ +00019240: 5e6e 206f 6e6c 790a 2020 2020 2020 2a20 ^n only. * │ │ │ │ +00019250: 4d65 7468 6f64 2028 6d69 7373 696e 6720 Method (missing │ │ │ │ +00019260: 646f 6375 6d65 6e74 6174 696f 6e29 203d documentation) = │ │ │ │ +00019270: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ +00019280: 616c 7565 0a20 2020 2020 2020 2049 6e63 alue. Inc │ │ │ │ +00019290: 6c75 7369 6f6e 4578 636c 7573 696f 6e2c lusionExclusion, │ │ │ │ +000192a0: 2049 6e63 6c75 7369 6f6e 4578 636c 7573 InclusionExclus │ │ │ │ +000192b0: 696f 6e2c 2061 7070 6c69 6361 626c 6520 ion, applicable │ │ │ │ +000192c0: 666f 7220 616c 6c20 696e 7075 7473 0a20 for all inputs. │ │ │ │ +000192d0: 2020 2020 202a 204d 6574 686f 6420 286d * Method (m │ │ │ │ +000192e0: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ +000192f0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ +00019300: 6661 756c 7420 7661 6c75 650a 2020 2020 fault value. │ │ │ │ +00019310: 2020 2020 496e 636c 7573 696f 6e45 7863 InclusionExc │ │ │ │ +00019320: 6c75 7369 6f6e 2c20 4469 7265 6374 436f lusion, DirectCo │ │ │ │ +00019330: 6d70 6c65 7465 496e 742c 2074 6869 7320 mpleteInt, this │ │ │ │ +00019340: 6d65 7468 6f64 206d 6179 2070 726f 7669 method may provi │ │ │ │ +00019350: 6465 2061 0a20 2020 2020 2020 2070 6572 de a. per │ │ │ │ +00019360: 666f 726d 616e 6365 2069 6d70 726f 7665 formance improve │ │ │ │ +00019370: 6d65 6e74 2077 6865 6e20 7468 6520 696e ment when the in │ │ │ │ +00019380: 7075 7420 6973 2061 2063 6f6d 706c 6574 put is a complet │ │ │ │ +00019390: 6520 696e 7465 7273 6563 7469 6f6e 2c20 e intersection, │ │ │ │ +000193a0: 6966 0a20 2020 2020 2020 2074 6865 2069 if. the i │ │ │ │ +000193b0: 6e70 7574 2069 7320 6e6f 7420 6120 636f nput is not a co │ │ │ │ +000193c0: 6d70 6c65 7465 2069 6e74 6572 7365 6374 mplete intersect │ │ │ │ +000193d0: 696f 6e20 696e 636c 7573 696f 6e2f 6578 ion inclusion/ex │ │ │ │ +000193e0: 636c 7573 696f 6e20 6974 2077 696c 6c0a clusion it will. │ │ │ │ +000193f0: 2020 2020 2020 2020 7265 7475 726e 2061 return a │ │ │ │ +00019400: 6e20 6572 726f 720a 2020 2020 2020 2a20 n error. * │ │ │ │ +00019410: 496e 7075 7449 7353 6d6f 6f74 6820 286d InputIsSmooth (m │ │ │ │ +00019420: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ +00019430: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ +00019440: 6661 756c 7420 7661 6c75 6520 6661 6c73 fault value fals │ │ │ │ +00019450: 652c 2074 6869 730a 2020 2020 2020 2020 e, this. │ │ │ │ +00019460: 6f70 7469 6f6e 2068 6173 2076 616c 7565 option has value │ │ │ │ +00019470: 7320 7472 7565 2f66 616c 7365 2061 6e64 s true/false and │ │ │ │ +00019480: 2074 656c 6c73 2074 6865 206d 6574 686f tells the metho │ │ │ │ +00019490: 6420 7768 6574 6865 7220 746f 2061 7373 d whether to ass │ │ │ │ +000194a0: 756d 6520 7468 650a 2020 2020 2020 2020 ume the. │ │ │ │ +000194b0: 696e 7075 7420 6964 6561 6c20 6465 6669 input ideal defi │ │ │ │ +000194c0: 6e65 7320 6120 736d 6f6f 7468 2073 6368 nes a smooth sch │ │ │ │ +000194d0: 656d 652c 2061 6e64 2068 656e 6365 2074 eme, and hence t │ │ │ │ +000194e0: 6f20 6361 6c6c 2074 6865 206d 6574 686f o call the metho │ │ │ │ +000194f0: 6420 4368 6572 6e0a 2020 2020 2020 2020 d Chern. │ │ │ │ +00019500: 696e 7374 6561 6420 666f 7220 7265 6475 instead for redu │ │ │ │ +00019510: 6365 6420 7275 6e20 7469 6d65 2c20 616c ced run time, al │ │ │ │ +00019520: 7465 726e 6174 6976 656c 7920 7468 6520 ternatively the │ │ │ │ +00019530: 4368 6572 6e20 6675 6e63 7469 6f6e 2063 Chern function c │ │ │ │ +00019540: 616e 2062 650a 2020 2020 2020 2020 7573 an be. us │ │ │ │ +00019550: 6564 2064 6972 6563 746c 790a 2020 2020 ed directly. │ │ │ │ +00019560: 2020 2a20 4f75 7470 7574 203d 3e20 2e2e * Output => .. │ │ │ │ +00019570: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ +00019580: 2043 686f 7752 696e 6745 6c65 6d65 6e74 ChowRingElement │ │ │ │ +00019590: 2c20 7468 6520 7479 7065 206f 6620 6f75 , the type of ou │ │ │ │ +000195a0: 7470 7574 2074 6f0a 2020 2020 2020 2020 tput to. │ │ │ │ +000195b0: 7265 7475 726e 2074 6865 2064 6566 6175 return the defau │ │ │ │ +000195c0: 6c74 206f 7574 7075 7420 6973 2061 6e20 lt output is an │ │ │ │ +000195d0: 696e 7465 6765 720a 2020 2020 2020 2a20 integer. * │ │ │ │ +000195e0: 4f75 7470 7574 203d 3e20 2e2e 2e2c 2064 Output => ..., d │ │ │ │ +000195f0: 6566 6175 6c74 2076 616c 7565 2043 686f efault value Cho │ │ │ │ +00019600: 7752 696e 6745 6c65 6d65 6e74 2c20 4861 wRingElement, Ha │ │ │ │ +00019610: 7368 466f 726d 2c20 7468 6520 7479 7065 shForm, the type │ │ │ │ +00019620: 206f 660a 2020 2020 2020 2020 6f75 7470 of. outp │ │ │ │ +00019630: 7574 2074 6f20 7265 7475 726e 2c20 4861 ut to return, Ha │ │ │ │ +00019640: 7368 466f 726d 2072 6574 7572 6e73 2061 shForm returns a │ │ │ │ +00019650: 204d 7574 6162 6c65 4861 7368 5461 626c MutableHashTabl │ │ │ │ +00019660: 6520 636f 6e74 6169 6e69 6e67 2074 6865 e containing the │ │ │ │ +00019670: 0a20 2020 2020 2020 206b 6579 2022 4353 . key "CS │ │ │ │ +00019680: 4d22 2028 7468 6520 4353 4d20 636c 6173 M" (the CSM clas │ │ │ │ +00019690: 7329 2c20 616e 6420 6b65 7973 206f 6620 s), and keys of │ │ │ │ +000196a0: 7468 6520 666f 726d 0a20 2020 2020 2020 the form. │ │ │ │ +000196b0: 205c 7b30 5c7d 2c5c 7b31 5c7d 2c5c 7b32 \{0\},\{1\},\{2 │ │ │ │ +000196c0: 5c7d 2c2e 2e2e 2c5c 7b30 2c31 5c7d 2c5c \},...,\{0,1\},\ │ │ │ │ +000196d0: 7b30 2c32 5c7d 202e 2e2e 2e5c 7b30 2c31 {0,2\} ....\{0,1 │ │ │ │ +000196e0: 2c32 5c7d 2e2e 2e20 616e 6420 736f 206f ,2\}... and so o │ │ │ │ +000196f0: 6e20 7768 6963 680a 2020 2020 2020 2020 n which. │ │ │ │ +00019700: 636f 7272 6573 706f 6e64 2074 6f20 7468 correspond to th │ │ │ │ +00019710: 6520 696e 6469 6365 7320 6f66 2074 6865 e indices of the │ │ │ │ +00019720: 2070 6f73 7369 626c 6520 7375 6273 6574 possible subset │ │ │ │ +00019730: 7320 6f66 2074 6865 2067 656e 6572 6174 s of the generat │ │ │ │ +00019740: 6f72 7320 6f66 0a20 2020 2020 2020 2074 ors of. t │ │ │ │ +00019750: 6865 2069 6e70 7574 2069 6465 616c 2c20 he input ideal, │ │ │ │ +00019760: 666f 7220 6561 6368 2073 6574 206f 6620 for each set of │ │ │ │ +00019770: 696e 6469 6365 7320 7468 6520 4353 4d20 indices the CSM │ │ │ │ +00019780: 636c 6173 7320 6f66 2074 6865 0a20 2020 class of the. │ │ │ │ +00019790: 2020 2020 2068 7970 6572 7375 7266 6163 hypersurfac │ │ │ │ +000197a0: 6520 6769 7665 6e20 6279 2074 6865 2070 e given by the p │ │ │ │ +000197b0: 726f 6475 6374 206f 6620 616c 6c20 706f roduct of all po │ │ │ │ +000197c0: 6c79 6e6f 6d69 616c 7320 696e 2074 6865 lynomials in the │ │ │ │ +000197d0: 0a20 2020 2020 2020 2063 6f72 7265 7370 . corresp │ │ │ │ +000197e0: 6f6e 6469 6e67 2073 6574 206f 6620 6765 onding set of ge │ │ │ │ +000197f0: 6e65 7261 746f 7273 2069 7320 7374 6f72 nerators is stor │ │ │ │ +00019800: 6564 2c20 7468 6572 6520 6973 206e 6f20 ed, there is no │ │ │ │ +00019810: 6578 7472 6120 636f 7374 2074 6f0a 2020 extra cost to. │ │ │ │ +00019820: 2020 2020 2020 7573 696e 6720 7468 6973 using this │ │ │ │ +00019830: 206f 7074 696f 6e0a 2020 2020 2020 2a20 option. * │ │ │ │ +00019840: 496e 6473 4f66 536d 6f6f 7468 2028 6d69 IndsOfSmooth (mi │ │ │ │ +00019850: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ +00019860: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ +00019870: 6175 6c74 2076 616c 7565 207b 7d2c 2074 ault value {}, t │ │ │ │ +00019880: 6869 730a 2020 2020 2020 2020 6f70 7469 his. opti │ │ │ │ +00019890: 6f6e 206d 6179 2073 7065 6564 2075 7020 on may speed up │ │ │ │ +000198a0: 7468 6520 7275 6e20 7469 6d65 2077 6865 the run time whe │ │ │ │ +000198b0: 6e20 7573 696e 6720 7468 6520 4469 7265 n using the Dire │ │ │ │ +000198c0: 6374 436f 6d70 6c65 7465 496e 740a 2020 ctCompleteInt. │ │ │ │ +000198d0: 2020 2020 2020 4d65 7468 6f64 2069 6620 Method if │ │ │ │ +000198e0: 7468 6520 7573 6572 206b 6e6f 7773 2061 the user knows a │ │ │ │ +000198f0: 6464 6974 696f 6e61 6c20 696e 666f 726d dditional inform │ │ │ │ +00019900: 6174 696f 6e20 6162 6f75 7420 7468 6520 ation about the │ │ │ │ +00019910: 696e 7075 7420 6964 6561 6c2c 0a20 2020 input ideal,. │ │ │ │ +00019920: 2020 2020 2073 6565 202a 6e6f 7465 2049 see *note I │ │ │ │ +00019930: 6e64 734f 6653 6d6f 6f74 683a 2049 6e64 ndsOfSmooth: Ind │ │ │ │ +00019940: 734f 6653 6d6f 6f74 682c 0a20 202a 204f sOfSmooth,. * O │ │ │ │ +00019950: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +00019960: 6120 2a6e 6f74 6520 7269 6e67 2065 6c65 a *note ring ele │ │ │ │ +00019970: 6d65 6e74 3a20 284d 6163 6175 6c61 7932 ment: (Macaulay2 │ │ │ │ +00019980: 446f 6329 5269 6e67 456c 656d 656e 742c Doc)RingElement, │ │ │ │ +00019990: 2c20 7468 6520 4575 6c65 720a 2020 2020 , the Euler. │ │ │ │ +000199a0: 2020 2020 6368 6172 6163 7465 7269 7374 characterist │ │ │ │ +000199b0: 6963 0a0a 4465 7363 7269 7074 696f 6e0a ic..Description. │ │ │ │ +000199c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a46 6f72 ===========..For │ │ │ │ +000199d0: 2061 2073 7562 7363 6865 6d65 2056 206f a subscheme V o │ │ │ │ +000199e0: 6620 616e 2061 7070 6c69 6361 626c 6520 f an applicable │ │ │ │ +000199f0: 746f 7269 6320 7661 7269 6574 7920 582c toric variety X, │ │ │ │ +00019a00: 2074 6869 7320 636f 6d6d 616e 6420 636f this command co │ │ │ │ +00019a10: 6d70 7574 6573 2074 6865 0a45 756c 6572 mputes the.Euler │ │ │ │ +00019a20: 2063 6861 7261 6374 6572 6973 7469 630a characteristic. │ │ │ │ +00019a30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00019a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00019a80: 0a7c 6931 203a 206b 6b3d 5a5a 2f33 3237 .|i1 : kk=ZZ/327 │ │ │ │ +00019a90: 3439 3b20 2020 2020 2020 2020 2020 2020 49; │ │ │ │ +00019aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019ac0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00019ad0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00019ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00019b20: 0a7c 6932 203a 2052 3d6b 6b5b 785f 302e .|i2 : R=kk[x_0. │ │ │ │ +00019b30: 2e78 5f34 5d20 2020 2020 2020 2020 2020 .x_4] │ │ │ │ +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 207c | │ │ │ │ +00019b70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00019b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00019bb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00019bc0: 0a7c 6f32 203d 2052 2020 2020 2020 2020 .|o2 = R │ │ │ │ +00019bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c20: 2020 2020 207c 0a7c 6f32 203d 2052 2020 |.|o2 = R │ │ │ │ +00019c00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00019c10: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00019c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00019c50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00019c60: 0a7c 6f32 203a 2050 6f6c 796e 6f6d 6961 .|o2 : Polynomia │ │ │ │ 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-00019d60: 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 3d69 -----+.|i3 : I=i │ │ │ │ -00019d70: 6465 616c 2872 616e 646f 6d28 312c 5229 deal(random(1,R) │ │ │ │ -00019d80: 2c72 616e 646f 6d28 322c 5229 2920 2020 ,random(2,R)) │ │ │ │ -00019d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019db0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00019ca0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00019cb0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00019cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00019d00: 0a7c 6933 203a 2049 3d69 6465 616c 2872 .|i3 : I=ideal(r │ │ │ │ +00019d10: 616e 646f 6d28 312c 5229 2c72 616e 646f andom(1,R),rando │ │ │ │ +00019d20: 6d28 322c 5229 2920 2020 2020 2020 2020 m(2,R)) │ │ │ │ +00019d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019d40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00019d50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00019d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019d90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00019da0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00019db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019e00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00019e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019e40: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00019e50: 2020 2020 207c 0a7c 6f33 203d 2069 6465 |.|o3 = ide │ │ │ │ -00019e60: 616c 2028 3130 3778 2020 2b20 3433 3736 al (107x + 4376 │ │ │ │ -00019e70: 7820 202d 2036 3331 3678 2020 2b20 3331 x - 6316x + 31 │ │ │ │ -00019e80: 3837 7820 202b 2033 3738 3378 202c 202d 87x + 3783x , - │ │ │ │ -00019e90: 2036 3035 3378 2020 2b20 3835 3730 7820 6053x + 8570x │ │ │ │ -00019ea0: 7820 202b 207c 0a7c 2020 2020 2020 2020 x + |.| │ │ │ │ -00019eb0: 2020 2020 2020 2020 3020 2020 2020 2020 0 │ │ │ │ -00019ec0: 2031 2020 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -00019ed0: 2020 2033 2020 2020 2020 2020 3420 2020 3 4 │ │ │ │ -00019ee0: 2020 2020 2020 3020 2020 2020 2020 2030 0 0 │ │ │ │ -00019ef0: 2031 2020 207c 0a7c 2020 2020 202d 2d2d 1 |.| --- │ │ │ │ -00019f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019f10: 2d2d 2d2d 2d2d 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│ │ │ -0001a000: 2020 2020 2020 2020 3120 3220 2020 2020 1 2 │ │ │ │ -0001a010: 2020 2032 2020 2020 2020 2020 3020 3320 2 0 3 │ │ │ │ -0001a020: 2020 2020 2020 2031 2033 2020 2020 2020 1 3 │ │ │ │ -0001a030: 2020 3220 337c 0a7c 2020 2020 202d 2d2d 2 3|.| --- │ │ │ │ -0001a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a080: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0001a090: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0001a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a0c0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -0001a0d0: 2020 2020 207c 0a7c 2020 2020 202d 2031 |.| - 1 │ │ │ │ -0001a0e0: 3432 3534 7820 202d 2031 3132 3236 7820 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+00019f20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00019f30: 0a7c 2020 2020 2031 3033 3539 7820 202d .| 10359x - │ │ │ │ +00019f40: 2031 3630 3930 7820 7820 202d 2038 3231 16090x x - 821 │ │ │ │ +00019f50: 3078 2078 2020 2b20 3530 3731 7820 202b 0x x + 5071x + │ │ │ │ +00019f60: 2038 3434 3478 2078 2020 2d20 3839 3937 8444x x - 8997 │ │ │ │ +00019f70: 7820 7820 202d 2036 3934 3978 2078 207c x x - 6949x x | │ │ │ │ +00019f80: 0a7c 2020 2020 2020 2020 2020 2031 2020 .| 1 │ │ │ │ +00019f90: 2020 2020 2020 2030 2032 2020 2020 2020 0 2 │ │ │ │ +00019fa0: 2020 3120 3220 2020 2020 2020 2032 2020 1 2 2 │ │ │ │ +00019fb0: 2020 2020 2020 3020 3320 2020 2020 2020 0 3 │ │ │ │ +00019fc0: 2031 2033 2020 2020 2020 2020 3220 337c 1 3 2 3| │ │ │ │ +00019fd0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +00019fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0001a020: 0a7c 2020 2020 2020 2020 2020 2020 2032 .| 2 │ │ │ │ +0001a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a060: 2020 2020 2020 2020 3220 2020 2020 207c 2 | │ │ │ │ +0001a070: 0a7c 2020 2020 202d 2031 3432 3534 7820 .| - 14254x │ │ │ │ +0001a080: 202d 2031 3132 3236 7820 7820 202b 2032 - 11226x x + 2 │ │ │ │ +0001a090: 3635 3378 2078 2020 2b20 3132 3336 3578 653x x + 12365x │ │ │ │ +0001a0a0: 2078 2020 2d20 3130 3232 3678 2078 2020 x - 10226x x │ │ │ │ +0001a0b0: 2d20 3132 3639 3678 2029 2020 2020 207c - 12696x ) | │ │ │ │ +0001a0c0: 0a7c 2020 2020 2020 2020 2020 2020 2033 .| 3 │ │ │ │ +0001a0d0: 2020 2020 2020 2020 2030 2034 2020 2020 0 4 │ │ │ │ +0001a0e0: 2020 2020 3120 3420 2020 2020 2020 2020 1 4 │ │ │ │ +0001a0f0: 3220 3420 2020 2020 2020 2020 3320 3420 2 4 3 4 │ │ │ │ +0001a100: 2020 2020 2020 2020 3420 2020 2020 207c 4 | │ │ │ │ +0001a110: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a150: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a160: 0a7c 6f33 203a 2049 6465 616c 206f 6620 .|o3 : Ideal of │ │ │ │ +0001a170: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0001a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a1c0: 2020 2020 207c 0a7c 6f33 203a 2049 6465 |.|o3 : Ide │ │ │ │ -0001a1d0: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ -0001a1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a210: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a260: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 696d -----+.|i4 : tim │ │ │ │ -0001a270: 6520 4575 6c65 7228 492c 496e 7075 7449 e Euler(I,InputI │ │ │ │ -0001a280: 7353 6d6f 6f74 683d 3e74 7275 6529 2020 sSmooth=>true) │ │ │ │ -0001a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a2b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001a2c0: 2030 2e30 3339 3532 3135 7320 2863 7075 0.0395215s (cpu │ │ │ │ -0001a2d0: 293b 2030 2e30 3337 3432 3336 7320 2874 ); 0.0374236s (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 |.| │ │ │ │ +0001a1a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a1b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001a200: 0a7c 6934 203a 2074 696d 6520 4575 6c65 .|i4 : time Eule │ │ │ │ +0001a210: 7228 492c 496e 7075 7449 7353 6d6f 6f74 r(I,InputIsSmoot │ │ │ │ +0001a220: 683d 3e74 7275 6529 2020 2020 2020 2020 h=>true) │ │ │ │ +0001a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a240: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a250: 0a7c 202d 2d20 7573 6564 2030 2e30 3930 .| -- used 0.090 │ │ │ │ +0001a260: 3834 3537 7320 2863 7075 293b 2030 2e30 8457s (cpu); 0.0 │ │ │ │ +0001a270: 3532 3537 3638 7320 2874 6872 6561 6429 525768s (thread) │ │ │ │ +0001a280: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0001a290: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a2a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a2e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a2f0: 0a7c 6f34 203d 2034 2020 2020 2020 2020 .|o4 = 4 │ │ │ │ +0001a300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a350: 2020 2020 207c 0a7c 6f34 203d 2034 2020 |.|o4 = 4 │ │ │ │ -0001a360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a3a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001a3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a3f0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 696d -----+.|i5 : tim │ │ │ │ -0001a400: 6520 4575 6c65 7220 4920 2020 2020 2020 e Euler I │ │ │ │ -0001a410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a440: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001a450: 2030 2e32 3036 3039 3373 2028 6370 7529 0.206093s (cpu) │ │ │ │ -0001a460: 3b20 302e 3134 3233 3839 7320 2874 6872 ; 0.142389s (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 |.| │ │ │ │ +0001a330: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a340: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001a350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001a390: 0a7c 6935 203a 2074 696d 6520 4575 6c65 .|i5 : time Eule │ │ │ │ +0001a3a0: 7220 4920 2020 2020 2020 2020 2020 2020 r I │ │ │ │ +0001a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a3d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a3e0: 0a7c 202d 2d20 7573 6564 2030 2e33 3632 .| -- used 0.362 │ │ │ │ +0001a3f0: 3530 3273 2028 6370 7529 3b20 302e 3139 502s (cpu); 0.19 │ │ │ │ +0001a400: 3838 3633 7320 2874 6872 6561 6429 3b20 8863s (thread); │ │ │ │ +0001a410: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +0001a420: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a430: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001a440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a470: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a480: 0a7c 6f35 203d 2034 2020 2020 2020 2020 .|o5 = 4 │ │ │ │ +0001a490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a4e0: 2020 2020 207c 0a7c 6f35 203d 2034 2020 |.|o5 = 4 │ │ │ │ -0001a4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a530: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a580: 2d2d 2d2d 2d2b 0a7c 6936 203a 2045 756c -----+.|i6 : Eul │ │ │ │ -0001a590: 6572 4948 6173 683d 4575 6c65 7228 492c erIHash=Euler(I, │ │ │ │ -0001a5a0: 4f75 7470 7574 3d3e 4861 7368 466f 726d Output=>HashForm │ │ │ │ -0001a5b0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ -0001a5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a5d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a620: 2d2d 2d2d 2d2b 0a7c 6937 203a 2041 3d72 -----+.|i7 : A=r │ │ │ │ -0001a630: 696e 6720 4575 6c65 7249 4861 7368 2322 ing EulerIHash#" │ │ │ │ -0001a640: 4353 4d22 2020 2020 2020 2020 2020 2020 CSM" │ │ │ │ -0001a650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a670: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a4c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a4d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001a520: 0a7c 6936 203a 2045 756c 6572 4948 6173 .|i6 : EulerIHas │ │ │ │ +0001a530: 683d 4575 6c65 7228 492c 4f75 7470 7574 h=Euler(I,Output │ │ │ │ +0001a540: 3d3e 4861 7368 466f 726d 293b 2020 2020 =>HashForm); │ │ │ │ +0001a550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a560: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a570: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001a5c0: 0a7c 6937 203a 2041 3d72 696e 6720 4575 .|i7 : A=ring Eu │ │ │ │ +0001a5d0: 6c65 7249 4861 7368 2322 4353 4d22 2020 lerIHash#"CSM" │ │ │ │ +0001a5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a600: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a610: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a650: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a660: 0a7c 6f37 203d 2041 2020 2020 2020 2020 .|o7 = A │ │ │ │ +0001a670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a6c0: 2020 2020 207c 0a7c 6f37 203d 2041 2020 |.|o7 = A │ │ │ │ +0001a6a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a6b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001a6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a710: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a6f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a700: 0a7c 6f37 203a 2051 756f 7469 656e 7452 .|o7 : QuotientR │ │ │ │ +0001a710: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ 0001a720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a760: 2020 2020 207c 0a7c 6f37 203a 2051 756f |.|o7 : Quo │ │ │ │ -0001a770: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ -0001a780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a7b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a800: 2d2d 2d2d 2d2b 0a7c 6938 203a 2045 756c -----+.|i8 : Eul │ │ │ │ -0001a810: 6572 4948 6173 6823 7b30 2c31 7d3d 3d43 erIHash#{0,1}==C │ │ │ │ -0001a820: 534d 2841 2c69 6465 616c 2849 5f30 2a49 SM(A,ideal(I_0*I │ │ │ │ -0001a830: 5f31 2929 2020 2020 2020 2020 2020 2020 _1)) │ │ │ │ -0001a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a850: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a740: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a750: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001a760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001a7a0: 0a7c 6938 203a 2045 756c 6572 4948 6173 .|i8 : EulerIHas │ │ │ │ +0001a7b0: 6823 7b30 2c31 7d3d 3d43 534d 2841 2c69 h#{0,1}==CSM(A,i │ │ │ │ +0001a7c0: 6465 616c 2849 5f30 2a49 5f31 2929 2020 deal(I_0*I_1)) │ │ │ │ +0001a7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a7e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a7f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001a800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a830: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a840: 0a7c 6f38 203d 2074 7275 6520 2020 2020 .|o8 = true │ │ │ │ +0001a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8a0: 2020 2020 207c 0a7c 6f38 203d 2074 7275 |.|o8 = tru │ │ │ │ -0001a8b0: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -0001a8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a940: 2d2d 2d2d 2d2b 0a7c 6939 203a 204a 3d49 -----+.|i9 : J=I │ │ │ │ -0001a950: 2b69 6465 616c 2878 5f30 2a78 5f32 2d78 +ideal(x_0*x_2-x │ │ │ │ -0001a960: 5f33 2a78 5f30 2920 2020 2020 2020 2020 _3*x_0) │ │ │ │ -0001a970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a990: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a880: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a890: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001a8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001a8e0: 0a7c 6939 203a 204a 3d49 2b69 6465 616c .|i9 : J=I+ideal │ │ │ │ +0001a8f0: 2878 5f30 2a78 5f32 2d78 5f33 2a78 5f30 (x_0*x_2-x_3*x_0 │ │ │ │ +0001a900: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0001a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a930: 0a7c 2020 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 2020 2020 207c | │ │ │ │ +0001a980: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001a990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a9e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa20: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0001aa30: 2020 2020 207c 0a7c 6f39 203d 2069 6465 |.|o9 = ide │ │ │ │ -0001aa40: 616c 2028 3130 3778 2020 2b20 3433 3736 al (107x + 4376 │ │ │ │ -0001aa50: 7820 202d 2036 3331 3678 2020 2b20 3331 x - 6316x + 31 │ │ │ │ -0001aa60: 3837 7820 202b 2033 3738 3378 202c 202d 87x + 3783x , - │ │ │ │ -0001aa70: 2036 3035 3378 2020 2b20 3835 3730 7820 6053x + 8570x │ │ │ │ -0001aa80: 7820 202b 207c 0a7c 2020 2020 2020 2020 x + |.| │ │ │ │ -0001aa90: 2020 2020 2020 2020 3020 2020 2020 2020 0 │ │ │ │ -0001aaa0: 2031 2020 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -0001aab0: 2020 2033 2020 2020 2020 2020 3420 2020 3 4 │ │ │ │ -0001aac0: 2020 2020 2020 3020 2020 2020 2020 2030 0 0 │ │ │ │ -0001aad0: 2031 2020 207c 0a7c 2020 2020 202d 2d2d 1 |.| --- │ │ │ │ -0001aae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001aaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab20: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0001ab30: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001ab40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ab50: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001ab60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ab70: 2020 2020 207c 0a7c 2020 2020 2031 3033 |.| 103 │ │ │ │ -0001ab80: 3539 7820 202d 2031 3630 3930 7820 7820 59x - 16090x x │ │ │ │ -0001ab90: 202d 2038 3231 3078 2078 2020 2b20 3530 - 8210x x + 50 │ │ │ │ -0001aba0: 3731 7820 202b 2038 3434 3478 2078 2020 71x + 8444x x │ │ │ │ -0001abb0: 2d20 3839 3937 7820 7820 202d 2036 3934 - 8997x x - 694 │ │ │ │ -0001abc0: 3978 2078 207c 0a7c 2020 2020 2020 2020 9x x |.| │ │ │ │ -0001abd0: 2020 2031 2020 2020 2020 2020 2030 2032 1 0 2 │ │ │ │ -0001abe0: 2020 2020 2020 2020 3120 3220 2020 2020 1 2 │ │ │ │ -0001abf0: 2020 2032 2020 2020 2020 2020 3020 3320 2 0 3 │ │ │ │ -0001ac00: 2020 2020 2020 2031 2033 2020 2020 2020 1 3 │ │ │ │ -0001ac10: 2020 3220 337c 0a7c 2020 2020 202d 2d2d 2 3|.| --- │ │ │ │ -0001ac20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ac30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ac40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ac50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ac60: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0001ac70: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aca0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -0001acb0: 2020 2020 207c 0a7c 2020 2020 202d 2031 |.| - 1 │ │ │ │ -0001acc0: 3432 3534 7820 202d 2031 3132 3236 7820 4254x - 11226x │ │ │ │ -0001acd0: 7820 202b 2032 3635 3378 2078 2020 2b20 x + 2653x x + │ │ │ │ -0001ace0: 3132 3336 3578 2078 2020 2d20 3130 3232 12365x x - 1022 │ │ │ │ -0001acf0: 3678 2078 2020 2d20 3132 3639 3678 202c 6x x - 12696x , │ │ │ │ -0001ad00: 2078 2078 207c 0a7c 2020 2020 2020 2020 x x |.| │ │ │ │ -0001ad10: 2020 2020 2033 2020 2020 2020 2020 2030 3 0 │ │ │ │ -0001ad20: 2034 2020 2020 2020 2020 3120 3420 2020 4 1 4 │ │ │ │ -0001ad30: 2020 2020 2020 3220 3420 2020 2020 2020 2 4 │ │ │ │ -0001ad40: 2020 3320 3420 2020 2020 2020 2020 3420 3 4 4 │ │ │ │ -0001ad50: 2020 3020 327c 0a7c 2020 2020 202d 2d2d 0 2|.| --- │ │ │ │ -0001ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ad80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ada0: 2d2d 2d2d 2d7c 0a7c 2020 2020 202d 2078 -----|.| - x │ │ │ │ -0001adb0: 2078 2029 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ +0001a9c0: 3220 2020 2020 2020 2020 2020 2020 207c 2 | │ │ │ │ +0001a9d0: 0a7c 6f39 203d 2069 6465 616c 2028 3130 .|o9 = ideal (10 │ │ │ │ +0001a9e0: 3778 2020 2b20 3433 3736 7820 202d 2036 7x + 4376x - 6 │ │ │ │ +0001a9f0: 3331 3678 2020 2b20 3331 3837 7820 202b 316x + 3187x + │ │ │ │ +0001aa00: 2033 3738 3378 202c 202d 2036 3035 3378 3783x , - 6053x │ │ │ │ +0001aa10: 2020 2b20 3835 3730 7820 7820 202b 207c + 8570x x + | │ │ │ │ +0001aa20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001aa30: 2020 3020 2020 2020 2020 2031 2020 2020 0 1 │ │ │ │ +0001aa40: 2020 2020 3220 2020 2020 2020 2033 2020 2 3 │ │ │ │ +0001aa50: 2020 2020 2020 3420 2020 2020 2020 2020 4 │ │ │ │ +0001aa60: 3020 2020 2020 2020 2030 2031 2020 207c 0 0 1 | │ │ │ │ +0001aa70: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +0001aa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aa90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0001aac0: 0a7c 2020 2020 2020 2020 2020 2032 2020 .| 2 │ │ │ │ +0001aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001aae0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +0001aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ab00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001ab10: 0a7c 2020 2020 2031 3033 3539 7820 202d .| 10359x - │ │ │ │ +0001ab20: 2031 3630 3930 7820 7820 202d 2038 3231 16090x x - 821 │ │ │ │ +0001ab30: 3078 2078 2020 2b20 3530 3731 7820 202b 0x x + 5071x + │ │ │ │ +0001ab40: 2038 3434 3478 2078 2020 2d20 3839 3937 8444x x - 8997 │ │ │ │ +0001ab50: 7820 7820 202d 2036 3934 3978 2078 207c x x - 6949x x | │ │ │ │ +0001ab60: 0a7c 2020 2020 2020 2020 2020 2031 2020 .| 1 │ │ │ │ +0001ab70: 2020 2020 2020 2030 2032 2020 2020 2020 0 2 │ │ │ │ +0001ab80: 2020 3120 3220 2020 2020 2020 2032 2020 1 2 2 │ │ │ │ +0001ab90: 2020 2020 2020 3020 3320 2020 2020 2020 0 3 │ │ │ │ +0001aba0: 2031 2033 2020 2020 2020 2020 3220 337c 1 3 2 3| │ │ │ │ +0001abb0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +0001abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001abd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001abe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0001ac00: 0a7c 2020 2020 2020 2020 2020 2020 2032 .| 2 │ │ │ │ +0001ac10: 2020 2020 2020 2020 2020 2020 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 3220 2020 2020 207c 2 | │ │ │ │ +0001ac50: 0a7c 2020 2020 202d 2031 3432 3534 7820 .| - 14254x │ │ │ │ +0001ac60: 202d 2031 3132 3236 7820 7820 202b 2032 - 11226x x + 2 │ │ │ │ +0001ac70: 3635 3378 2078 2020 2b20 3132 3336 3578 653x x + 12365x │ │ │ │ +0001ac80: 2078 2020 2d20 3130 3232 3678 2078 2020 x - 10226x x │ │ │ │ +0001ac90: 2d20 3132 3639 3678 202c 2078 2078 207c - 12696x , x x | │ │ │ │ +0001aca0: 0a7c 2020 2020 2020 2020 2020 2020 2033 .| 3 │ │ │ │ +0001acb0: 2020 2020 2020 2020 2030 2034 2020 2020 0 4 │ │ │ │ +0001acc0: 2020 2020 3120 3420 2020 2020 2020 2020 1 4 │ │ │ │ +0001acd0: 3220 3420 2020 2020 2020 2020 3320 3420 2 4 3 4 │ │ │ │ +0001ace0: 2020 2020 2020 2020 3420 2020 3020 327c 4 0 2| │ │ │ │ +0001acf0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +0001ad00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ad10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ad20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0001ad40: 0a7c 2020 2020 202d 2078 2078 2029 2020 .| - x x ) │ │ │ │ +0001ad50: 2020 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 207c | │ │ │ │ +0001ad90: 0a7c 2020 2020 2020 2020 3020 3320 2020 .| 0 3 │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -0001ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001adf0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001ae00: 3020 3320 2020 2020 2020 2020 2020 2020 0 3 │ │ │ │ +0001add0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001ade0: 0a7c 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001ae20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001ae30: 0a7c 6f39 203a 2049 6465 616c 206f 6620 .|o9 : Ideal of │ │ │ │ +0001ae40: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0001ae50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae90: 2020 2020 207c 0a7c 6f39 203a 2049 6465 |.|o9 : Ide │ │ │ │ -0001aea0: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ -0001aeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aee0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001af10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001af20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001af30: 2d2d 2d2d 2d2b 0a0a 4e6f 7465 2074 6861 -----+..Note tha │ │ │ │ -0001af40: 7420 7468 6520 6964 6561 6c20 4a20 6162 t the ideal J ab │ │ │ │ -0001af50: 6f76 6520 6973 2061 2063 6f6d 706c 6574 ove is a complet │ │ │ │ -0001af60: 6520 696e 7465 7273 6563 7469 6f6e 2c20 e intersection, │ │ │ │ -0001af70: 7468 7573 2077 6520 6d61 7920 6368 616e thus we may chan │ │ │ │ -0001af80: 6765 2074 6865 0a6d 6574 686f 6420 6f70 ge the.method op │ │ │ │ -0001af90: 7469 6f6e 2077 6869 6368 206d 6179 2073 tion which may s │ │ │ │ -0001afa0: 7065 6564 2063 6f6d 7075 7461 7469 6f6e peed computation │ │ │ │ -0001afb0: 2069 6e20 736f 6d65 2063 6173 6573 2e20 in some cases. │ │ │ │ -0001afc0: 5765 206d 6179 2061 6c73 6f20 6e6f 7465 We may also note │ │ │ │ -0001afd0: 2074 6861 740a 7468 6520 6964 6561 6c20 that.the ideal │ │ │ │ -0001afe0: 6765 6e65 7261 7465 6420 6279 2074 6865 generated by the │ │ │ │ -0001aff0: 2066 6972 7374 2032 2067 656e 6572 6174 first 2 generat │ │ │ │ -0001b000: 6f72 7320 6f66 2049 2064 6566 696e 6573 ors of I defines │ │ │ │ -0001b010: 2061 2073 6d6f 6f74 6820 7363 6865 6d65 a smooth scheme │ │ │ │ -0001b020: 2061 6e64 0a69 6e70 7574 2074 6869 7320 and.input this │ │ │ │ -0001b030: 696e 666f 726d 6174 696f 6e20 696e 746f information into │ │ │ │ -0001b040: 2074 6865 206d 6574 686f 642e 2054 6869 the method. Thi │ │ │ │ -0001b050: 7320 6d61 7920 616c 736f 2069 6d70 726f s may also impro │ │ │ │ -0001b060: 7665 2063 6f6d 7075 7461 7469 6f6e 0a73 ve computation.s │ │ │ │ -0001b070: 7065 6564 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d peed...+-------- │ │ │ │ -0001b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b0b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 ---------+.|i10 │ │ │ │ -0001b0c0: 3a20 7469 6d65 2045 756c 6572 284a 2c4d : time Euler(J,M │ │ │ │ -0001b0d0: 6574 686f 643d 3e44 6972 6563 7443 6f6d ethod=>DirectCom │ │ │ │ -0001b0e0: 706c 6574 6549 6e74 2920 2020 2020 2020 pleteInt) │ │ │ │ -0001b0f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001b100: 202d 2d20 7573 6564 2030 2e30 3739 3332 -- used 0.07932 │ │ │ │ -0001b110: 3373 2028 6370 7529 3b20 302e 3037 3533 3s (cpu); 0.0753 │ │ │ │ -0001b120: 3437 3773 2028 7468 7265 6164 293b 2030 477s (thread); 0 │ │ │ │ -0001b130: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ -0001b140: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001ae70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001ae80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001ae90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001aed0: 0a0a 4e6f 7465 2074 6861 7420 7468 6520 ..Note that the │ │ │ │ +0001aee0: 6964 6561 6c20 4a20 6162 6f76 6520 6973 ideal J above is │ │ │ │ +0001aef0: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ +0001af00: 7273 6563 7469 6f6e 2c20 7468 7573 2077 rsection, thus w │ │ │ │ +0001af10: 6520 6d61 7920 6368 616e 6765 2074 6865 e may change the │ │ │ │ +0001af20: 0a6d 6574 686f 6420 6f70 7469 6f6e 2077 .method option w │ │ │ │ +0001af30: 6869 6368 206d 6179 2073 7065 6564 2063 hich may speed c │ │ │ │ +0001af40: 6f6d 7075 7461 7469 6f6e 2069 6e20 736f omputation in so │ │ │ │ +0001af50: 6d65 2063 6173 6573 2e20 5765 206d 6179 me cases. We may │ │ │ │ +0001af60: 2061 6c73 6f20 6e6f 7465 2074 6861 740a also note that. │ │ │ │ +0001af70: 7468 6520 6964 6561 6c20 6765 6e65 7261 the ideal genera │ │ │ │ +0001af80: 7465 6420 6279 2074 6865 2066 6972 7374 ted by the first │ │ │ │ +0001af90: 2032 2067 656e 6572 6174 6f72 7320 6f66 2 generators of │ │ │ │ +0001afa0: 2049 2064 6566 696e 6573 2061 2073 6d6f I defines a smo │ │ │ │ +0001afb0: 6f74 6820 7363 6865 6d65 2061 6e64 0a69 oth scheme and.i │ │ │ │ +0001afc0: 6e70 7574 2074 6869 7320 696e 666f 726d nput this inform │ │ │ │ +0001afd0: 6174 696f 6e20 696e 746f 2074 6865 206d ation into the m │ │ │ │ +0001afe0: 6574 686f 642e 2054 6869 7320 6d61 7920 ethod. This may │ │ │ │ +0001aff0: 616c 736f 2069 6d70 726f 7665 2063 6f6d also improve com │ │ │ │ +0001b000: 7075 7461 7469 6f6e 0a73 7065 6564 2e0a putation.speed.. │ │ │ │ +0001b010: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001b020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b050: 2d2d 2d2b 0a7c 6931 3020 3a20 7469 6d65 ---+.|i10 : time │ │ │ │ +0001b060: 2045 756c 6572 284a 2c4d 6574 686f 643d Euler(J,Method= │ │ │ │ +0001b070: 3e44 6972 6563 7443 6f6d 706c 6574 6549 >DirectCompleteI │ │ │ │ +0001b080: 6e74 2920 2020 2020 2020 2020 2020 2020 nt) │ │ │ │ +0001b090: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ +0001b0a0: 6564 2030 2e32 3037 3831 7320 2863 7075 ed 0.20781s (cpu │ │ │ │ +0001b0b0: 293b 2030 2e31 3132 3138 3473 2028 7468 ); 0.112184s (th │ │ │ │ +0001b0c0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +0001b0d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001b0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b110: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b120: 0a7c 6f31 3020 3d20 3220 2020 2020 2020 .|o10 = 2 │ │ │ │ +0001b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b150: 2020 2020 2020 2020 2020 2020 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 │ │ │ │ -0001b1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b1c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0001b210: 6931 3120 3a20 7469 6d65 2045 756c 6572 i11 : time Euler │ │ │ │ -0001b220: 284a 2c4d 6574 686f 643d 3e44 6972 6563 (J,Method=>Direc │ │ │ │ -0001b230: 7443 6f6d 706c 6574 6549 6e74 2c49 6e64 tCompleteInt,Ind │ │ │ │ -0001b240: 734f 6653 6d6f 6f74 683d 3e7b 302c 317d sOfSmooth=>{0,1} │ │ │ │ -0001b250: 297c 0a7c 202d 2d20 7573 6564 2030 2e31 )|.| -- used 0.1 │ │ │ │ -0001b260: 3637 3531 3873 2028 6370 7529 3b20 302e 67518s (cpu); 0. │ │ │ │ -0001b270: 3039 3030 3936 3973 2028 7468 7265 6164 0900969s (thread │ │ │ │ -0001b280: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ -0001b290: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001b160: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001b170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b1a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ +0001b1b0: 7469 6d65 2045 756c 6572 284a 2c4d 6574 time Euler(J,Met │ │ │ │ +0001b1c0: 686f 643d 3e44 6972 6563 7443 6f6d 706c hod=>DirectCompl │ │ │ │ +0001b1d0: 6574 6549 6e74 2c49 6e64 734f 6653 6d6f eteInt,IndsOfSmo │ │ │ │ +0001b1e0: 6f74 683d 3e7b 302c 317d 297c 0a7c 202d oth=>{0,1})|.| - │ │ │ │ +0001b1f0: 2d20 7573 6564 2030 2e32 3733 3331 3973 - used 0.273319s │ │ │ │ +0001b200: 2028 6370 7529 3b20 302e 3131 3236 3332 (cpu); 0.112632 │ │ │ │ +0001b210: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +0001b220: 6763 2920 2020 2020 2020 2020 2020 207c gc) | │ │ │ │ +0001b230: 0a7c 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: 2020 207c 0a7c 6f31 3120 3d20 3220 2020 |.|o11 = 2 │ │ │ │ +0001b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b290: 2020 2020 2020 2020 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 │ │ │ │ -0001b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b310: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0001b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b2b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001b2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4e6f -----------+..No │ │ │ │ +0001b300: 7720 636f 6e73 6964 6572 2061 6e20 6578 w consider an ex │ │ │ │ +0001b310: 616d 706c 6520 696e 205c 5050 5e32 205c ample in \PP^2 \ │ │ │ │ +0001b320: 7469 6d65 7320 5c50 505e 322e 0a0a 2b2d times \PP^2...+- │ │ │ │ 0001b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b360: 2d2b 0a0a 4e6f 7720 636f 6e73 6964 6572 -+..Now consider │ │ │ │ -0001b370: 2061 6e20 6578 616d 706c 6520 696e 205c an example in \ │ │ │ │ -0001b380: 5050 5e32 205c 7469 6d65 7320 5c50 505e PP^2 \times \PP^ │ │ │ │ -0001b390: 322e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2...+----------- │ │ │ │ -0001b3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b3c0: 2d2d 2d2b 0a7c 6931 3220 3a20 523d 4d75 ---+.|i12 : R=Mu │ │ │ │ -0001b3d0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ -0001b3e0: 287b 322c 327d 2920 2020 2020 2020 2020 ({2,2}) │ │ │ │ -0001b3f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001b360: 6931 3220 3a20 523d 4d75 6c74 6950 726f i12 : R=MultiPro │ │ │ │ +0001b370: 6a43 6f6f 7264 5269 6e67 287b 322c 327d jCoordRing({2,2} │ │ │ │ +0001b380: 2920 2020 2020 2020 2020 2020 2020 7c0a ) |. │ │ │ │ +0001b390: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001b3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b3b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b3c0: 0a7c 6f31 3220 3d20 5220 2020 2020 2020 .|o12 = R │ │ │ │ +0001b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b3f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b420: 2020 2020 207c 0a7c 6f31 3220 3d20 5220 |.|o12 = R │ │ │ │ -0001b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b420: 207c 0a7c 6f31 3220 3a20 506f 6c79 6e6f |.|o12 : Polyno │ │ │ │ +0001b430: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ 0001b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b450: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b480: 2020 2020 2020 207c 0a7c 6f31 3220 3a20 |.|o12 : │ │ │ │ -0001b490: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ +0001b450: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001b460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b480: 2d2d 2d2b 0a7c 6931 3320 3a20 723d 6765 ---+.|i13 : r=ge │ │ │ │ +0001b490: 6e73 2052 2020 2020 2020 2020 2020 2020 ns R │ │ │ │ 0001b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b4b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001b4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b4e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 ---------+.|i13 │ │ │ │ -0001b4f0: 3a20 723d 6765 6e73 2052 2020 2020 2020 : r=gens R │ │ │ │ -0001b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b510: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001b520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b540: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0001b550: 3320 3d20 7b78 202c 2078 202c 2078 202c 3 = {x , x , x , │ │ │ │ -0001b560: 2078 202c 2078 202c 2078 207d 2020 2020 x , x , x } │ │ │ │ -0001b570: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001b580: 2020 2020 2020 2030 2020 2031 2020 2032 0 1 2 │ │ │ │ -0001b590: 2020 2033 2020 2034 2020 2035 2020 2020 3 4 5 │ │ │ │ -0001b5a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001b5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b5d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b5e0: 7c6f 3133 203a 204c 6973 7420 2020 2020 |o13 : List │ │ │ │ -0001b5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b600: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001b610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0001b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b640: 2b0a 7c69 3134 203a 204b 3d69 6465 616c +.|i14 : K=ideal │ │ │ │ -0001b650: 2872 5f30 5e32 2a72 5f33 2d72 5f34 2a72 (r_0^2*r_3-r_4*r │ │ │ │ -0001b660: 5f31 2a72 5f32 2c72 5f32 5e32 2a72 5f35 _1*r_2,r_2^2*r_5 │ │ │ │ -0001b670: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ -0001b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b6a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b6b0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001b6c0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001b6d0: 2020 207c 0a7c 6f31 3420 3d20 6964 6561 |.|o14 = idea │ │ │ │ -0001b6e0: 6c20 2878 2078 2020 2d20 7820 7820 7820 l (x x - x x x │ │ │ │ -0001b6f0: 2c20 7820 7820 2920 2020 2020 2020 2020 , x x ) │ │ │ │ -0001b700: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001b710: 2020 2020 2030 2033 2020 2020 3120 3220 0 3 1 2 │ │ │ │ -0001b720: 3420 2020 3220 3520 2020 2020 2020 2020 4 2 5 │ │ │ │ -0001b730: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001b740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b760: 2020 2020 2020 7c0a 7c6f 3134 203a 2049 |.|o14 : I │ │ │ │ -0001b770: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ +0001b4b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001b4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b4e0: 2020 2020 207c 0a7c 6f31 3320 3d20 7b78 |.|o13 = {x │ │ │ │ +0001b4f0: 202c 2078 202c 2078 202c 2078 202c 2078 , x , x , x , x │ │ │ │ +0001b500: 202c 2078 207d 2020 2020 2020 2020 2020 , x } │ │ │ │ +0001b510: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001b520: 2030 2020 2031 2020 2032 2020 2033 2020 0 1 2 3 │ │ │ │ +0001b530: 2034 2020 2035 2020 2020 2020 2020 2020 4 5 │ │ │ │ +0001b540: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b570: 2020 2020 2020 2020 7c0a 7c6f 3133 203a |.|o13 : │ │ │ │ +0001b580: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ +0001b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b5a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001b5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 ----------+.|i14 │ │ │ │ +0001b5e0: 203a 204b 3d69 6465 616c 2872 5f30 5e32 : K=ideal(r_0^2 │ │ │ │ +0001b5f0: 2a72 5f33 2d72 5f34 2a72 5f31 2a72 5f32 *r_3-r_4*r_1*r_2 │ │ │ │ +0001b600: 2c72 5f32 5e32 2a72 5f35 297c 0a7c 2020 ,r_2^2*r_5)|.| │ │ │ │ +0001b610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b630: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001b640: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +0001b650: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +0001b660: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001b670: 6f31 3420 3d20 6964 6561 6c20 2878 2078 o14 = ideal (x x │ │ │ │ +0001b680: 2020 2d20 7820 7820 7820 2c20 7820 7820 - x x x , x x │ │ │ │ +0001b690: 2920 2020 2020 2020 2020 2020 2020 7c0a ) |. │ │ │ │ +0001b6a0: 7c20 2020 2020 2020 2020 2020 2020 2030 | 0 │ │ │ │ +0001b6b0: 2033 2020 2020 3120 3220 3420 2020 3220 3 1 2 4 2 │ │ │ │ +0001b6c0: 3520 2020 2020 2020 2020 2020 2020 207c 5 | │ │ │ │ +0001b6d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001b6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b700: 7c0a 7c6f 3134 203a 2049 6465 616c 206f |.|o14 : Ideal o │ │ │ │ +0001b710: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ +0001b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b730: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0001b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b760: 2d2d 2b0a 7c69 3135 203a 2045 756c 6572 --+.|i15 : Euler │ │ │ │ +0001b770: 4b3d 4575 6c65 7228 4b29 2020 2020 2020 K=Euler(K) │ │ │ │ 0001b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b790: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0001b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b7c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a --------+.|i15 : │ │ │ │ -0001b7d0: 2045 756c 6572 4b3d 4575 6c65 7228 4b29 EulerK=Euler(K) │ │ │ │ +0001b790: 2020 207c 0a7c 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 7c6f 3135 203d 2037 2020 |.|o15 = 7 │ │ │ │ +0001b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b7f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0001b800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b820: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ -0001b830: 203d 2037 2020 2020 2020 2020 2020 2020 = 7 │ │ │ │ +0001b7f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b820: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2063 ------+.|i16 : c │ │ │ │ +0001b830: 736d 4b3d 2043 534d 284b 2920 2020 2020 smK= CSM(K) │ │ │ │ 0001b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b850: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0001b860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001b890: 3136 203a 2063 736d 4b3d 2043 534d 284b 16 : csmK= CSM(K │ │ │ │ -0001b8a0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0001b8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001b8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b8e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b8f0: 7c20 2020 2020 2020 2032 2032 2020 2020 | 2 2 │ │ │ │ -0001b900: 2032 2020 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ -0001b910: 3220 2020 2020 2020 2020 2020 2032 207c 2 2 | │ │ │ │ -0001b920: 0a7c 6f31 3620 3d20 3768 2068 2020 2b20 .|o16 = 7h h + │ │ │ │ -0001b930: 3568 2068 2020 2b20 3468 2068 2020 2b20 5h h + 4h h + │ │ │ │ -0001b940: 6820 202b 2033 6820 6820 202b 2068 2020 h + 3h h + h │ │ │ │ -0001b950: 7c0a 7c20 2020 2020 2020 2031 2032 2020 |.| 1 2 │ │ │ │ -0001b960: 2020 2031 2032 2020 2020 2031 2032 2020 1 2 1 2 │ │ │ │ -0001b970: 2020 3120 2020 2020 3120 3220 2020 2032 1 1 2 2 │ │ │ │ -0001b980: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b850: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b880: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001b890: 2020 2032 2032 2020 2020 2032 2020 2020 2 2 2 │ │ │ │ +0001b8a0: 2020 2020 2032 2020 2020 3220 2020 2020 2 2 │ │ │ │ +0001b8b0: 2020 2020 2020 2032 207c 0a7c 6f31 3620 2 |.|o16 │ │ │ │ +0001b8c0: 3d20 3768 2068 2020 2b20 3568 2068 2020 = 7h h + 5h h │ │ │ │ +0001b8d0: 2b20 3468 2068 2020 2b20 6820 202b 2033 + 4h h + h + 3 │ │ │ │ +0001b8e0: 6820 6820 202b 2068 2020 7c0a 7c20 2020 h h + h |.| │ │ │ │ +0001b8f0: 2020 2020 2031 2032 2020 2020 2031 2032 1 2 1 2 │ │ │ │ +0001b900: 2020 2020 2031 2032 2020 2020 3120 2020 1 2 1 │ │ │ │ +0001b910: 2020 3120 3220 2020 2032 207c 0a7c 2020 1 2 2 |.| │ │ │ │ +0001b920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b940: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001b950: 2020 2020 205a 5a5b 6820 2e2e 6820 5d20 ZZ[h ..h ] │ │ │ │ +0001b960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001b980: 2020 2020 2020 2020 2020 3120 2020 3220 1 2 │ │ │ │ 0001b990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9b0: 2020 7c0a 7c20 2020 2020 205a 5a5b 6820 |.| ZZ[h │ │ │ │ -0001b9c0: 2e2e 6820 5d20 2020 2020 2020 2020 2020 ..h ] │ │ │ │ -0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b9f0: 3120 2020 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ +0001b9a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001b9b0: 7c6f 3136 203a 202d 2d2d 2d2d 2d2d 2d2d |o16 : --------- │ │ │ │ +0001b9c0: 2d20 2020 2020 2020 2020 2020 2020 2020 - │ │ │ │ +0001b9d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b9e0: 0a7c 2020 2020 2020 2020 2033 2020 2033 .| 3 3 │ │ │ │ +0001b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba10: 2020 2020 7c0a 7c6f 3136 203a 202d 2d2d |.|o16 : --- │ │ │ │ -0001ba20: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020 ------- │ │ │ │ +0001ba10: 7c0a 7c20 2020 2020 2020 2868 202c 2068 |.| (h , h │ │ │ │ +0001ba20: 2029 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0001ba30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001ba50: 2033 2020 2033 2020 2020 2020 2020 2020 3 3 │ │ │ │ +0001ba40: 207c 0a7c 2020 2020 2020 2020 2031 2020 |.| 1 │ │ │ │ +0001ba50: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0001ba60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba70: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001ba80: 2868 202c 2068 2029 2020 2020 2020 2020 (h , h ) │ │ │ │ -0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001baa0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0001bab0: 2020 2031 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ +0001ba70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001baa0: 2d2d 2d2b 0a7c 6931 3720 3a20 4575 6c65 ---+.|i17 : Eule │ │ │ │ +0001bab0: 724b 3d3d 4575 6c65 7228 6373 6d4b 2920 rK==Euler(csmK) │ │ │ │ 0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bad0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001baf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bb00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720 ---------+.|i17 │ │ │ │ -0001bb10: 3a20 4575 6c65 724b 3d3d 4575 6c65 7228 : EulerK==Euler( │ │ │ │ -0001bb20: 6373 6d4b 2920 2020 2020 2020 2020 2020 csmK) │ │ │ │ -0001bb30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb60: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0001bb70: 3720 3d20 7472 7565 2020 2020 2020 2020 7 = true │ │ │ │ -0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0001bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0001bbd0: 496e 2074 6865 2063 6173 6520 7768 6572 In the case wher │ │ │ │ -0001bbe0: 6520 7468 6520 616d 6269 656e 7420 7370 e the ambient sp │ │ │ │ -0001bbf0: 6163 6520 6973 2061 2074 6f72 6963 2076 ace is a toric v │ │ │ │ -0001bc00: 6172 6965 7479 2077 6869 6368 2069 7320 ariety which is │ │ │ │ -0001bc10: 6e6f 7420 6120 7072 6f64 7563 740a 6f66 not a product.of │ │ │ │ -0001bc20: 2070 726f 6a65 6374 6976 6520 7370 6163 projective spac │ │ │ │ -0001bc30: 6573 2077 6520 6d75 7374 206c 6f61 6420 es we must load │ │ │ │ -0001bc40: 7468 6520 4e6f 726d 616c 546f 7269 6356 the NormalToricV │ │ │ │ -0001bc50: 6172 6965 7469 6573 2070 6163 6b61 6765 arieties package │ │ │ │ -0001bc60: 2061 6e64 206d 7573 740a 616c 736f 2069 and must.also i │ │ │ │ -0001bc70: 6e70 7574 2074 6865 2074 6f72 6963 2076 nput the toric v │ │ │ │ -0001bc80: 6172 6965 7479 2e20 4966 2074 6865 2074 ariety. If the t │ │ │ │ -0001bc90: 6f72 6963 2076 6172 6965 7479 2069 7320 oric variety is │ │ │ │ -0001bca0: 6120 7072 6f64 7563 7420 6f66 2070 726f a product of pro │ │ │ │ -0001bcb0: 6a65 6374 6976 650a 7370 6163 6520 6974 jective.space it │ │ │ │ -0001bcc0: 2069 7320 7265 636f 6d6d 656e 6465 6420 is recommended │ │ │ │ -0001bcd0: 746f 2075 7365 2074 6865 2066 6f72 6d20 to use the form │ │ │ │ -0001bce0: 6162 6f76 6520 7261 7468 6572 2074 6861 above rather tha │ │ │ │ -0001bcf0: 6e20 696e 7075 7474 696e 6720 7468 6520 n inputting the │ │ │ │ -0001bd00: 746f 7269 630a 7661 7269 6574 7920 666f toric.variety fo │ │ │ │ -0001bd10: 7220 6566 6669 6369 656e 6379 2072 6561 r efficiency rea │ │ │ │ -0001bd20: 736f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d sons...+-------- │ │ │ │ -0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd70: 2b0a 7c69 3138 203a 206e 6565 6473 5061 +.|i18 : needsPa │ │ │ │ -0001bd80: 636b 6167 6520 224e 6f72 6d61 6c54 6f72 ckage "NormalTor │ │ │ │ -0001bd90: 6963 5661 7269 6574 6965 7322 2020 2020 icVarieties" │ │ │ │ -0001bda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bdb0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001bad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001bae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001baf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb00: 2020 2020 207c 0a7c 6f31 3720 3d20 7472 |.|o17 = tr │ │ │ │ +0001bb10: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ +0001bb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb30: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001bb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bb60: 2d2d 2d2d 2d2d 2d2b 0a0a 496e 2074 6865 -------+..In the │ │ │ │ +0001bb70: 2063 6173 6520 7768 6572 6520 7468 6520 case where the │ │ │ │ +0001bb80: 616d 6269 656e 7420 7370 6163 6520 6973 ambient space is │ │ │ │ +0001bb90: 2061 2074 6f72 6963 2076 6172 6965 7479 a toric variety │ │ │ │ +0001bba0: 2077 6869 6368 2069 7320 6e6f 7420 6120 which is not a │ │ │ │ +0001bbb0: 7072 6f64 7563 740a 6f66 2070 726f 6a65 product.of proje │ │ │ │ +0001bbc0: 6374 6976 6520 7370 6163 6573 2077 6520 ctive spaces we │ │ │ │ +0001bbd0: 6d75 7374 206c 6f61 6420 7468 6520 4e6f must load the No │ │ │ │ +0001bbe0: 726d 616c 546f 7269 6356 6172 6965 7469 rmalToricVarieti │ │ │ │ +0001bbf0: 6573 2070 6163 6b61 6765 2061 6e64 206d es package and m │ │ │ │ +0001bc00: 7573 740a 616c 736f 2069 6e70 7574 2074 ust.also input t │ │ │ │ +0001bc10: 6865 2074 6f72 6963 2076 6172 6965 7479 he toric variety │ │ │ │ +0001bc20: 2e20 4966 2074 6865 2074 6f72 6963 2076 . If the toric v │ │ │ │ +0001bc30: 6172 6965 7479 2069 7320 6120 7072 6f64 ariety is a prod │ │ │ │ +0001bc40: 7563 7420 6f66 2070 726f 6a65 6374 6976 uct of projectiv │ │ │ │ +0001bc50: 650a 7370 6163 6520 6974 2069 7320 7265 e.space it is re │ │ │ │ +0001bc60: 636f 6d6d 656e 6465 6420 746f 2075 7365 commended to use │ │ │ │ +0001bc70: 2074 6865 2066 6f72 6d20 6162 6f76 6520 the form above │ │ │ │ +0001bc80: 7261 7468 6572 2074 6861 6e20 696e 7075 rather than inpu │ │ │ │ +0001bc90: 7474 696e 6720 7468 6520 746f 7269 630a tting the toric. │ │ │ │ +0001bca0: 7661 7269 6574 7920 666f 7220 6566 6669 variety for effi │ │ │ │ +0001bcb0: 6369 656e 6379 2072 6561 736f 6e73 2e0a ciency reasons.. │ │ │ │ +0001bcc0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001bcd0: 2d2d 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 2b0a 7c69 3138 ----------+.|i18 │ │ │ │ +0001bd10: 203a 206e 6565 6473 5061 636b 6167 6520 : needsPackage │ │ │ │ +0001bd20: 224e 6f72 6d61 6c54 6f72 6963 5661 7269 "NormalToricVari │ │ │ │ +0001bd30: 6574 6965 7322 2020 2020 2020 2020 2020 eties" │ │ │ │ +0001bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bd50: 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bda0: 7c0a 7c6f 3138 203d 204e 6f72 6d61 6c54 |.|o18 = NormalT │ │ │ │ +0001bdb0: 6f72 6963 5661 7269 6574 6965 7320 2020 oricVarieties │ │ │ │ 0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bde0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0001bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be00: 2020 2020 2020 7c0a 7c6f 3138 203d 204e |.|o18 = N │ │ │ │ -0001be10: 6f72 6d61 6c54 6f72 6963 5661 7269 6574 ormalToricVariet │ │ │ │ -0001be20: 6965 7320 2020 2020 2020 2020 2020 2020 ies │ │ │ │ -0001be30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be50: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001be00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be30: 2020 2020 2020 7c0a 7c6f 3138 203a 2050 |.|o18 : P │ │ │ │ +0001be40: 6163 6b61 6765 2020 2020 2020 2020 2020 ackage │ │ │ │ +0001be50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001bea0: 3138 203a 2050 6163 6b61 6765 2020 2020 18 : Package │ │ │ │ -0001beb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bee0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0001bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bf30: 2d2d 2b0a 7c69 3139 203a 2052 686f 203d --+.|i19 : Rho = │ │ │ │ -0001bf40: 207b 7b31 2c30 2c30 7d2c 7b30 2c31 2c30 {{1,0,0},{0,1,0 │ │ │ │ -0001bf50: 7d2c 7b30 2c30 2c31 7d2c 7b2d 312c 2d31 },{0,0,1},{-1,-1 │ │ │ │ -0001bf60: 2c30 7d2c 7b30 2c30 2c2d 317d 7d20 2020 ,0},{0,0,-1}} │ │ │ │ -0001bf70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001bf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be80: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0001be90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001beb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0001bed0: 3139 203a 2052 686f 203d 207b 7b31 2c30 19 : Rho = {{1,0 │ │ │ │ +0001bee0: 2c30 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30 ,0},{0,1,0},{0,0 │ │ │ │ +0001bef0: 2c31 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30 ,1},{-1,-1,0},{0 │ │ │ │ +0001bf00: 2c30 2c2d 317d 7d20 2020 2020 2020 2020 ,0,-1}} │ │ │ │ +0001bf10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bf60: 2020 7c0a 7c6f 3139 203d 207b 7b31 2c20 |.|o19 = {{1, │ │ │ │ +0001bf70: 302c 2030 7d2c 207b 302c 2031 2c20 307d 0, 0}, {0, 1, 0} │ │ │ │ +0001bf80: 2c20 7b30 2c20 302c 2031 7d2c 207b 2d31 , {0, 0, 1}, {-1 │ │ │ │ +0001bf90: 2c20 2d31 2c20 307d 2c20 7b30 2c20 302c , -1, 0}, {0, 0, │ │ │ │ +0001bfa0: 202d 317d 7d20 2020 2020 2020 207c 0a7c -1}} |.| │ │ │ │ 0001bfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bfc0: 2020 2020 2020 2020 7c0a 7c6f 3139 203d |.|o19 = │ │ │ │ -0001bfd0: 207b 7b31 2c20 302c 2030 7d2c 207b 302c {{1, 0, 0}, {0, │ │ │ │ -0001bfe0: 2031 2c20 307d 2c20 7b30 2c20 302c 2031 1, 0}, {0, 0, 1 │ │ │ │ -0001bff0: 7d2c 207b 2d31 2c20 2d31 2c20 307d 2c20 }, {-1, -1, 0}, │ │ │ │ -0001c000: 7b30 2c20 302c 202d 317d 7d20 2020 2020 {0, 0, -1}} │ │ │ │ -0001c010: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001bfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bff0: 2020 2020 2020 2020 7c0a 7c6f 3139 203a |.|o19 : │ │ │ │ +0001c000: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ +0001c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001c060: 7c6f 3139 203a 204c 6973 7420 2020 2020 |o19 : List │ │ │ │ -0001c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c0a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0001c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c0f0: 2d2d 2d2d 2b0a 7c69 3230 203a 2053 6967 ----+.|i20 : Sig │ │ │ │ -0001c100: 6d61 203d 207b 7b30 2c31 2c32 7d2c 7b31 ma = {{0,1,2},{1 │ │ │ │ -0001c110: 2c32 2c33 7d2c 7b30 2c32 2c33 7d2c 7b30 ,2,3},{0,2,3},{0 │ │ │ │ -0001c120: 2c31 2c34 7d2c 7b31 2c33 2c34 7d2c 7b30 ,1,4},{1,3,4},{0 │ │ │ │ -0001c130: 2c33 2c34 7d7d 2020 2020 2020 2020 207c ,3,4}} | │ │ │ │ -0001c140: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c180: 2020 2020 2020 2020 2020 7c0a 7c6f 3230 |.|o20 │ │ │ │ -0001c190: 203d 207b 7b30 2c20 312c 2032 7d2c 207b = {{0, 1, 2}, { │ │ │ │ -0001c1a0: 312c 2032 2c20 337d 2c20 7b30 2c20 322c 1, 2, 3}, {0, 2, │ │ │ │ -0001c1b0: 2033 7d2c 207b 302c 2031 2c20 347d 2c20 3}, {0, 1, 4}, │ │ │ │ -0001c1c0: 7b31 2c20 332c 2034 7d2c 207b 302c 2033 {1, 3, 4}, {0, 3 │ │ │ │ -0001c1d0: 2c20 347d 7d7c 0a7c 2020 2020 2020 2020 , 4}}|.| │ │ │ │ +0001c040: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001c050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0001c090: 7c69 3230 203a 2053 6967 6d61 203d 207b |i20 : Sigma = { │ │ │ │ +0001c0a0: 7b30 2c31 2c32 7d2c 7b31 2c32 2c33 7d2c {0,1,2},{1,2,3}, │ │ │ │ +0001c0b0: 7b30 2c32 2c33 7d2c 7b30 2c31 2c34 7d2c {0,2,3},{0,1,4}, │ │ │ │ +0001c0c0: 7b31 2c33 2c34 7d2c 7b30 2c33 2c34 7d7d {1,3,4},{0,3,4}} │ │ │ │ +0001c0d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001c0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c120: 2020 2020 7c0a 7c6f 3230 203d 207b 7b30 |.|o20 = {{0 │ │ │ │ +0001c130: 2c20 312c 2032 7d2c 207b 312c 2032 2c20 , 1, 2}, {1, 2, │ │ │ │ +0001c140: 337d 2c20 7b30 2c20 322c 2033 7d2c 207b 3}, {0, 2, 3}, { │ │ │ │ +0001c150: 302c 2031 2c20 347d 2c20 7b31 2c20 332c 0, 1, 4}, {1, 3, │ │ │ │ +0001c160: 2034 7d2c 207b 302c 2033 2c20 347d 7d7c 4}, {0, 3, 4}}| │ │ │ │ +0001c170: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +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 2020 2020 2020 2020 7c0a 7c6f 3230 |.|o20 │ │ │ │ +0001c1c0: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +0001c1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c220: 7c0a 7c6f 3230 203a 204c 6973 7420 2020 |.|o20 : List │ │ │ │ -0001c230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c260: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0001c270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c2b0: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 2058 ------+.|i21 : X │ │ │ │ -0001c2c0: 203d 206e 6f72 6d61 6c54 6f72 6963 5661 = normalToricVa │ │ │ │ -0001c2d0: 7269 6574 7928 5268 6f2c 5369 676d 612c riety(Rho,Sigma, │ │ │ │ -0001c2e0: 436f 6566 6669 6369 656e 7452 696e 6720 CoefficientRing │ │ │ │ -0001c2f0: 3d3e 5a5a 2f33 3237 3439 2920 2020 2020 =>ZZ/32749) │ │ │ │ -0001c300: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c200: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001c210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c250: 2b0a 7c69 3231 203a 2058 203d 206e 6f72 +.|i21 : X = nor │ │ │ │ +0001c260: 6d61 6c54 6f72 6963 5661 7269 6574 7928 malToricVariety( │ │ │ │ +0001c270: 5268 6f2c 5369 676d 612c 436f 6566 6669 Rho,Sigma,Coeffi │ │ │ │ +0001c280: 6369 656e 7452 696e 6720 3d3e 5a5a 2f33 cientRing =>ZZ/3 │ │ │ │ +0001c290: 3237 3439 2920 2020 2020 207c 0a7c 2020 2749) |.| │ │ │ │ +0001c2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c2e0: 2020 2020 2020 7c0a 7c6f 3231 203d 2058 |.|o21 = X │ │ │ │ +0001c2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c340: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001c350: 3231 203d 2058 2020 2020 2020 2020 2020 21 = X │ │ │ │ +0001c330: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c390: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001c370: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0001c380: 3231 203a 204e 6f72 6d61 6c54 6f72 6963 21 : NormalToric │ │ │ │ +0001c390: 5661 7269 6574 7920 2020 2020 2020 2020 Variety │ │ │ │ 0001c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c3e0: 2020 7c0a 7c6f 3231 203a 204e 6f72 6d61 |.|o21 : Norma │ │ │ │ -0001c3f0: 6c54 6f72 6963 5661 7269 6574 7920 2020 lToricVariety │ │ │ │ -0001c400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c420: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0001c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c470: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a --------+.|i22 : │ │ │ │ -0001c480: 2043 6865 636b 546f 7269 6356 6172 6965 CheckToricVarie │ │ │ │ -0001c490: 7479 5661 6c69 6428 5829 2020 2020 2020 tyValid(X) │ │ │ │ -0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c4c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c3c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001c3d0: 2d2d 2d2d 2d2d 2d2d 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 2b0a 7c69 3232 203a 2043 6865 636b --+.|i22 : Check │ │ │ │ +0001c420: 546f 7269 6356 6172 6965 7479 5661 6c69 ToricVarietyVali │ │ │ │ +0001c430: 6428 5829 2020 2020 2020 2020 2020 2020 d(X) │ │ │ │ +0001c440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c450: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 │ │ │ │ +0001c4a0: 2020 2020 2020 2020 7c0a 7c6f 3232 203d |.|o22 = │ │ │ │ +0001c4b0: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ +0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c500: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001c510: 7c6f 3232 203d 2074 7275 6520 2020 2020 |o22 = true │ │ │ │ -0001c520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c550: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0001c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c5a0: 2d2d 2d2d 2b0a 7c69 3233 203a 2052 3d72 ----+.|i23 : R=r │ │ │ │ -0001c5b0: 696e 6728 5829 2020 2020 2020 2020 2020 ing(X) │ │ │ │ +0001c4f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001c500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0001c540: 7c69 3233 203a 2052 3d72 696e 6728 5829 |i23 : R=ring(X) │ │ │ │ +0001c550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c580: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001c590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c5e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001c5f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001c5d0: 2020 2020 7c0a 7c6f 3233 203d 2052 2020 |.|o23 = R │ │ │ │ +0001c5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c630: 2020 2020 2020 2020 2020 7c0a 7c6f 3233 |.|o23 │ │ │ │ -0001c640: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +0001c610: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c620: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001c630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c680: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001c660: 2020 2020 2020 2020 2020 7c0a 7c6f 3233 |.|o23 │ │ │ │ +0001c670: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ +0001c680: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 0001c690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c6d0: 7c0a 7c6f 3233 203a 2050 6f6c 796e 6f6d |.|o23 : Polynom │ │ │ │ -0001c6e0: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ -0001c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c710: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c760: 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a 2049 ------+.|i24 : I │ │ │ │ -0001c770: 3d69 6465 616c 2852 5f30 5e34 2a52 5f31 =ideal(R_0^4*R_1 │ │ │ │ -0001c780: 2c52 5f30 2a52 5f33 2a52 5f34 2a52 5f32 ,R_0*R_3*R_4*R_2 │ │ │ │ -0001c790: 2d52 5f32 5e32 2a52 5f30 5e32 2920 2020 -R_2^2*R_0^2) │ │ │ │ -0001c7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c6b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c700: 2b0a 7c69 3234 203a 2049 3d69 6465 616c +.|i24 : I=ideal │ │ │ │ +0001c710: 2852 5f30 5e34 2a52 5f31 2c52 5f30 2a52 (R_0^4*R_1,R_0*R │ │ │ │ +0001c720: 5f33 2a52 5f34 2a52 5f32 2d52 5f32 5e32 _3*R_4*R_2-R_2^2 │ │ │ │ +0001c730: 2a52 5f30 5e32 2920 2020 2020 2020 2020 *R_0^2) │ │ │ │ +0001c740: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001c750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c790: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001c7a0: 2020 2020 2020 2034 2020 2020 2020 2032 4 2 │ │ │ │ +0001c7b0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001c800: 2020 2020 2020 2020 2020 2020 2034 2020 4 │ │ │ │ -0001c810: 2020 2020 2032 2032 2020 2020 2020 2020 2 2 │ │ │ │ -0001c820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c840: 2020 2020 2020 207c 0a7c 6f32 3420 3d20 |.|o24 = │ │ │ │ -0001c850: 6964 6561 6c20 2878 2078 202c 202d 2078 ideal (x x , - x │ │ │ │ -0001c860: 2078 2020 2b20 7820 7820 7820 7820 2920 x + x x x x ) │ │ │ │ -0001c870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c7e0: 207c 0a7c 6f32 3420 3d20 6964 6561 6c20 |.|o24 = ideal │ │ │ │ +0001c7f0: 2878 2078 202c 202d 2078 2078 2020 2b20 (x x , - x x + │ │ │ │ +0001c800: 7820 7820 7820 7820 2920 2020 2020 2020 x x x x ) │ │ │ │ +0001c810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c820: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001c830: 2020 2020 2020 2020 2020 2020 2030 2031 0 1 │ │ │ │ +0001c840: 2020 2020 2030 2032 2020 2020 3020 3220 0 2 0 2 │ │ │ │ +0001c850: 3320 3420 2020 2020 2020 2020 2020 2020 3 4 │ │ │ │ +0001c860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c870: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0001c880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c890: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c8a0: 2020 2030 2031 2020 2020 2030 2032 2020 0 1 0 2 │ │ │ │ -0001c8b0: 2020 3020 3220 3320 3420 2020 2020 2020 0 2 3 4 │ │ │ │ -0001c8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c8d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c8c0: 2020 7c0a 7c6f 3234 203a 2049 6465 616c |.|o24 : Ideal │ │ │ │ +0001c8d0: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ 0001c8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c920: 2020 2020 2020 2020 7c0a 7c6f 3234 203a |.|o24 : │ │ │ │ -0001c930: 2049 6465 616c 206f 6620 5220 2020 2020 Ideal of R │ │ │ │ -0001c940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c970: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0001c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001c9c0: 7c69 3235 203a 2063 736d 493d 4353 4d28 |i25 : csmI=CSM( │ │ │ │ -0001c9d0: 582c 4929 2020 2020 2020 2020 2020 2020 X,I) │ │ │ │ -0001c9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001c900: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001c910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c950: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3235 203a --------+.|i25 : │ │ │ │ +0001c960: 2063 736d 493d 4353 4d28 582c 4929 2020 csmI=CSM(X,I) │ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c9e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001c9f0: 7c20 2020 2020 2020 2032 2020 2020 2020 | 2 │ │ │ │ +0001ca00: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0001ca10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ca20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca50: 2020 2020 7c0a 7c20 2020 2020 2020 2032 |.| 2 │ │ │ │ -0001ca60: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0001ca30: 2020 2020 2020 2020 207c 0a7c 6f32 3520 |.|o25 │ │ │ │ +0001ca40: 3d20 3578 2078 2020 2b20 3378 2020 2b20 = 5x x + 3x + │ │ │ │ +0001ca50: 3478 2078 2020 2b20 7820 2020 2020 2020 4x x + x │ │ │ │ +0001ca60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ca70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001caa0: 0a7c 6f32 3520 3d20 3578 2078 2020 2b20 .|o25 = 5x x + │ │ │ │ -0001cab0: 3378 2020 2b20 3478 2078 2020 2b20 7820 3x + 4x x + x │ │ │ │ -0001cac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001caf0: 2020 2020 2033 2034 2020 2020 2033 2020 3 4 3 │ │ │ │ -0001cb00: 2020 2033 2034 2020 2020 3320 2020 2020 3 4 3 │ │ │ │ -0001cb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ca80: 2020 2020 7c0a 7c20 2020 2020 2020 2033 |.| 3 │ │ │ │ +0001ca90: 2034 2020 2020 2033 2020 2020 2033 2034 4 3 3 4 │ │ │ │ +0001caa0: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ +0001cab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cac0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001cad0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001cae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001caf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 3020 2020 3420 2020 0 4 │ │ │ │ -0001cbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc10: 2020 2020 2020 7c0a 7c6f 3235 203a 202d |.|o25 : - │ │ │ │ -0001cc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cc40: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020 -------- │ │ │ │ -0001cc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc60: 207c 0a7c 2020 2020 2020 2878 2078 202c |.| (x x , │ │ │ │ -0001cc70: 2078 2078 2078 202c 2078 2020 2d20 7820 x x x , x - x │ │ │ │ -0001cc80: 2c20 7820 202d 2078 202c 2078 2020 2d20 , x - x , x - │ │ │ │ -0001cc90: 7820 2920 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ -0001cca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001ccb0: 2020 2020 2020 2032 2034 2020 2030 2031 2 4 0 1 │ │ │ │ -0001ccc0: 2033 2020 2030 2020 2020 3320 2020 3120 3 0 3 1 │ │ │ │ -0001ccd0: 2020 2033 2020 2032 2020 2020 3420 2020 3 2 4 │ │ │ │ -0001cce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ccf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cd40: 2d2d 2b0a 7c69 3236 203a 2045 756c 6572 --+.|i26 : Euler │ │ │ │ -0001cd50: 493d 4575 6c65 7228 582c 4929 2020 2020 I=Euler(X,I) │ │ │ │ +0001cb80: 2020 3020 2020 3420 2020 2020 2020 2020 0 4 │ │ │ │ +0001cb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cbb0: 7c0a 7c6f 3235 203a 202d 2d2d 2d2d 2d2d |.|o25 : ------- │ │ │ │ +0001cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cbe0: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ +0001cbf0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001cc00: 2020 2020 2878 2078 202c 2078 2078 2078 (x x , x x x │ │ │ │ +0001cc10: 202c 2078 2020 2d20 7820 2c20 7820 202d , x - x , x - │ │ │ │ +0001cc20: 2078 202c 2078 2020 2d20 7820 2920 2020 x , x - x ) │ │ │ │ +0001cc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cc40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001cc50: 2032 2034 2020 2030 2031 2033 2020 2030 2 4 0 1 3 0 │ │ │ │ +0001cc60: 2020 2020 3320 2020 3120 2020 2033 2020 3 1 3 │ │ │ │ +0001cc70: 2032 2020 2020 3420 2020 2020 2020 2020 2 4 │ │ │ │ +0001cc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cc90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0001cca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ccb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ccc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ccd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0001cce0: 3236 203a 2045 756c 6572 493d 4575 6c65 26 : EulerI=Eule │ │ │ │ +0001ccf0: 7228 582c 4929 2020 2020 2020 2020 2020 r(X,I) │ │ │ │ +0001cd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cd20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001cd70: 2020 7c0a 7c6f 3236 203d 2035 2020 2020 |.|o26 = 5 │ │ │ │ +0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cdd0: 2020 2020 2020 2020 7c0a 7c6f 3236 203d |.|o26 = │ │ │ │ -0001cde0: 2035 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ -0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0001ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ce50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ce60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001ce70: 7c69 3237 203a 2045 756c 6572 2863 736d |i27 : Euler(csm │ │ │ │ -0001ce80: 4929 3d3d 4575 6c65 7249 2020 2020 2020 I)==EulerI │ │ │ │ -0001ce90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ceb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001cdb0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cdd0: 2d2d 2d2d 2d2d 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 2b0a 7c69 3237 203a --------+.|i27 : │ │ │ │ +0001ce10: 2045 756c 6572 2863 736d 4929 3d3d 4575 Euler(csmI)==Eu │ │ │ │ +0001ce20: 6c65 7249 2020 2020 2020 2020 2020 2020 lerI │ │ │ │ +0001ce30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ce40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ce50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001ce60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ce70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ce80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ce90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001cea0: 7c6f 3237 203d 2074 7275 6520 2020 2020 |o27 = true │ │ │ │ +0001ceb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ced0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf00: 2020 2020 7c0a 7c6f 3237 203d 2074 7275 |.|o27 = tru │ │ │ │ -0001cf10: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -0001cf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001cf50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0001cf60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 6c6c ----------+..All │ │ │ │ -0001cfa0: 2074 6865 2065 7861 6d70 6c65 7320 7765 the examples we │ │ │ │ -0001cfb0: 7265 2064 6f6e 6520 7573 696e 6720 7379 re done using sy │ │ │ │ -0001cfc0: 6d62 6f6c 6963 2063 6f6d 7075 7461 7469 mbolic computati │ │ │ │ -0001cfd0: 6f6e 7320 7769 7468 2047 725c 226f 626e ons with Gr\"obn │ │ │ │ -0001cfe0: 6572 2062 6173 6573 2e0a 4368 616e 6769 er bases..Changi │ │ │ │ -0001cff0: 6e67 2074 6865 206f 7074 696f 6e20 2a6e ng the option *n │ │ │ │ -0001d000: 6f74 6520 436f 6d70 4d65 7468 6f64 3a20 ote CompMethod: │ │ │ │ -0001d010: 436f 6d70 4d65 7468 6f64 2c20 746f 2062 CompMethod, to b │ │ │ │ -0001d020: 6572 7469 6e69 2077 696c 6c20 646f 2074 ertini will do t │ │ │ │ -0001d030: 6865 206d 6169 6e0a 636f 6d70 7574 6174 he main.computat │ │ │ │ -0001d040: 696f 6e73 206e 756d 6572 6963 616c 6c79 ions numerically │ │ │ │ -0001d050: 2c20 7072 6f76 6964 6564 2042 6572 7469 , provided Berti │ │ │ │ -0001d060: 6e69 2069 7320 2a6e 6f74 6520 696e 7374 ni is *note inst │ │ │ │ -0001d070: 616c 6c65 6420 616e 6420 636f 6e66 6967 alled and config │ │ │ │ -0001d080: 7572 6564 3a0a 636f 6e66 6967 7572 696e ured:.configurin │ │ │ │ -0001d090: 6720 4265 7274 696e 692c 2e20 4e6f 7465 g Bertini,. Note │ │ │ │ -0001d0a0: 2074 6861 7420 7468 6520 6265 7274 696e that the bertin │ │ │ │ -0001d0b0: 6920 616e 6420 506e 5265 7369 6475 616c i and PnResidual │ │ │ │ -0001d0c0: 206f 7074 696f 6e73 206d 6179 206f 6e6c options may onl │ │ │ │ -0001d0d0: 7920 6265 0a75 7365 6420 666f 7220 7375 y be.used for su │ │ │ │ -0001d0e0: 6273 6368 656d 6573 206f 6620 5c50 505e bschemes of \PP^ │ │ │ │ -0001d0f0: 6e2e 0a0a 4f62 7365 7276 6520 7468 6174 n...Observe that │ │ │ │ -0001d100: 2074 6865 2061 6c67 6f72 6974 686d 2069 the algorithm i │ │ │ │ -0001d110: 7320 6120 7072 6f62 6162 696c 6973 7469 s a probabilisti │ │ │ │ -0001d120: 6320 616c 676f 7269 7468 6d20 616e 6420 c algorithm and │ │ │ │ -0001d130: 6d61 7920 6769 7665 2061 2077 726f 6e67 may give a wrong │ │ │ │ -0001d140: 0a61 6e73 7765 7220 7769 7468 2061 2073 .answer with a s │ │ │ │ -0001d150: 6d61 6c6c 2062 7574 206e 6f6e 7a65 726f mall but nonzero │ │ │ │ -0001d160: 2070 726f 6261 6269 6c69 7479 2e20 5265 probability. Re │ │ │ │ -0001d170: 6164 206d 6f72 6520 756e 6465 7220 2a6e ad more under *n │ │ │ │ -0001d180: 6f74 650a 7072 6f62 6162 696c 6973 7469 ote.probabilisti │ │ │ │ -0001d190: 6320 616c 676f 7269 7468 6d3a 2070 726f c algorithm: pro │ │ │ │ -0001d1a0: 6261 6269 6c69 7374 6963 2061 6c67 6f72 babilistic algor │ │ │ │ -0001d1b0: 6974 686d 2c2e 0a0a 5761 7973 2074 6f20 ithm,...Ways to │ │ │ │ -0001d1c0: 7573 6520 4575 6c65 723a 0a3d 3d3d 3d3d use Euler:.===== │ │ │ │ -0001d1d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -0001d1e0: 202a 2022 4575 6c65 7228 4964 6561 6c29 * "Euler(Ideal) │ │ │ │ -0001d1f0: 220a 2020 2a20 2245 756c 6572 2852 696e ". * "Euler(Rin │ │ │ │ -0001d200: 6745 6c65 6d65 6e74 2922 0a0a 466f 7220 gElement)"..For │ │ │ │ -0001d210: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0001d220: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001d230: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0001d240: 6f74 6520 4575 6c65 723a 2045 756c 6572 ote Euler: Euler │ │ │ │ -0001d250: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -0001d260: 686f 6420 6675 6e63 7469 6f6e 2077 6974 hod function wit │ │ │ │ -0001d270: 6820 6f70 7469 6f6e 733a 0a28 4d61 6361 h options:.(Maca │ │ │ │ -0001d280: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -0001d290: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ -0001d2a0: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ -0001d2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d2f0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -0001d300: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -0001d310: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -0001d320: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -0001d330: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ -0001d340: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+ds/M2/Macaul │ │ │ │ -0001d350: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ -0001d360: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ -0001d370: 7365 732e 6d32 3a32 3331 323a 302e 0a1f ses.m2:2312:0... │ │ │ │ -0001d380: 0a46 696c 653a 2043 6861 7261 6374 6572 .File: Character │ │ │ │ -0001d390: 6973 7469 6343 6c61 7373 6573 2e69 6e66 isticClasses.inf │ │ │ │ -0001d3a0: 6f2c 204e 6f64 653a 2045 756c 6572 4166 o, Node: EulerAf │ │ │ │ -0001d3b0: 6669 6e65 2c20 4e65 7874 3a20 496e 6473 fine, Next: Inds │ │ │ │ -0001d3c0: 4f66 536d 6f6f 7468 2c20 5072 6576 3a20 OfSmooth, Prev: │ │ │ │ -0001d3d0: 4575 6c65 722c 2055 703a 2054 6f70 0a0a Euler, Up: Top.. │ │ │ │ -0001d3e0: 4575 6c65 7241 6666 696e 6520 2d2d 2054 EulerAffine -- T │ │ │ │ -0001d3f0: 6865 2045 756c 6572 2043 6861 7261 6374 he Euler Charact │ │ │ │ -0001d400: 6572 6973 7469 6320 6f66 2061 6e20 6166 eristic of an af │ │ │ │ -0001d410: 6669 6e65 2076 6172 6965 7479 2e0a 2a2a fine variety..** │ │ │ │ -0001d420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0001d460: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0001d470: 2045 756c 6572 4166 6669 6e65 2049 0a20 EulerAffine I. │ │ │ │ -0001d480: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -0001d490: 202a 2049 2c20 616e 202a 6e6f 7465 2069 * I, an *note i │ │ │ │ -0001d4a0: 6465 616c 3a20 284d 6163 6175 6c61 7932 deal: (Macaulay2 │ │ │ │ -0001d4b0: 446f 6329 4964 6561 6c2c 2c20 616e 2069 Doc)Ideal,, an i │ │ │ │ -0001d4c0: 6465 616c 2069 6e20 6120 706f 6c79 6e6f deal in a polyno │ │ │ │ -0001d4d0: 6d69 616c 2072 696e 670a 2020 2020 2020 mial ring. │ │ │ │ -0001d4e0: 2020 6f76 6572 2061 2066 6965 6c64 2064 over a field d │ │ │ │ -0001d4f0: 6566 696e 696e 6720 616e 2061 6666 696e efining an affin │ │ │ │ -0001d500: 6520 7661 7269 6574 792e 0a20 202a 204f e variety.. * O │ │ │ │ -0001d510: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -0001d520: 6120 2a6e 6f74 6520 7269 6e67 2065 6c65 a *note ring ele │ │ │ │ -0001d530: 6d65 6e74 3a20 284d 6163 6175 6c61 7932 ment: (Macaulay2 │ │ │ │ -0001d540: 446f 6329 5269 6e67 456c 656d 656e 742c Doc)RingElement, │ │ │ │ -0001d550: 2c20 7468 6520 4575 6c65 720a 2020 2020 , the Euler. │ │ │ │ -0001d560: 2020 2020 6368 6172 6163 7465 7269 7374 characterist │ │ │ │ -0001d570: 6963 0a0a 4465 7363 7269 7074 696f 6e0a ic..Description. │ │ │ │ -0001d580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ -0001d590: 7320 636f 6d6d 616e 6420 636f 6d70 7574 s command comput │ │ │ │ -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 ---------------- │ │ │ │ +0001cee0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001cef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cf30: 2d2d 2d2d 2b0a 0a41 6c6c 2074 6865 2065 ----+..All the e │ │ │ │ +0001cf40: 7861 6d70 6c65 7320 7765 7265 2064 6f6e xamples were don │ │ │ │ +0001cf50: 6520 7573 696e 6720 7379 6d62 6f6c 6963 e using symbolic │ │ │ │ +0001cf60: 2063 6f6d 7075 7461 7469 6f6e 7320 7769 computations wi │ │ │ │ +0001cf70: 7468 2047 725c 226f 626e 6572 2062 6173 th Gr\"obner bas │ │ │ │ +0001cf80: 6573 2e0a 4368 616e 6769 6e67 2074 6865 es..Changing the │ │ │ │ +0001cf90: 206f 7074 696f 6e20 2a6e 6f74 6520 436f option *note Co │ │ │ │ +0001cfa0: 6d70 4d65 7468 6f64 3a20 436f 6d70 4d65 mpMethod: CompMe │ │ │ │ +0001cfb0: 7468 6f64 2c20 746f 2062 6572 7469 6e69 thod, to bertini │ │ │ │ +0001cfc0: 2077 696c 6c20 646f 2074 6865 206d 6169 will do the mai │ │ │ │ +0001cfd0: 6e0a 636f 6d70 7574 6174 696f 6e73 206e n.computations n │ │ │ │ +0001cfe0: 756d 6572 6963 616c 6c79 2c20 7072 6f76 umerically, prov │ │ │ │ +0001cff0: 6964 6564 2042 6572 7469 6e69 2069 7320 ided Bertini is │ │ │ │ +0001d000: 2a6e 6f74 6520 696e 7374 616c 6c65 6420 *note installed │ │ │ │ +0001d010: 616e 6420 636f 6e66 6967 7572 6564 3a0a and configured:. │ │ │ │ +0001d020: 636f 6e66 6967 7572 696e 6720 4265 7274 configuring Bert │ │ │ │ +0001d030: 696e 692c 2e20 4e6f 7465 2074 6861 7420 ini,. Note that │ │ │ │ +0001d040: 7468 6520 6265 7274 696e 6920 616e 6420 the bertini and │ │ │ │ +0001d050: 506e 5265 7369 6475 616c 206f 7074 696f PnResidual optio │ │ │ │ +0001d060: 6e73 206d 6179 206f 6e6c 7920 6265 0a75 ns may only be.u │ │ │ │ +0001d070: 7365 6420 666f 7220 7375 6273 6368 656d sed for subschem │ │ │ │ +0001d080: 6573 206f 6620 5c50 505e 6e2e 0a0a 4f62 es of \PP^n...Ob │ │ │ │ +0001d090: 7365 7276 6520 7468 6174 2074 6865 2061 serve that the a │ │ │ │ +0001d0a0: 6c67 6f72 6974 686d 2069 7320 6120 7072 lgorithm is a pr │ │ │ │ +0001d0b0: 6f62 6162 696c 6973 7469 6320 616c 676f obabilistic algo │ │ │ │ +0001d0c0: 7269 7468 6d20 616e 6420 6d61 7920 6769 rithm and may gi │ │ │ │ +0001d0d0: 7665 2061 2077 726f 6e67 0a61 6e73 7765 ve a wrong.answe │ │ │ │ +0001d0e0: 7220 7769 7468 2061 2073 6d61 6c6c 2062 r with a small b │ │ │ │ +0001d0f0: 7574 206e 6f6e 7a65 726f 2070 726f 6261 ut nonzero proba │ │ │ │ +0001d100: 6269 6c69 7479 2e20 5265 6164 206d 6f72 bility. Read mor │ │ │ │ +0001d110: 6520 756e 6465 7220 2a6e 6f74 650a 7072 e under *note.pr │ │ │ │ +0001d120: 6f62 6162 696c 6973 7469 6320 616c 676f obabilistic algo │ │ │ │ +0001d130: 7269 7468 6d3a 2070 726f 6261 6269 6c69 rithm: probabili │ │ │ │ +0001d140: 7374 6963 2061 6c67 6f72 6974 686d 2c2e stic algorithm,. │ │ │ │ +0001d150: 0a0a 5761 7973 2074 6f20 7573 6520 4575 ..Ways to use Eu │ │ │ │ +0001d160: 6c65 723a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ler:.=========== │ │ │ │ +0001d170: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4575 =======.. * "Eu │ │ │ │ +0001d180: 6c65 7228 4964 6561 6c29 220a 2020 2a20 ler(Ideal)". * │ │ │ │ +0001d190: 2245 756c 6572 2852 696e 6745 6c65 6d65 "Euler(RingEleme │ │ │ │ +0001d1a0: 6e74 2922 0a0a 466f 7220 7468 6520 7072 nt)"..For the pr │ │ │ │ +0001d1b0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +0001d1c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +0001d1d0: 206f 626a 6563 7420 2a6e 6f74 6520 4575 object *note Eu │ │ │ │ +0001d1e0: 6c65 723a 2045 756c 6572 2c20 6973 2061 ler: Euler, is a │ │ │ │ +0001d1f0: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ +0001d200: 6e63 7469 6f6e 2077 6974 6820 6f70 7469 nction with opti │ │ │ │ +0001d210: 6f6e 733a 0a28 4d61 6361 756c 6179 3244 ons:.(Macaulay2D │ │ │ │ +0001d220: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +0001d230: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +0001d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +0001d290: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +0001d2a0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +0001d2b0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +0001d2c0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +0001d2d0: 756c 6179 322d 312e 3235 2e31 312b 6473 ulay2-1.25.11+ds │ │ │ │ +0001d2e0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +0001d2f0: 636b 6167 6573 2f0a 4368 6172 6163 7465 ckages/.Characte │ │ │ │ +0001d300: 7269 7374 6963 436c 6173 7365 732e 6d32 risticClasses.m2 │ │ │ │ +0001d310: 3a32 3331 323a 302e 0a1f 0a46 696c 653a :2312:0....File: │ │ │ │ +0001d320: 2043 6861 7261 6374 6572 6973 7469 6343 CharacteristicC │ │ │ │ +0001d330: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64 lasses.info, Nod │ │ │ │ +0001d340: 653a 2045 756c 6572 4166 6669 6e65 2c20 e: EulerAffine, │ │ │ │ +0001d350: 4e65 7874 3a20 496e 6473 4f66 536d 6f6f Next: IndsOfSmoo │ │ │ │ +0001d360: 7468 2c20 5072 6576 3a20 4575 6c65 722c th, Prev: Euler, │ │ │ │ +0001d370: 2055 703a 2054 6f70 0a0a 4575 6c65 7241 Up: Top..EulerA │ │ │ │ +0001d380: 6666 696e 6520 2d2d 2054 6865 2045 756c ffine -- The Eul │ │ │ │ +0001d390: 6572 2043 6861 7261 6374 6572 6973 7469 er Characteristi │ │ │ │ +0001d3a0: 6320 6f66 2061 6e20 6166 6669 6e65 2076 c of an affine v │ │ │ │ +0001d3b0: 6172 6965 7479 2e0a 2a2a 2a2a 2a2a 2a2a ariety..******** │ │ │ │ +0001d3c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001d3d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001d3e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001d3f0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +0001d400: 3a20 0a20 2020 2020 2020 2045 756c 6572 : . Euler │ │ │ │ +0001d410: 4166 6669 6e65 2049 0a20 202a 2049 6e70 Affine I. * Inp │ │ │ │ +0001d420: 7574 733a 0a20 2020 2020 202a 2049 2c20 uts:. * I, │ │ │ │ +0001d430: 616e 202a 6e6f 7465 2069 6465 616c 3a20 an *note ideal: │ │ │ │ +0001d440: 284d 6163 6175 6c61 7932 446f 6329 4964 (Macaulay2Doc)Id │ │ │ │ +0001d450: 6561 6c2c 2c20 616e 2069 6465 616c 2069 eal,, an ideal i │ │ │ │ +0001d460: 6e20 6120 706f 6c79 6e6f 6d69 616c 2072 n a polynomial r │ │ │ │ +0001d470: 696e 670a 2020 2020 2020 2020 6f76 6572 ing. over │ │ │ │ +0001d480: 2061 2066 6965 6c64 2064 6566 696e 696e a field definin │ │ │ │ +0001d490: 6720 616e 2061 6666 696e 6520 7661 7269 g an affine vari │ │ │ │ +0001d4a0: 6574 792e 0a20 202a 204f 7574 7075 7473 ety.. * Outputs │ │ │ │ +0001d4b0: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ +0001d4c0: 6520 7269 6e67 2065 6c65 6d65 6e74 3a20 e ring element: │ │ │ │ +0001d4d0: 284d 6163 6175 6c61 7932 446f 6329 5269 (Macaulay2Doc)Ri │ │ │ │ +0001d4e0: 6e67 456c 656d 656e 742c 2c20 7468 6520 ngElement,, the │ │ │ │ +0001d4f0: 4575 6c65 720a 2020 2020 2020 2020 6368 Euler. ch │ │ │ │ +0001d500: 6172 6163 7465 7269 7374 6963 0a0a 4465 aracteristic..De │ │ │ │ +0001d510: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +0001d520: 3d3d 3d3d 3d0a 0a54 6869 7320 636f 6d6d =====..This comm │ │ │ │ +0001d530: 616e 6420 636f 6d70 7574 6573 2074 6865 and computes the │ │ │ │ +0001d540: 2045 756c 6572 2063 6861 7261 6374 6572 Euler character │ │ │ │ +0001d550: 6973 7469 6320 6f66 2061 2063 6f6d 706c istic of a compl │ │ │ │ +0001d560: 6578 2061 6666 696e 6520 7661 7269 6574 ex affine variet │ │ │ │ +0001d570: 792e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d y...+----------- │ │ │ │ +0001d580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0001d5b0: 3120 3a20 6b6b 3d5a 5a2f 3332 3734 393b 1 : kk=ZZ/32749; │ │ │ │ +0001d5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d5e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0001d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d610: 2d2d 2b0a 7c69 3120 3a20 6b6b 3d5a 5a2f --+.|i1 : kk=ZZ/ │ │ │ │ -0001d620: 3332 3734 393b 2020 2020 2020 2020 2020 32749; │ │ │ │ -0001d630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d640: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0001d650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d680: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 523d ------+.|i2 : R= │ │ │ │ -0001d690: 6b6b 5b78 5f31 2e2e 785f 335d 2020 2020 kk[x_1..x_3] │ │ │ │ +0001d610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d620: 2b0a 7c69 3220 3a20 523d 6b6b 5b78 5f31 +.|i2 : R=kk[x_1 │ │ │ │ +0001d630: 2e2e 785f 335d 2020 2020 2020 2020 2020 ..x_3] │ │ │ │ +0001d640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d650: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d690: 2020 2020 7c0a 7c6f 3220 3d20 5220 2020 |.|o2 = R │ │ │ │ 0001d6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d6c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001d6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001d6d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0001d6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d6f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -0001d700: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ -0001d710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d700: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ +0001d710: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ 0001d720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d730: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001d740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d760: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d770: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c |o2 : Polynomial │ │ │ │ -0001d780: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ -0001d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d7a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d7e0: 2d2d 2b0a 7c69 3320 3a20 493d 6964 6561 --+.|i3 : I=idea │ │ │ │ -0001d7f0: 6c28 785f 315e 322b 785f 325e 322b 785f l(x_1^2+x_2^2+x_ │ │ │ │ -0001d800: 335e 322d 3129 2020 2020 2020 2020 2020 3^2-1) │ │ │ │ -0001d810: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001d820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +0001d730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d740: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001d750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0001d780: 3320 3a20 493d 6964 6561 6c28 785f 315e 3 : I=ideal(x_1^ │ │ │ │ +0001d790: 322b 785f 325e 322b 785f 335e 322d 3129 2+x_2^2+x_3^2-1) │ │ │ │ +0001d7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d7b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001d7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d7f0: 7c0a 7c20 2020 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ +0001d800: 2020 2020 3220 2020 2032 2020 2020 2020 2 2 │ │ │ │ +0001d810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d820: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +0001d830: 3d20 6964 6561 6c28 7820 202b 2078 2020 = ideal(x + x │ │ │ │ +0001d840: 2b20 7820 202d 2031 2920 2020 2020 2020 + x - 1) │ │ │ │ +0001d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d860: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001d870: 2020 2031 2020 2020 3220 2020 2033 2020 1 2 3 │ │ │ │ 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) │ │ │ │ +0001d890: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001d8a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 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 │ │ │ │ +0001d8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d8d0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ +0001d8e0: 4964 6561 6c20 6f66 2052 2020 2020 2020 Ideal of R │ │ │ │ 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 2020 │ │ │ │ -0001d930: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d940: 7c6f 3320 3a20 4964 6561 6c20 6f66 2052 |o3 : Ideal of R │ │ │ │ -0001d950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d970: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d9b0: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2045 --+.|i4 : time E │ │ │ │ -0001d9c0: 756c 6572 4166 6669 6e65 2049 2020 2020 ulerAffine I │ │ │ │ +0001d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d910: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0001d950: 3420 3a20 7469 6d65 2045 756c 6572 4166 4 : time EulerAf │ │ │ │ +0001d960: 6669 6e65 2049 2020 2020 2020 2020 2020 fine I │ │ │ │ +0001d970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d980: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ +0001d990: 6420 302e 3037 3735 3631 3173 2028 6370 d 0.0775611s (cp │ │ │ │ +0001d9a0: 7529 3b20 302e 3036 3634 3637 3573 2028 u); 0.0664675s ( │ │ │ │ +0001d9b0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +0001d9c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d9e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001d9f0: 2d2d 2075 7365 6420 302e 3035 3232 3932 -- used 0.052292 │ │ │ │ -0001da00: 3673 2028 6370 7529 3b20 302e 3035 3137 6s (cpu); 0.0517 │ │ │ │ -0001da10: 3535 3173 2028 7468 7265 6164 293b 2030 551s (thread); 0 │ │ │ │ -0001da20: 7320 2867 6329 7c0a 7c20 2020 2020 2020 s (gc)|.| │ │ │ │ -0001da30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da60: 7c0a 7c6f 3420 3d20 3220 2020 2020 2020 |.|o4 = 2 │ │ │ │ -0001da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0001daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dad0: 2d2d 2d2d 2b0a 0a4f 6273 6572 7665 2074 ----+..Observe t │ │ │ │ -0001dae0: 6861 7420 7468 6520 616c 676f 7269 7468 hat the algorith │ │ │ │ -0001daf0: 6d20 6973 2061 2070 726f 6261 6269 6c69 m is a probabili │ │ │ │ -0001db00: 7374 6963 2061 6c67 6f72 6974 686d 2061 stic algorithm a │ │ │ │ -0001db10: 6e64 206d 6179 2067 6976 6520 6120 7772 nd may give a wr │ │ │ │ -0001db20: 6f6e 670a 616e 7377 6572 2077 6974 6820 ong.answer with │ │ │ │ -0001db30: 6120 736d 616c 6c20 6275 7420 6e6f 6e7a a small but nonz │ │ │ │ -0001db40: 6572 6f20 7072 6f62 6162 696c 6974 792e ero probability. │ │ │ │ -0001db50: 2052 6561 6420 6d6f 7265 2075 6e64 6572 Read more under │ │ │ │ -0001db60: 202a 6e6f 7465 0a70 726f 6261 6269 6c69 *note.probabili │ │ │ │ -0001db70: 7374 6963 2061 6c67 6f72 6974 686d 3a20 stic algorithm: │ │ │ │ -0001db80: 7072 6f62 6162 696c 6973 7469 6320 616c probabilistic al │ │ │ │ -0001db90: 676f 7269 7468 6d2c 2e0a 0a57 6179 7320 gorithm,...Ways │ │ │ │ -0001dba0: 746f 2075 7365 2045 756c 6572 4166 6669 to use EulerAffi │ │ │ │ -0001dbb0: 6e65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ne:.============ │ │ │ │ -0001dbc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0001dbd0: 2a20 2245 756c 6572 4166 6669 6e65 2849 * "EulerAffine(I │ │ │ │ -0001dbe0: 6465 616c 2922 0a0a 466f 7220 7468 6520 deal)"..For the │ │ │ │ -0001dbf0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0001dc00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0001dc10: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0001dc20: 4575 6c65 7241 6666 696e 653a 2045 756c EulerAffine: Eul │ │ │ │ -0001dc30: 6572 4166 6669 6e65 2c20 6973 2061 202a erAffine, is a * │ │ │ │ -0001dc40: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ -0001dc50: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ -0001dc60: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -0001dc70: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ -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 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 3235 h/macaulay2-1.25 │ │ │ │ -0001dd10: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+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 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 |.| │ │ │ │ +0001d9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d9f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ +0001da00: 3d20 3220 2020 2020 2020 2020 2020 2020 = 2 │ │ │ │ +0001da10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0001da70: 0a4f 6273 6572 7665 2074 6861 7420 7468 .Observe that th │ │ │ │ +0001da80: 6520 616c 676f 7269 7468 6d20 6973 2061 e algorithm is a │ │ │ │ +0001da90: 2070 726f 6261 6269 6c69 7374 6963 2061 probabilistic a │ │ │ │ +0001daa0: 6c67 6f72 6974 686d 2061 6e64 206d 6179 lgorithm and may │ │ │ │ +0001dab0: 2067 6976 6520 6120 7772 6f6e 670a 616e give a wrong.an │ │ │ │ +0001dac0: 7377 6572 2077 6974 6820 6120 736d 616c swer with a smal │ │ │ │ +0001dad0: 6c20 6275 7420 6e6f 6e7a 6572 6f20 7072 l but nonzero pr │ │ │ │ +0001dae0: 6f62 6162 696c 6974 792e 2052 6561 6420 obability. Read │ │ │ │ +0001daf0: 6d6f 7265 2075 6e64 6572 202a 6e6f 7465 more under *note │ │ │ │ +0001db00: 0a70 726f 6261 6269 6c69 7374 6963 2061 .probabilistic a │ │ │ │ +0001db10: 6c67 6f72 6974 686d 3a20 7072 6f62 6162 lgorithm: probab │ │ │ │ +0001db20: 696c 6973 7469 6320 616c 676f 7269 7468 ilistic algorith │ │ │ │ +0001db30: 6d2c 2e0a 0a57 6179 7320 746f 2075 7365 m,...Ways to use │ │ │ │ +0001db40: 2045 756c 6572 4166 6669 6e65 3a0a 3d3d EulerAffine:.== │ │ │ │ +0001db50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001db60: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 756c ======.. * "Eul │ │ │ │ +0001db70: 6572 4166 6669 6e65 2849 6465 616c 2922 erAffine(Ideal)" │ │ │ │ +0001db80: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ +0001db90: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ +0001dba0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ +0001dbb0: 6563 7420 2a6e 6f74 6520 4575 6c65 7241 ect *note EulerA │ │ │ │ +0001dbc0: 6666 696e 653a 2045 756c 6572 4166 6669 ffine: EulerAffi │ │ │ │ +0001dbd0: 6e65 2c20 6973 2061 202a 6e6f 7465 206d ne, is a *note m │ │ │ │ +0001dbe0: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ +0001dbf0: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +0001dc00: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +0001dc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +0001dc60: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +0001dc70: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +0001dc80: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +0001dc90: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +0001dca0: 756c 6179 322d 312e 3235 2e31 312b 6473 ulay2-1.25.11+ds │ │ │ │ +0001dcb0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +0001dcc0: 636b 6167 6573 2f0a 4368 6172 6163 7465 ckages/.Characte │ │ │ │ +0001dcd0: 7269 7374 6963 436c 6173 7365 732e 6d32 risticClasses.m2 │ │ │ │ +0001dce0: 3a32 3534 313a 302e 0a1f 0a46 696c 653a :2541:0....File: │ │ │ │ +0001dcf0: 2043 6861 7261 6374 6572 6973 7469 6343 CharacteristicC │ │ │ │ +0001dd00: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64 lasses.info, Nod │ │ │ │ +0001dd10: 653a 2049 6e64 734f 6653 6d6f 6f74 682c e: IndsOfSmooth, │ │ │ │ +0001dd20: 204e 6578 743a 2049 6e70 7574 4973 536d Next: InputIsSm │ │ │ │ +0001dd30: 6f6f 7468 2c20 5072 6576 3a20 4575 6c65 ooth, Prev: Eule │ │ │ │ +0001dd40: 7241 6666 696e 652c 2055 703a 2054 6f70 rAffine, Up: Top │ │ │ │ +0001dd50: 0a0a 496e 6473 4f66 536d 6f6f 7468 0a2a ..IndsOfSmooth.* │ │ │ │ +0001dd60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 ***********..Des │ │ │ │ +0001dd70: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +0001dd80: 3d3d 3d3d 0a0a 5468 6520 6f70 7469 6f6e ====..The option │ │ │ │ +0001dd90: 2049 6e64 734f 6653 6d6f 6f74 6820 6973 IndsOfSmooth is │ │ │ │ +0001dda0: 206f 6e6c 7920 7573 6564 2062 7920 7468 only used by th │ │ │ │ +0001ddb0: 6520 636f 6d6d 616e 6473 202a 6e6f 7465 e commands *note │ │ │ │ +0001ddc0: 2043 534d 3a20 4353 4d2c 2c20 616e 6420 CSM: CSM,, and │ │ │ │ +0001ddd0: 2a6e 6f74 650a 4575 6c65 723a 2045 756c *note.Euler: Eul │ │ │ │ +0001dde0: 6572 2c20 696e 2063 6f6d 6269 6e61 7469 er, in combinati │ │ │ │ +0001ddf0: 6f6e 2077 6974 6820 7468 6520 6f70 7469 on with the opti │ │ │ │ +0001de00: 6f6e 204d 6574 686f 643d 3e44 6972 6563 on Method=>Direc │ │ │ │ +0001de10: 7443 6f6d 706c 6574 496e 742e 2057 6865 tCompletInt. Whe │ │ │ │ +0001de20: 6e0a 7573 6564 2074 6869 7320 6f70 7469 n.used this opti │ │ │ │ +0001de30: 6f6e 206d 6179 2061 6c6c 6f77 2074 6865 on may allow the │ │ │ │ +0001de40: 2075 7365 7220 746f 2073 7065 6564 2075 user to speed u │ │ │ │ +0001de50: 7020 7468 6520 636f 6d70 7574 6174 696f p the computatio │ │ │ │ +0001de60: 6e20 6279 2074 656c 6c69 6e67 0a67 6976 n by telling.giv │ │ │ │ +0001de70: 696e 6720 7468 6520 6d65 7468 6f64 2061 ing the method a │ │ │ │ +0001de80: 206c 6973 7420 6f66 2069 6e64 6963 6573 list of indices │ │ │ │ +0001de90: 2066 6f72 2074 6865 2067 656e 6572 6174 for the generat │ │ │ │ +0001dea0: 6f72 7320 6f66 2074 6865 2069 6e70 7574 ors of the input │ │ │ │ +0001deb0: 2069 6465 616c 2074 6861 742c 0a77 6865 ideal that,.whe │ │ │ │ +0001dec0: 6e20 7461 6b65 6e20 746f 6765 7468 6572 n taken together │ │ │ │ +0001ded0: 2c20 6465 6669 6e65 2061 2073 6d6f 6f74 , define a smoot │ │ │ │ +0001dee0: 6820 7375 6273 6368 656d 6520 6f66 2074 h subscheme of t │ │ │ │ +0001def0: 6865 2061 6d62 6965 6e74 2073 7061 6365 he ambient space │ │ │ │ +0001df00: 2e20 5468 6973 0a6f 7074 696f 6e20 7769 . This.option wi │ │ │ │ +0001df10: 6c6c 2062 6520 6967 6e6f 7265 6420 6f74 ll be ignored ot │ │ │ │ +0001df20: 6865 7277 6973 652e 0a0a 2b2d 2d2d 2d2d herwise...+----- │ │ │ │ +0001df30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001df40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001df50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001df60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001df70: 0a7c 6931 203a 2052 203d 204d 756c 7469 .|i1 : R = Multi │ │ │ │ +0001df80: 5072 6f6a 436f 6f72 6452 696e 6728 7b32 ProjCoordRing({2 │ │ │ │ +0001df90: 2c32 7d29 2020 2020 2020 2020 2020 2020 ,2}) │ │ │ │ +0001dfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001dfb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001dfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001dff0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001e000: 6f31 203d 2052 2020 2020 2020 2020 2020 o1 = R │ │ │ │ +0001e010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 2020 2020 │ │ │ │ +0001e040: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0001e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e060: 2020 207c 0a7c 6f31 203d 2052 2020 2020 |.|o1 = R │ │ │ │ +0001e060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e0a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001e080: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0001e090: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ +0001e0a0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 0001e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e130: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0001e180: 0a7c 6932 203a 2049 3d69 6465 616c 2852 .|i2 : I=ideal(R │ │ │ │ -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));|.| │ │ │ │ +0001e0d0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001e0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e110: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ +0001e120: 2049 3d69 6465 616c 2852 5f30 2a52 5f31 I=ideal(R_0*R_1 │ │ │ │ +0001e130: 2a52 5f33 2d52 5f30 5e32 2a52 5f33 2c72 *R_3-R_0^2*R_3,r │ │ │ │ +0001e140: 616e 646f 6d28 7b30 2c31 7d2c 5229 2c72 andom({0,1},R),r │ │ │ │ +0001e150: 616e 646f 6d28 7b31 2c32 7d2c 5229 293b andom({1,2},R)); │ │ │ │ +0001e160: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e1a0: 2020 2020 2020 207c 0a7c 6f32 203a 2049 |.|o2 : I │ │ │ │ +0001e1b0: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ +0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e200: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001e210: 6f32 203a 2049 6465 616c 206f 6620 5220 o2 : Ideal of R │ │ │ │ -0001e220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e250: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0001e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ -0001e2a0: 203a 2074 696d 6520 4353 4d28 492c 4d65 : time CSM(I,Me │ │ │ │ -0001e2b0: 7468 6f64 3d3e 4469 7265 6374 436f 6d70 thod=>DirectComp │ │ │ │ -0001e2c0: 6c65 7449 6e74 2920 2020 2020 2020 2020 letInt) │ │ │ │ +0001e1e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001e1f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001e200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e230: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 696d -----+.|i3 : tim │ │ │ │ +0001e240: 6520 4353 4d28 492c 4d65 7468 6f64 3d3e e CSM(I,Method=> │ │ │ │ +0001e250: 4469 7265 6374 436f 6d70 6c65 7449 6e74 DirectCompletInt │ │ │ │ +0001e260: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0001e270: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001e280: 2d2d 2075 7365 6420 352e 3734 3039 3473 -- used 5.74094s │ │ │ │ +0001e290: 2028 6370 7529 3b20 312e 3531 3538 3873 (cpu); 1.51588s │ │ │ │ +0001e2a0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +0001e2b0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +0001e2c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e2e0: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e |.| -- used 1. │ │ │ │ -0001e2f0: 3536 3433 3173 2028 6370 7529 3b20 312e 56431s (cpu); 1. │ │ │ │ -0001e300: 3231 3037 3373 2028 7468 7265 6164 293b 21073s (thread); │ │ │ │ -0001e310: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ -0001e320: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001e2e0: 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+0001e3e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +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 7c0a 7c20 2020 2020 5a5a |.| ZZ │ │ │ │ +0001e430: 5b68 202e 2e68 205d 2020 2020 2020 2020 [h ..h ] │ │ │ │ +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 2020 2020 │ │ │ │ -0001e480: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001e490: 2020 2020 5a5a 5b68 202e 2e68 205d 2020 ZZ[h ..h ] │ │ │ │ +0001e460: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001e470: 2020 2020 2020 2020 2031 2020 2032 2020 1 2 │ │ │ │ +0001e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 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 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 : ---------- │ │ │ │ +0001e4b0: 2020 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d |.|o3 : ---- │ │ │ │ +0001e4c0: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020 ------ │ │ │ │ +0001e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e4f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001e500: 2020 2020 2020 3320 2020 3320 2020 2020 3 3 │ │ │ │ +0001e510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 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 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 ) │ │ │ │ +0001e540: 2020 7c0a 7c20 2020 2020 2028 6820 2c20 |.| (h , │ │ │ │ +0001e550: 6820 2920 2020 2020 2020 2020 2020 2020 h ) │ │ │ │ +0001e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e580: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001e590: 2020 2020 3120 2020 3220 2020 2020 2020 1 2 │ │ │ │ +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 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: 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0001e680: 6934 203a 2074 696d 6520 4353 4d28 492c i4 : time CSM(I, │ │ │ │ -0001e690: 4d65 7468 6f64 3d3e 4469 7265 6374 436f Method=>DirectCo │ │ │ │ -0001e6a0: 6d70 6c65 7449 6e74 2c49 6e64 734f 6653 mpletInt,IndsOfS │ │ │ │ -0001e6b0: 6d6f 6f74 683d 3e7b 312c 327d 2920 2020 mooth=>{1,2}) │ │ │ │ -0001e6c0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -0001e6d0: 312e 3735 3635 3173 2028 6370 7529 3b20 1.75651s (cpu); │ │ │ │ -0001e6e0: 312e 3334 3835 3373 2028 7468 7265 6164 1.34853s (thread │ │ │ │ -0001e6f0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ -0001e700: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001e5d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001e5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e610: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 -------+.|i4 : t │ │ │ │ +0001e620: 696d 6520 4353 4d28 492c 4d65 7468 6f64 ime CSM(I,Method │ │ │ │ +0001e630: 3d3e 4469 7265 6374 436f 6d70 6c65 7449 =>DirectCompletI │ │ │ │ +0001e640: 6e74 2c49 6e64 734f 6653 6d6f 6f74 683d nt,IndsOfSmooth= │ │ │ │ +0001e650: 3e7b 312c 327d 2920 2020 2020 2020 7c0a >{1,2}) |. │ │ │ │ +0001e660: 7c20 2d2d 2075 7365 6420 362e 3038 3431 | -- used 6.0841 │ │ │ │ +0001e670: 3373 2028 6370 7529 3b20 312e 3531 3635 3s (cpu); 1.5165 │ │ │ │ +0001e680: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ +0001e690: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0001e6a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e6e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001e6f0: 2020 2020 2020 3220 3220 2020 2020 3220 2 2 2 │ │ │ │ +0001e700: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ 0001e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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|.| │ │ │ │ +0001e780: 2020 2020 3120 3220 2020 2020 3120 3220 1 2 1 2 │ │ │ │ +0001e790: 2020 2020 3120 3220 2020 2020 2020 2020 1 2 │ │ │ │ +0001e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e7c0: 207c 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e820: 2020 2020 2020 207c 0a7c 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 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001e810: 5a5a 5b68 202e 2e68 205d 2020 2020 2020 ZZ[h ..h ] │ │ │ │ +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: 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 ] │ │ │ │ +0001e840: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001e850: 0a7c 2020 2020 2020 2020 2031 2020 2032 .| 1 2 │ │ │ │ +0001e860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e880: 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 2020 2020 2020 2020 2020 │ │ │ │ -0001e8f0: 2020 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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 ) │ │ │ │ +0001e920: 2020 2020 7c0a 7c20 2020 2020 2028 6820 |.| (h │ │ │ │ +0001e930: 2c20 6820 2920 2020 2020 2020 2020 2020 , h ) │ │ │ │ +0001e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e960: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001e970: 2020 2020 2020 3120 2020 3220 2020 2020 1 2 │ │ │ │ +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 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 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 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 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 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 352e 3131 2b64 732f 4d32 2f4d 6163 .25.11+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 ---------------- │ │ │ │ +0001e9b0: 2020 7c0a 2b2d 2d2d 2d2d 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 2d2b 0a0a 4675 6e63 ---------+..Func │ │ │ │ +0001ea00: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ +0001ea10: 6e61 6c20 6172 6775 6d65 6e74 206e 616d nal argument nam │ │ │ │ +0001ea20: 6564 2049 6e64 734f 6653 6d6f 6f74 683a ed IndsOfSmooth: │ │ │ │ +0001ea30: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +0001ea40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001ea50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001ea60: 3d3d 3d3d 3d0a 0a20 202a 2022 4353 4d28 =====.. * "CSM( │ │ │ │ +0001ea70: 2e2e 2e2c 496e 6473 4f66 536d 6f6f 7468 ...,IndsOfSmooth │ │ │ │ +0001ea80: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ +0001ea90: 6e6f 7465 2043 534d 3a20 4353 4d2c 202d note CSM: CSM, - │ │ │ │ +0001eaa0: 2d20 5468 650a 2020 2020 4368 6572 6e2d - The. Chern- │ │ │ │ +0001eab0: 5363 6877 6172 747a 2d4d 6163 5068 6572 Schwartz-MacPher │ │ │ │ +0001eac0: 736f 6e20 636c 6173 730a 2020 2a20 4575 son class. * Eu │ │ │ │ +0001ead0: 6c65 7228 2e2e 2e2c 496e 6473 4f66 536d ler(...,IndsOfSm │ │ │ │ +0001eae0: 6f6f 7468 3d3e 2e2e 2e29 2028 6d69 7373 ooth=>...) (miss │ │ │ │ +0001eaf0: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ +0001eb00: 6e29 0a0a 466f 7220 7468 6520 7072 6f67 n)..For the prog │ │ │ │ +0001eb10: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +0001eb20: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +0001eb30: 626a 6563 7420 2a6e 6f74 6520 496e 6473 bject *note Inds │ │ │ │ +0001eb40: 4f66 536d 6f6f 7468 3a20 496e 6473 4f66 OfSmooth: IndsOf │ │ │ │ +0001eb50: 536d 6f6f 7468 2c20 6973 2061 202a 6e6f Smooth, is a *no │ │ │ │ +0001eb60: 7465 2073 796d 626f 6c3a 0a28 4d61 6361 te symbol:.(Maca │ │ │ │ +0001eb70: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ +0001eb80: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0001eb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001eba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ebb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ebc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ebd0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0001ebe0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0001ebf0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0001ec00: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0001ec10: 6163 6175 6c61 7932 2d31 2e32 352e 3131 acaulay2-1.25.11 │ │ │ │ +0001ec20: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0001ec30: 2f70 6163 6b61 6765 732f 0a43 6861 7261 /packages/.Chara │ │ │ │ +0001ec40: 6374 6572 6973 7469 6343 6c61 7373 6573 cteristicClasses │ │ │ │ +0001ec50: 2e6d 323a 3234 3832 3a30 2e0a 1f0a 4669 .m2:2482:0....Fi │ │ │ │ +0001ec60: 6c65 3a20 4368 6172 6163 7465 7269 7374 le: Characterist │ │ │ │ +0001ec70: 6963 436c 6173 7365 732e 696e 666f 2c20 icClasses.info, │ │ │ │ +0001ec80: 4e6f 6465 3a20 496e 7075 7449 7353 6d6f Node: InputIsSmo │ │ │ │ +0001ec90: 6f74 682c 204e 6578 743a 2069 734d 756c oth, Next: isMul │ │ │ │ +0001eca0: 7469 486f 6d6f 6765 6e65 6f75 732c 2050 tiHomogeneous, P │ │ │ │ +0001ecb0: 7265 763a 2049 6e64 734f 6653 6d6f 6f74 rev: IndsOfSmoot │ │ │ │ +0001ecc0: 682c 2055 703a 2054 6f70 0a0a 496e 7075 h, Up: Top..Inpu │ │ │ │ +0001ecd0: 7449 7353 6d6f 6f74 680a 2a2a 2a2a 2a2a tIsSmooth.****** │ │ │ │ +0001ece0: 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 6970 *******..Descrip │ │ │ │ +0001ecf0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +0001ed00: 0a0a 5468 6520 6f70 7469 6f6e 2049 6e70 ..The option Inp │ │ │ │ +0001ed10: 7574 4973 536d 6f6f 7468 2069 7320 6f6e utIsSmooth is on │ │ │ │ +0001ed20: 6c79 2075 7365 6420 6279 2074 6865 2063 ly used by the c │ │ │ │ +0001ed30: 6f6d 6d61 6e64 7320 2a6e 6f74 6520 4353 ommands *note CS │ │ │ │ +0001ed40: 4d3a 2043 534d 2c2c 2061 6e64 0a2a 6e6f M: CSM,, and.*no │ │ │ │ +0001ed50: 7465 2045 756c 6572 3a20 4575 6c65 722c te Euler: Euler, │ │ │ │ +0001ed60: 2e20 4966 2074 6865 2069 6e70 7574 2069 . If the input i │ │ │ │ +0001ed70: 6465 616c 2069 7320 6b6e 6f77 6e20 746f deal is known to │ │ │ │ +0001ed80: 2064 6566 696e 6520 6120 736d 6f6f 7468 define a smooth │ │ │ │ +0001ed90: 2073 7562 7363 6865 6d65 0a73 6574 7469 subscheme.setti │ │ │ │ +0001eda0: 6e67 2074 6869 7320 6f70 7469 6f6e 2074 ng this option t │ │ │ │ +0001edb0: 6f20 7472 7565 2077 696c 6c20 7370 6565 o true will spee │ │ │ │ +0001edc0: 6420 7570 2063 6f6d 7075 7461 7469 6f6e d up computation │ │ │ │ +0001edd0: 7320 2869 7420 6973 2073 6574 2074 6f20 s (it is set to │ │ │ │ +0001ede0: 6661 6c73 6520 6279 0a64 6566 6175 6c74 false by.default │ │ │ │ +0001edf0: 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d )...+----------- │ │ │ │ +0001ee00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ee10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ee20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0001ee30: 203a 2052 203d 205a 5a2f 3332 3734 395b : R = ZZ/32749[ │ │ │ │ +0001ee40: 785f 302e 2e78 5f34 5d3b 2020 2020 2020 x_0..x_4]; │ │ │ │ +0001ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ee60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0001ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ee90: 2d2b 0a7c 6931 203a 2052 203d 205a 5a2f -+.|i1 : R = ZZ/ │ │ │ │ -0001eea0: 3332 3734 395b 785f 302e 2e78 5f34 5d3b 32749[x_0..x_4]; │ │ │ │ -0001eeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eec0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ef00: 2d2d 2d2b 0a7c 6932 203a 2049 3d69 6465 ---+.|i2 : I=ide │ │ │ │ -0001ef10: 616c 2872 616e 646f 6d28 322c 5229 2c72 al(random(2,R),r │ │ │ │ -0001ef20: 616e 646f 6d28 322c 5229 2c72 616e 646f andom(2,R),rando │ │ │ │ -0001ef30: 6d28 312c 5229 293b 2020 2020 7c0a 7c20 m(1,R)); |.| │ │ │ │ -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 2020 2020 │ │ │ │ -0001ef70: 2020 2020 207c 0a7c 6f32 203a 2049 6465 |.|o2 : Ide │ │ │ │ -0001ef80: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ -0001ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001efa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001efb0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0001efc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001efd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001efe0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 -------+.|i3 : t │ │ │ │ -0001eff0: 696d 6520 4353 4d20 4920 2020 2020 2020 ime CSM I │ │ │ │ +0001ee90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001eea0: 6932 203a 2049 3d69 6465 616c 2872 616e i2 : I=ideal(ran │ │ │ │ +0001eeb0: 646f 6d28 322c 5229 2c72 616e 646f 6d28 dom(2,R),random( │ │ │ │ +0001eec0: 322c 5229 2c72 616e 646f 6d28 312c 5229 2,R),random(1,R) │ │ │ │ +0001eed0: 293b 2020 2020 7c0a 7c20 2020 2020 2020 ); |.| │ │ │ │ +0001eee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001eef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ef00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001ef10: 0a7c 6f32 203a 2049 6465 616c 206f 6620 .|o2 : Ideal of │ │ │ │ +0001ef20: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0001ef30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ef40: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001ef50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ef60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ef70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ef80: 2d2b 0a7c 6933 203a 2074 696d 6520 4353 -+.|i3 : time CS │ │ │ │ +0001ef90: 4d20 4920 2020 2020 2020 2020 2020 2020 M I │ │ │ │ +0001efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001efb0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ +0001efc0: 2075 7365 6420 312e 3230 3130 3673 2028 used 1.20106s ( │ │ │ │ +0001efd0: 6370 7529 3b20 302e 3536 3334 3532 7320 cpu); 0.563452s │ │ │ │ +0001efe0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +0001eff0: 2920 207c 0a7c 2020 2020 2020 2020 2020 ) |.| │ │ │ │ 0001f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f020: 7c0a 7c20 2d2d 2075 7365 6420 302e 3532 |.| -- used 0.52 │ │ │ │ -0001f030: 3239 3733 7320 2863 7075 293b 2030 2e33 2973s (cpu); 0.3 │ │ │ │ -0001f040: 3933 3937 3273 2028 7468 7265 6164 293b 93972s (thread); │ │ │ │ -0001f050: 2030 7320 2867 6329 207c 0a7c 2020 2020 0s (gc) |.| │ │ │ │ -0001f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f020: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f030: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ +0001f040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f060: 2020 2020 207c 0a7c 6f33 203d 2034 6820 |.|o3 = 4h │ │ │ │ 0001f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f090: 2020 7c0a 7c20 2020 2020 2020 3320 2020 |.| 3 │ │ │ │ -0001f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f090: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001f0a0: 7c20 2020 2020 2020 3120 2020 2020 2020 | 1 │ │ │ │ 0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f0c0: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -0001f0d0: 203d 2034 6820 2020 2020 2020 2020 2020 = 4h │ │ │ │ +0001f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f0d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0001f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f100: 2020 2020 7c0a 7c20 2020 2020 2020 3120 |.| 1 │ │ │ │ -0001f110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f110: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020 |.| ZZ[h ] │ │ │ │ 0001f120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f130: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001f140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f140: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001f150: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ 0001f160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f170: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a |.| ZZ │ │ │ │ -0001f180: 5b68 205d 2020 2020 2020 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2020 2020 7c0a |. │ │ │ │ +0001f430: 7c20 2020 2020 5a5a 5b68 205d 2020 2020 | ZZ[h ] │ │ │ │ 0001f440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f450: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0001f460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f460: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f470: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0001f480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f490: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b68 |.| ZZ[h │ │ │ │ -0001f4a0: 205d 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +0001f490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f4a0: 7c0a 7c6f 3420 3a20 2d2d 2d2d 2d2d 2020 |.|o4 : ------ │ │ │ │ 0001f4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f4c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001f4d0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ -0001f4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f4d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001f4e0: 2020 2020 3520 2020 2020 2020 2020 2020 5 │ │ │ │ 0001f4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f500: 2020 2020 2020 7c0a 7c6f 3420 3a20 2d2d |.|o4 : -- │ │ │ │ -0001f510: 2d2d 2d2d 2020 2020 2020 2020 2020 2020 ---- │ │ │ │ +0001f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f510: 2020 7c0a 7c20 2020 2020 2020 6820 2020 |.| h │ │ │ │ 0001f520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f530: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001f540: 0a7c 2020 2020 2020 2020 3520 2020 2020 .| 5 │ │ │ │ -0001f550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f540: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001f550: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ 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│ │ │ -0001f650: 2063 6f6d 6d61 6e64 202a 6e6f 7465 2043 command *note C │ │ │ │ -0001f660: 6865 726e 3a20 4368 6572 6e2c 2069 6e73 hern: Chern, ins │ │ │ │ -0001f670: 7465 6164 0a69 6e20 7468 6973 2063 6173 tead.in this cas │ │ │ │ -0001f680: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d e...+----------- │ │ │ │ -0001f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001f6c0: 3520 3a20 7469 6d65 2043 6865 726e 2049 5 : time Chern I │ │ │ │ +0001f570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f580: 2020 2020 7c0a 2b2d 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 2d2b 0a0a -------------+.. │ │ │ │ +0001f5c0: 4e6f 7465 2074 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2d2d 2d2d 2020 |.|o5 : ------ │ │ │ │ 0001f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f880: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001f890: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ -0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f890: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001f8a0: 2020 2020 2035 2020 2020 2020 2020 2020 5 │ │ │ │ 0001f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8c0: 2020 2020 2020 7c0a 7c6f 3520 3a20 2d2d |.|o5 : -- │ │ │ │ -0001f8d0: 2d2d 2d2d 2020 2020 2020 2020 2020 2020 ---- │ │ │ │ +0001f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f8d0: 2020 2020 7c0a 7c20 2020 2020 2020 6820 |.| h │ │ │ │ 0001f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f900: 7c0a 7c20 2020 2020 2020 2035 2020 2020 |.| 5 │ │ │ │ -0001f910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f900: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001f910: 7c20 2020 2020 2020 2031 2020 2020 2020 | 1 │ │ │ │ 0001f920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f930: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001f940: 2020 2020 6820 2020 2020 2020 2020 2020 h │ │ │ │ -0001f950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f970: 2020 2020 7c0a 7c20 2020 2020 2020 2031 |.| 1 │ │ │ │ -0001f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f9a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f9b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0001f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f9e0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 756e 6374 --------+..Funct │ │ │ │ -0001f9f0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ -0001fa00: 616c 2061 7267 756d 656e 7420 6e61 6d65 al argument name │ │ │ │ -0001fa10: 6420 496e 7075 7449 7353 6d6f 6f74 683a d InputIsSmooth: │ │ │ │ -0001fa20: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0001fa30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001fa40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001fa50: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2243 534d ======.. * "CSM │ │ │ │ -0001fa60: 282e 2e2e 2c49 6e70 7574 4973 536d 6f6f (...,InputIsSmoo │ │ │ │ -0001fa70: 7468 3d3e 2e2e 2e29 2220 2d2d 2073 6565 th=>...)" -- see │ │ │ │ -0001fa80: 202a 6e6f 7465 2043 534d 3a20 4353 4d2c *note CSM: CSM, │ │ │ │ -0001fa90: 202d 2d20 5468 650a 2020 2020 4368 6572 -- The. Cher │ │ │ │ -0001faa0: 6e2d 5363 6877 6172 747a 2d4d 6163 5068 n-Schwartz-MacPh │ │ │ │ -0001fab0: 6572 736f 6e20 636c 6173 730a 2020 2a20 erson class. * │ │ │ │ -0001fac0: 4575 6c65 7228 2e2e 2e2c 496e 7075 7449 Euler(...,InputI │ │ │ │ -0001fad0: 7353 6d6f 6f74 683d 3e2e 2e2e 2920 286d sSmooth=>...) (m │ │ │ │ -0001fae0: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ -0001faf0: 7469 6f6e 290a 0a46 6f72 2074 6865 2070 tion)..For the p │ │ │ │ -0001fb00: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -0001fb10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -0001fb20: 6520 6f62 6a65 6374 202a 6e6f 7465 2049 e object *note I │ │ │ │ -0001fb30: 6e70 7574 4973 536d 6f6f 7468 3a20 496e nputIsSmooth: In │ │ │ │ -0001fb40: 7075 7449 7353 6d6f 6f74 682c 2069 7320 putIsSmooth, is │ │ │ │ -0001fb50: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a0a a *note symbol:. │ │ │ │ -0001fb60: 284d 6163 6175 6c61 7932 446f 6329 5379 (Macaulay2Doc)Sy │ │ │ │ -0001fb70: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d mbol,...-------- │ │ │ │ -0001fb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fbc0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -0001fbd0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -0001fbe0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -0001fbf0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -0001fc00: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -0001fc10: 3235 2e31 312b 6473 2f4d 322f 4d61 6361 25.11+ds/M2/Maca │ │ │ │ -0001fc20: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -0001fc30: 4368 6172 6163 7465 7269 7374 6963 436c CharacteristicCl │ │ │ │ -0001fc40: 6173 7365 732e 6d32 3a32 3530 303a 302e asses.m2:2500:0. │ │ │ │ -0001fc50: 0a1f 0a46 696c 653a 2043 6861 7261 6374 ...File: Charact │ │ │ │ -0001fc60: 6572 6973 7469 6343 6c61 7373 6573 2e69 eristicClasses.i │ │ │ │ -0001fc70: 6e66 6f2c 204e 6f64 653a 2069 734d 756c nfo, Node: isMul │ │ │ │ -0001fc80: 7469 486f 6d6f 6765 6e65 6f75 732c 204e tiHomogeneous, N │ │ │ │ -0001fc90: 6578 743a 204d 6574 686f 642c 2050 7265 ext: Method, Pre │ │ │ │ -0001fca0: 763a 2049 6e70 7574 4973 536d 6f6f 7468 v: InputIsSmooth │ │ │ │ -0001fcb0: 2c20 5570 3a20 546f 700a 0a69 734d 756c , Up: Top..isMul │ │ │ │ -0001fcc0: 7469 486f 6d6f 6765 6e65 6f75 7320 2d2d tiHomogeneous -- │ │ │ │ -0001fcd0: 2043 6865 636b 7320 6966 2061 6e20 6964 Checks if an id │ │ │ │ -0001fce0: 6561 6c20 6973 2068 6f6d 6f67 656e 656f eal is homogeneo │ │ │ │ -0001fcf0: 7573 2077 6974 6820 7265 7370 6563 7420 us with respect │ │ │ │ -0001fd00: 746f 2074 6865 2067 7261 6469 6e67 206f to the grading o │ │ │ │ -0001fd10: 6e20 6974 7320 7269 6e67 2028 692e 652e n its ring (i.e. │ │ │ │ -0001fd20: 206d 756c 7469 2d68 6f6d 6f67 656e 656f multi-homogeneo │ │ │ │ -0001fd30: 7573 2069 6e20 7468 6520 6d75 6c74 692d us in the multi- │ │ │ │ -0001fd40: 6772 6164 6564 2063 6173 6529 0a2a 2a2a graded case).*** │ │ │ │ +0001f930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f940: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001f950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f980: 2d2d 2b0a 0a46 756e 6374 696f 6e73 2077 --+..Functions w │ │ │ │ +0001f990: 6974 6820 6f70 7469 6f6e 616c 2061 7267 ith optional arg │ │ │ │ +0001f9a0: 756d 656e 7420 6e61 6d65 6420 496e 7075 ument named Inpu │ │ │ │ +0001f9b0: 7449 7353 6d6f 6f74 683a 0a3d 3d3d 3d3d tIsSmooth:.===== │ │ │ │ +0001f9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001f9d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001f9e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001f9f0: 0a0a 2020 2a20 2243 534d 282e 2e2e 2c49 .. * "CSM(...,I │ │ │ │ +0001fa00: 6e70 7574 4973 536d 6f6f 7468 3d3e 2e2e nputIsSmooth=>.. │ │ │ │ +0001fa10: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465 .)" -- see *note │ │ │ │ +0001fa20: 2043 534d 3a20 4353 4d2c 202d 2d20 5468 CSM: CSM, -- Th │ │ │ │ +0001fa30: 650a 2020 2020 4368 6572 6e2d 5363 6877 e. Chern-Schw │ │ │ │ +0001fa40: 6172 747a 2d4d 6163 5068 6572 736f 6e20 artz-MacPherson │ │ │ │ +0001fa50: 636c 6173 730a 2020 2a20 4575 6c65 7228 class. * Euler( │ │ │ │ +0001fa60: 2e2e 2e2c 496e 7075 7449 7353 6d6f 6f74 ...,InputIsSmoot │ │ │ │ +0001fa70: 683d 3e2e 2e2e 2920 286d 6973 7369 6e67 h=>...) (missing │ │ │ │ +0001fa80: 2064 6f63 756d 656e 7461 7469 6f6e 290a documentation). │ │ │ │ +0001fa90: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0001faa0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0001fab0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0001fac0: 6374 202a 6e6f 7465 2049 6e70 7574 4973 ct *note InputIs │ │ │ │ +0001fad0: 536d 6f6f 7468 3a20 496e 7075 7449 7353 Smooth: InputIsS │ │ │ │ +0001fae0: 6d6f 6f74 682c 2069 7320 6120 2a6e 6f74 mooth, is a *not │ │ │ │ +0001faf0: 6520 7379 6d62 6f6c 3a0a 284d 6163 6175 e symbol:.(Macau │ │ │ │ +0001fb00: 6c61 7932 446f 6329 5379 6d62 6f6c 2c2e lay2Doc)Symbol,. │ │ │ │ +0001fb10: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +0001fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fb60: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +0001fb70: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +0001fb80: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +0001fb90: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +0001fba0: 6361 756c 6179 322d 312e 3235 2e31 312b caulay2-1.25.11+ │ │ │ │ +0001fbb0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +0001fbc0: 7061 636b 6167 6573 2f0a 4368 6172 6163 packages/.Charac │ │ │ │ +0001fbd0: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +0001fbe0: 6d32 3a32 3530 303a 302e 0a1f 0a46 696c m2:2500:0....Fil │ │ │ │ +0001fbf0: 653a 2043 6861 7261 6374 6572 6973 7469 e: Characteristi │ │ │ │ +0001fc00: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e cClasses.info, N │ │ │ │ +0001fc10: 6f64 653a 2069 734d 756c 7469 486f 6d6f ode: isMultiHomo │ │ │ │ +0001fc20: 6765 6e65 6f75 732c 204e 6578 743a 204d geneous, Next: M │ │ │ │ +0001fc30: 6574 686f 642c 2050 7265 763a 2049 6e70 ethod, Prev: Inp │ │ │ │ +0001fc40: 7574 4973 536d 6f6f 7468 2c20 5570 3a20 utIsSmooth, Up: │ │ │ │ +0001fc50: 546f 700a 0a69 734d 756c 7469 486f 6d6f Top..isMultiHomo │ │ │ │ +0001fc60: 6765 6e65 6f75 7320 2d2d 2043 6865 636b geneous -- Check │ │ │ │ +0001fc70: 7320 6966 2061 6e20 6964 6561 6c20 6973 s if an ideal is │ │ │ │ +0001fc80: 2068 6f6d 6f67 656e 656f 7573 2077 6974 homogeneous wit │ │ │ │ +0001fc90: 6820 7265 7370 6563 7420 746f 2074 6865 h respect to the │ │ │ │ +0001fca0: 2067 7261 6469 6e67 206f 6e20 6974 7320 grading on its │ │ │ │ +0001fcb0: 7269 6e67 2028 692e 652e 206d 756c 7469 ring (i.e. multi │ │ │ │ +0001fcc0: 2d68 6f6d 6f67 656e 656f 7573 2069 6e20 -homogeneous in │ │ │ │ +0001fcd0: 7468 6520 6d75 6c74 692d 6772 6164 6564 the multi-graded │ │ │ │ +0001fce0: 2063 6173 6529 0a2a 2a2a 2a2a 2a2a 2a2a case).********* │ │ │ │ +0001fcf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fd00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fd10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fd20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fd30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fd40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -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 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fdd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -0001fde0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -0001fdf0: 2020 2020 6973 4d75 6c74 6948 6f6d 6f67 isMultiHomog │ │ │ │ -0001fe00: 656e 656f 7573 2049 0a20 2020 2020 2020 eneous I. │ │ │ │ -0001fe10: 2069 734d 756c 7469 486f 6d6f 6765 6e65 isMultiHomogene │ │ │ │ -0001fe20: 6f75 7320 660a 2020 2a20 496e 7075 7473 ous f. * Inputs │ │ │ │ -0001fe30: 3a0a 2020 2020 2020 2a20 492c 2061 6e20 :. * I, an │ │ │ │ -0001fe40: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61 *note ideal: (Ma │ │ │ │ -0001fe50: 6361 756c 6179 3244 6f63 2949 6465 616c caulay2Doc)Ideal │ │ │ │ -0001fe60: 2c2c 2061 6e20 6964 6561 6c20 696e 2061 ,, an ideal in a │ │ │ │ -0001fe70: 2067 7261 6465 6420 6f72 0a20 2020 2020 graded or. │ │ │ │ -0001fe80: 2020 206d 756c 7469 2d67 7261 6465 6420 multi-graded │ │ │ │ -0001fe90: 7269 6e67 0a20 2020 2020 202a 2066 2c20 ring. * f, │ │ │ │ -0001fea0: 6120 2a6e 6f74 6520 7269 6e67 2065 6c65 a *note ring ele │ │ │ │ -0001feb0: 6d65 6e74 3a20 284d 6163 6175 6c61 7932 ment: (Macaulay2 │ │ │ │ -0001fec0: 446f 6329 5269 6e67 456c 656d 656e 742c Doc)RingElement, │ │ │ │ -0001fed0: 2c20 6120 656c 656d 656e 7420 696e 2061 , a element in a │ │ │ │ -0001fee0: 0a20 2020 2020 2020 2067 7261 6465 6420 . graded │ │ │ │ -0001fef0: 6f72 206d 756c 7469 2d67 7261 6465 6420 or multi-graded │ │ │ │ -0001ff00: 7269 6e67 0a20 202a 204f 7574 7075 7473 ring. * Outputs │ │ │ │ -0001ff10: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ -0001ff20: 6520 426f 6f6c 6561 6e20 7661 6c75 653a e Boolean value: │ │ │ │ -0001ff30: 2028 4d61 6361 756c 6179 3244 6f63 2942 (Macaulay2Doc)B │ │ │ │ -0001ff40: 6f6f 6c65 616e 2c2c 200a 0a44 6573 6372 oolean,, ..Descr │ │ │ │ -0001ff50: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -0001ff60: 3d3d 0a0a 5465 7374 7320 6966 2074 6865 ==..Tests if the │ │ │ │ -0001ff70: 2069 6e70 7574 2049 6465 616c 206f 7220 input Ideal or │ │ │ │ -0001ff80: 5269 6e67 456c 656d 656e 7420 6973 2048 RingElement is H │ │ │ │ -0001ff90: 6f6d 6f67 656e 656f 7573 2077 6974 6820 omogeneous with │ │ │ │ -0001ffa0: 7265 7370 6563 7420 746f 2074 6865 0a67 respect to the.g │ │ │ │ -0001ffb0: 7261 6469 6e67 206f 6e20 7468 6520 7269 rading on the ri │ │ │ │ -0001ffc0: 6e67 2e20 486f 6d6f 6765 6e65 6f75 7320 ng. Homogeneous │ │ │ │ -0001ffd0: 696e 7075 7420 6973 2072 6571 7569 7265 input is require │ │ │ │ -0001ffe0: 6420 666f 7220 616c 6c20 6d65 7468 6f64 d for all method │ │ │ │ -0001fff0: 7320 746f 2063 6f6d 7075 7465 0a63 6861 s to compute.cha │ │ │ │ -00020000: 7261 6374 6572 6973 7469 6320 636c 6173 racteristic clas │ │ │ │ -00020010: 7365 732e 0a0a 5468 6973 206d 6574 686f ses...This metho │ │ │ │ -00020020: 6420 776f 726b 7320 666f 7220 6964 6561 d works for idea │ │ │ │ -00020030: 6c73 2069 6e20 7468 6520 6772 6164 6564 ls in the graded │ │ │ │ -00020040: 2063 6f6f 7264 696e 6174 6520 7269 6e67 coordinate ring │ │ │ │ -00020050: 7320 6f66 2074 6f72 6963 2076 6172 6965 s of toric varie │ │ │ │ -00020060: 7469 6573 2c0a 616e 6420 6865 6e63 6520 ties,.and hence │ │ │ │ -00020070: 666f 7220 7072 6f64 7563 7473 206f 6620 for products of │ │ │ │ -00020080: 7072 6f6a 6563 7469 7665 2073 7061 6365 projective space │ │ │ │ -00020090: 732e 2054 6865 7365 2063 616e 2062 6520 s. These can be │ │ │ │ -000200a0: 6372 6561 7465 6420 6469 7265 6374 6c79 created directly │ │ │ │ -000200b0: 2c20 6f72 0a75 7369 6e67 206d 6574 686f , or.using metho │ │ │ │ -000200c0: 6473 2074 6865 202a 6e6f 7465 204d 756c ds the *note Mul │ │ │ │ -000200d0: 7469 5072 6f6a 436f 6f72 6452 696e 673a tiProjCoordRing: │ │ │ │ -000200e0: 204d 756c 7469 5072 6f6a 436f 6f72 6452 MultiProjCoordR │ │ │ │ -000200f0: 696e 672c 206d 6574 686f 6420 6f66 2074 ing, method of t │ │ │ │ -00020100: 6869 730a 7061 636b 6167 652c 206f 7220 his.package, or │ │ │ │ -00020110: 7769 7468 206d 6574 686f 6473 2066 726f with methods fro │ │ │ │ -00020120: 6d20 7468 6520 4e6f 726d 616c 546f 7269 m the NormalTori │ │ │ │ -00020130: 6356 6172 6965 7469 6573 2050 6163 6b61 cVarieties Packa │ │ │ │ -00020140: 6765 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ge...+---------- │ │ │ │ -00020150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020180: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 3d4d -----+.|i1 : R=M │ │ │ │ -00020190: 756c 7469 5072 6f6a 436f 6f72 6452 696e ultiProjCoordRin │ │ │ │ -000201a0: 6728 7b31 2c32 2c31 7d29 2020 2020 2020 g({1,2,1}) │ │ │ │ +0001fd70: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +0001fd80: 6167 653a 200a 2020 2020 2020 2020 6973 age: . is │ │ │ │ +0001fd90: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ +0001fda0: 2049 0a20 2020 2020 2020 2069 734d 756c I. isMul │ │ │ │ +0001fdb0: 7469 486f 6d6f 6765 6e65 6f75 7320 660a tiHomogeneous f. │ │ │ │ +0001fdc0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0001fdd0: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520 * I, an *note │ │ │ │ +0001fde0: 6964 6561 6c3a 2028 4d61 6361 756c 6179 ideal: (Macaulay │ │ │ │ +0001fdf0: 3244 6f63 2949 6465 616c 2c2c 2061 6e20 2Doc)Ideal,, an │ │ │ │ +0001fe00: 6964 6561 6c20 696e 2061 2067 7261 6465 ideal in a grade │ │ │ │ +0001fe10: 6420 6f72 0a20 2020 2020 2020 206d 756c d or. mul │ │ │ │ +0001fe20: 7469 2d67 7261 6465 6420 7269 6e67 0a20 ti-graded ring. │ │ │ │ +0001fe30: 2020 2020 202a 2066 2c20 6120 2a6e 6f74 * f, a *not │ │ │ │ +0001fe40: 6520 7269 6e67 2065 6c65 6d65 6e74 3a20 e ring element: │ │ │ │ +0001fe50: 284d 6163 6175 6c61 7932 446f 6329 5269 (Macaulay2Doc)Ri │ │ │ │ +0001fe60: 6e67 456c 656d 656e 742c 2c20 6120 656c ngElement,, a el │ │ │ │ +0001fe70: 656d 656e 7420 696e 2061 0a20 2020 2020 ement in a. │ │ │ │ +0001fe80: 2020 2067 7261 6465 6420 6f72 206d 756c graded or mul │ │ │ │ +0001fe90: 7469 2d67 7261 6465 6420 7269 6e67 0a20 ti-graded ring. │ │ │ │ +0001fea0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +0001feb0: 2020 2a20 6120 2a6e 6f74 6520 426f 6f6c * a *note Bool │ │ │ │ +0001fec0: 6561 6e20 7661 6c75 653a 2028 4d61 6361 ean value: (Maca │ │ │ │ +0001fed0: 756c 6179 3244 6f63 2942 6f6f 6c65 616e ulay2Doc)Boolean │ │ │ │ +0001fee0: 2c2c 200a 0a44 6573 6372 6970 7469 6f6e ,, ..Description │ │ │ │ +0001fef0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5465 .===========..Te │ │ │ │ +0001ff00: 7374 7320 6966 2074 6865 2069 6e70 7574 sts if the input │ │ │ │ +0001ff10: 2049 6465 616c 206f 7220 5269 6e67 456c Ideal or RingEl │ │ │ │ +0001ff20: 656d 656e 7420 6973 2048 6f6d 6f67 656e ement is Homogen │ │ │ │ +0001ff30: 656f 7573 2077 6974 6820 7265 7370 6563 eous with respec │ │ │ │ +0001ff40: 7420 746f 2074 6865 0a67 7261 6469 6e67 t to the.grading │ │ │ │ +0001ff50: 206f 6e20 7468 6520 7269 6e67 2e20 486f on the ring. Ho │ │ │ │ +0001ff60: 6d6f 6765 6e65 6f75 7320 696e 7075 7420 mogeneous input │ │ │ │ +0001ff70: 6973 2072 6571 7569 7265 6420 666f 7220 is required for │ │ │ │ +0001ff80: 616c 6c20 6d65 7468 6f64 7320 746f 2063 all methods to c │ │ │ │ +0001ff90: 6f6d 7075 7465 0a63 6861 7261 6374 6572 ompute.character │ │ │ │ +0001ffa0: 6973 7469 6320 636c 6173 7365 732e 0a0a istic classes... │ │ │ │ +0001ffb0: 5468 6973 206d 6574 686f 6420 776f 726b This method work │ │ │ │ +0001ffc0: 7320 666f 7220 6964 6561 6c73 2069 6e20 s for ideals in │ │ │ │ +0001ffd0: 7468 6520 6772 6164 6564 2063 6f6f 7264 the graded coord │ │ │ │ +0001ffe0: 696e 6174 6520 7269 6e67 7320 6f66 2074 inate rings of t │ │ │ │ +0001fff0: 6f72 6963 2076 6172 6965 7469 6573 2c0a oric varieties,. │ │ │ │ +00020000: 616e 6420 6865 6e63 6520 666f 7220 7072 and hence for pr │ │ │ │ +00020010: 6f64 7563 7473 206f 6620 7072 6f6a 6563 oducts of projec │ │ │ │ +00020020: 7469 7665 2073 7061 6365 732e 2054 6865 tive spaces. The │ │ │ │ +00020030: 7365 2063 616e 2062 6520 6372 6561 7465 se can be create │ │ │ │ +00020040: 6420 6469 7265 6374 6c79 2c20 6f72 0a75 d directly, or.u │ │ │ │ +00020050: 7369 6e67 206d 6574 686f 6473 2074 6865 sing methods the │ │ │ │ +00020060: 202a 6e6f 7465 204d 756c 7469 5072 6f6a *note MultiProj │ │ │ │ +00020070: 436f 6f72 6452 696e 673a 204d 756c 7469 CoordRing: Multi │ │ │ │ +00020080: 5072 6f6a 436f 6f72 6452 696e 672c 206d ProjCoordRing, m │ │ │ │ +00020090: 6574 686f 6420 6f66 2074 6869 730a 7061 ethod of this.pa │ │ │ │ +000200a0: 636b 6167 652c 206f 7220 7769 7468 206d ckage, or with m │ │ │ │ +000200b0: 6574 686f 6473 2066 726f 6d20 7468 6520 ethods from the │ │ │ │ +000200c0: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ +000200d0: 7469 6573 2050 6163 6b61 6765 2e0a 0a2b ties Package...+ │ │ │ │ +000200e0: 2d2d 2d2d 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 2d2b ---------------+ │ │ │ │ +00020120: 0a7c 6931 203a 2052 3d4d 756c 7469 5072 .|i1 : R=MultiPr │ │ │ │ +00020130: 6f6a 436f 6f72 6452 696e 6728 7b31 2c32 ojCoordRing({1,2 │ │ │ │ +00020140: 2c31 7d29 2020 2020 2020 2020 2020 2020 ,1}) │ │ │ │ +00020150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020160: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000201a0: 2020 207c 0a7c 6f31 203d 2052 2020 2020 |.|o1 = R │ │ │ │ 000201b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000201c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000201d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000201e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000201f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020200: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ -00020210: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -00020220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020240: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00020200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020220: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ +00020230: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -00020280: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00020290: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ -000202a0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -000202b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000202c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000202d0: 0a2b 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: 2d2b 0a7c 6932 203a 2078 3d67 656e 7328 -+.|i2 : x=gens( │ │ │ │ -00020320: 5229 2020 2020 2020 2020 2020 2020 2020 R) │ │ │ │ -00020330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020350: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020260: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00020270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000202a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +000202b0: 203a 2078 3d67 656e 7328 5229 2020 2020 : x=gens(R) │ │ │ │ +000202c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000202d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000202e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000202f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020320: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00020330: 0a7c 6f32 203d 207b 7820 2c20 7820 2c20 .|o2 = {x , x , │ │ │ │ +00020340: 7820 2c20 7820 2c20 7820 2c20 7820 2c20 x , x , x , x , │ │ │ │ +00020350: 7820 7d20 2020 2020 2020 2020 2020 2020 x } │ │ │ │ 00020360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020390: 2020 2020 207c 0a7c 6f32 203d 207b 7820 |.|o2 = {x │ │ │ │ -000203a0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820 , x , x , x , x │ │ │ │ -000203b0: 2c20 7820 2c20 7820 7d20 2020 2020 2020 , x , x } │ │ │ │ +00020370: 207c 0a7c 2020 2020 2020 2030 2020 2031 |.| 0 1 │ │ │ │ +00020380: 2020 2032 2020 2033 2020 2034 2020 2035 2 3 4 5 │ │ │ │ +00020390: 2020 2036 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +000203a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000203b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000203c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000203d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000203e0: 2030 2020 2031 2020 2032 2020 2033 2020 0 1 2 3 │ │ │ │ -000203f0: 2034 2020 2035 2020 2036 2020 2020 2020 4 5 6 │ │ │ │ -00020400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020410: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000203d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000203e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000203f0: 2020 2020 207c 0a7c 6f32 203a 204c 6973 |.|o2 : Lis │ │ │ │ +00020400: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00020410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020450: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00020460: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ -00020470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020490: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -000204a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000204b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000204c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000204d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000204e0: 0a7c 6933 203a 2049 3d69 6465 616c 2878 .|i3 : I=ideal(x │ │ │ │ -000204f0: 5f30 5e32 2a78 5f33 2d78 5f31 2a78 5f30 _0^2*x_3-x_1*x_0 │ │ │ │ -00020500: 2a78 5f34 2c78 5f36 5e33 2920 2020 2020 *x_4,x_6^3) │ │ │ │ -00020510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020520: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00020530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020560: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00020570: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00020580: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -00020590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000205a0: 2020 2020 207c 0a7c 6f33 203d 2069 6465 |.|o3 = ide │ │ │ │ -000205b0: 616c 2028 7820 7820 202d 2078 2078 2078 al (x x - x x x │ │ │ │ -000205c0: 202c 2078 2029 2020 2020 2020 2020 2020 , x ) │ │ │ │ +00020430: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00020440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020470: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +00020480: 2049 3d69 6465 616c 2878 5f30 5e32 2a78 I=ideal(x_0^2*x │ │ │ │ +00020490: 5f33 2d78 5f31 2a78 5f30 2a78 5f34 2c78 _3-x_1*x_0*x_4,x │ │ │ │ +000204a0: 5f36 5e33 2920 2020 2020 2020 2020 2020 _6^3) │ │ │ │ +000204b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000204c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000204d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000204e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000204f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00020500: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00020510: 2020 2020 2020 2020 2020 2020 3320 2020 3 │ │ │ │ +00020520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020530: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00020540: 0a7c 6f33 203d 2069 6465 616c 2028 7820 .|o3 = ideal (x │ │ │ │ +00020550: 7820 202d 2078 2078 2078 202c 2078 2029 x - x x x , x ) │ │ │ │ +00020560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020580: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020590: 2030 2033 2020 2020 3020 3120 3420 2020 0 3 0 1 4 │ │ │ │ +000205a0: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +000205b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000205c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000205d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000205e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000205f0: 2020 2020 2020 2030 2033 2020 2020 3020 0 3 0 │ │ │ │ -00020600: 3120 3420 2020 3620 2020 2020 2020 2020 1 4 6 │ │ │ │ -00020610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020620: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000205e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000205f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020600: 2020 2020 207c 0a7c 6f33 203a 2049 6465 |.|o3 : Ide │ │ │ │ +00020610: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ +00020620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020660: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -00020670: 203a 2049 6465 616c 206f 6620 5220 2020 : Ideal of R │ │ │ │ -00020680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000206a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -000206b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000206c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000206d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000206e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000206f0: 0a7c 6934 203a 2069 734d 756c 7469 486f .|i4 : isMultiHo │ │ │ │ -00020700: 6d6f 6765 6e65 6f75 7320 4920 2020 2020 mogeneous I │ │ │ │ -00020710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020640: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00020650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020680: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ +00020690: 2069 734d 756c 7469 486f 6d6f 6765 6e65 isMultiHomogene │ │ │ │ +000206a0: 6f75 7320 4920 2020 2020 2020 2020 2020 ous I │ │ │ │ +000206b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000206c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000206d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000206e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000206f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020700: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00020710: 6f34 203d 2074 7275 6520 2020 2020 2020 o4 = true │ │ │ │ 00020720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020730: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00020740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020770: 2020 207c 0a7c 6f34 203d 2074 7275 6520 |.|o4 = true │ │ │ │ -00020780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000207a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000207b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -000207c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2069 -------+.|i5 : i │ │ │ │ -00020800: 734d 756c 7469 486f 6d6f 6765 6e65 6f75 sMultiHomogeneou │ │ │ │ -00020810: 7320 6964 6561 6c28 785f 302a 785f 332d s ideal(x_0*x_3- │ │ │ │ -00020820: 785f 312a 785f 302a 785f 342c 785f 365e x_1*x_0*x_4,x_6^ │ │ │ │ -00020830: 3329 2020 2020 2020 207c 0a7c 496e 7075 3) |.|Inpu │ │ │ │ -00020840: 7420 7465 726d 2062 656c 6f77 2069 7320 t term below is │ │ │ │ -00020850: 6e6f 7420 686f 6d6f 6765 6e65 6f75 7320 not homogeneous │ │ │ │ -00020860: 7769 7468 2072 6573 7065 6374 2074 6f20 with respect to │ │ │ │ -00020870: 7468 6520 6772 6164 696e 677c 0a7c 2d20 the grading|.|- │ │ │ │ -00020880: 7820 7820 7820 202b 2078 2078 2020 2020 x x x + x x │ │ │ │ -00020890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020740: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00020750: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00020760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020790: 2d2b 0a7c 6935 203a 2069 734d 756c 7469 -+.|i5 : isMulti │ │ │ │ +000207a0: 486f 6d6f 6765 6e65 6f75 7320 6964 6561 Homogeneous idea │ │ │ │ +000207b0: 6c28 785f 302a 785f 332d 785f 312a 785f l(x_0*x_3-x_1*x_ │ │ │ │ +000207c0: 302a 785f 342c 785f 365e 3329 2020 2020 0*x_4,x_6^3) │ │ │ │ +000207d0: 2020 207c 0a7c 496e 7075 7420 7465 726d |.|Input term │ │ │ │ +000207e0: 2062 656c 6f77 2069 7320 6e6f 7420 686f below is not ho │ │ │ │ +000207f0: 6d6f 6765 6e65 6f75 7320 7769 7468 2072 mogeneous with r │ │ │ │ +00020800: 6573 7065 6374 2074 6f20 7468 6520 6772 espect to the gr │ │ │ │ +00020810: 6164 696e 677c 0a7c 2d20 7820 7820 7820 ading|.|- x x x │ │ │ │ +00020820: 202b 2078 2078 2020 2020 2020 2020 2020 + x x │ │ │ │ +00020830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020850: 2020 2020 2020 207c 0a7c 2020 2030 2031 |.| 0 1 │ │ │ │ +00020860: 2034 2020 2020 3020 3320 2020 2020 2020 4 0 3 │ │ │ │ +00020870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020890: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000208a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000208b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000208c0: 2020 2030 2031 2034 2020 2020 3020 3320 0 1 4 0 3 │ │ │ │ -000208d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000208e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000208f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020900: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00020910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020940: 207c 0a7c 6f35 203d 2066 616c 7365 2020 |.|o5 = false │ │ │ │ -00020950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020980: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00020990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000209a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000209b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000209c0: 2d2d 2d2d 2d2b 0a0a 4e6f 7465 2074 6861 -----+..Note tha │ │ │ │ -000209d0: 7420 666f 7220 616e 2069 6465 616c 2074 t for an ideal t │ │ │ │ -000209e0: 6f20 6265 206d 756c 7469 2d68 6f6d 6f67 o be multi-homog │ │ │ │ -000209f0: 656e 656f 7573 2074 6865 2064 6567 7265 eneous the degre │ │ │ │ -00020a00: 6520 7665 6374 6f72 206f 6620 616c 6c0a e vector of all. │ │ │ │ -00020a10: 6d6f 6e6f 6d69 616c 7320 696e 2061 2067 monomials in a g │ │ │ │ -00020a20: 6976 656e 2067 656e 6572 6174 6f72 206d iven generator m │ │ │ │ -00020a30: 7573 7420 6265 2074 6865 2073 616d 652e ust be the same. │ │ │ │ -00020a40: 0a0a 5761 7973 2074 6f20 7573 6520 6973 ..Ways to use is │ │ │ │ -00020a50: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ -00020a60: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -00020a70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00020a80: 3d0a 0a20 202a 2022 6973 4d75 6c74 6948 =.. * "isMultiH │ │ │ │ -00020a90: 6f6d 6f67 656e 656f 7573 2849 6465 616c omogeneous(Ideal │ │ │ │ -00020aa0: 2922 0a20 202a 2022 6973 4d75 6c74 6948 )". * "isMultiH │ │ │ │ -00020ab0: 6f6d 6f67 656e 656f 7573 2852 696e 6745 omogeneous(RingE │ │ │ │ -00020ac0: 6c65 6d65 6e74 2922 0a0a 466f 7220 7468 lement)"..For th │ │ │ │ -00020ad0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00020ae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00020af0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00020b00: 6520 6973 4d75 6c74 6948 6f6d 6f67 656e e isMultiHomogen │ │ │ │ -00020b10: 656f 7573 3a20 6973 4d75 6c74 6948 6f6d eous: isMultiHom │ │ │ │ -00020b20: 6f67 656e 656f 7573 2c20 6973 2061 202a ogeneous, is a * │ │ │ │ -00020b30: 6e6f 7465 206d 6574 686f 640a 6675 6e63 note method.func │ │ │ │ -00020b40: 7469 6f6e 3a20 284d 6163 6175 6c61 7932 tion: (Macaulay2 │ │ │ │ -00020b50: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -00020b60: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ -00020b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020bb0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -00020bc0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -00020bd0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -00020be0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -00020bf0: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ -00020c00: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+ds/M2/Macaul │ │ │ │ -00020c10: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ -00020c20: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ -00020c30: 7365 732e 6d32 3a32 3031 323a 302e 0a1f ses.m2:2012:0... │ │ │ │ -00020c40: 0a46 696c 653a 2043 6861 7261 6374 6572 .File: Character │ │ │ │ -00020c50: 6973 7469 6343 6c61 7373 6573 2e69 6e66 isticClasses.inf │ │ │ │ -00020c60: 6f2c 204e 6f64 653a 204d 6574 686f 642c o, Node: Method, │ │ │ │ -00020c70: 204e 6578 743a 204d 756c 7469 5072 6f6a Next: MultiProj │ │ │ │ -00020c80: 436f 6f72 6452 696e 672c 2050 7265 763a CoordRing, Prev: │ │ │ │ -00020c90: 2069 734d 756c 7469 486f 6d6f 6765 6e65 isMultiHomogene │ │ │ │ -00020ca0: 6f75 732c 2055 703a 2054 6f70 0a0a 4d65 ous, Up: Top..Me │ │ │ │ -00020cb0: 7468 6f64 0a2a 2a2a 2a2a 2a0a 0a44 6573 thod.******..Des │ │ │ │ -00020cc0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00020cd0: 3d3d 3d3d 0a0a 5468 6520 6f70 7469 6f6e ====..The option │ │ │ │ -00020ce0: 204d 6574 686f 6420 6973 206f 6e6c 7920 Method is only │ │ │ │ -00020cf0: 7573 6564 2062 7920 7468 6520 636f 6d6d used by the comm │ │ │ │ -00020d00: 616e 6473 202a 6e6f 7465 2043 534d 3a20 ands *note CSM: │ │ │ │ -00020d10: 4353 4d2c 2061 6e64 202a 6e6f 7465 2045 CSM, and *note E │ │ │ │ -00020d20: 756c 6572 3a0a 4575 6c65 722c 2061 6e64 uler:.Euler, and │ │ │ │ -00020d30: 206f 6e6c 7920 696e 2063 6f6d 6269 6e61 only in combina │ │ │ │ -00020d40: 7469 6f6e 2077 6974 6820 2a6e 6f74 6520 tion with *note │ │ │ │ -00020d50: 436f 6d70 4d65 7468 6f64 3a0a 436f 6d70 CompMethod:.Comp │ │ │ │ -00020d60: 4d65 7468 6f64 2c3d 3e50 726f 6a65 6374 Method,=>Project │ │ │ │ -00020d70: 6976 6544 6567 7265 652e 2054 6865 204d iveDegree. The M │ │ │ │ -00020d80: 6574 686f 6420 496e 636c 7573 696f 6e45 ethod InclusionE │ │ │ │ -00020d90: 7863 6c75 7369 6f6e 2077 696c 6c20 616c xclusion will al │ │ │ │ -00020da0: 7761 7973 2062 650a 7573 6564 2077 6974 ways be.used wit │ │ │ │ -00020db0: 6820 2a6e 6f74 6520 436f 6d70 4d65 7468 h *note CompMeth │ │ │ │ -00020dc0: 6f64 3a20 436f 6d70 4d65 7468 6f64 2c20 od: CompMethod, │ │ │ │ -00020dd0: 506e 5265 7369 6475 616c 206f 7220 6265 PnResidual or be │ │ │ │ -00020de0: 7274 696e 692e 2057 6865 6e20 7468 6520 rtini. When the │ │ │ │ -00020df0: 696e 7075 740a 6964 6561 6c20 6973 2061 input.ideal is a │ │ │ │ -00020e00: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -00020e10: 6563 7469 6f6e 206f 6e65 206d 6179 2c20 ection one may, │ │ │ │ -00020e20: 706f 7465 6e74 6961 6c6c 792c 2073 7065 potentially, spe │ │ │ │ -00020e30: 6564 2075 7020 7468 6520 636f 6d70 7574 ed up the comput │ │ │ │ -00020e40: 6174 696f 6e0a 6279 2073 6574 7469 6e67 ation.by setting │ │ │ │ -00020e50: 204d 6574 686f 643d 3e20 4469 7265 6374 Method=> Direct │ │ │ │ -00020e60: 436f 6d70 6c65 7465 496e 742e 2054 6865 CompleteInt. The │ │ │ │ -00020e70: 206f 7074 696f 6e20 4d65 7468 6f64 2069 option Method i │ │ │ │ -00020e80: 7320 6f6e 6c79 2075 7365 6420 6279 2074 s only used by t │ │ │ │ -00020e90: 6865 0a63 6f6d 6d61 6e64 7320 2a6e 6f74 he.commands *not │ │ │ │ -00020ea0: 6520 4353 4d3a 2043 534d 2c20 616e 6420 e CSM: CSM, and │ │ │ │ -00020eb0: 2a6e 6f74 6520 4575 6c65 723a 2045 756c *note Euler: Eul │ │ │ │ -00020ec0: 6572 2c20 616e 6420 6f6e 6c79 2069 6e20 er, and only in │ │ │ │ -00020ed0: 636f 6d62 696e 6174 696f 6e20 7769 7468 combination with │ │ │ │ -00020ee0: 0a2a 6e6f 7465 2043 6f6d 704d 6574 686f .*note CompMetho │ │ │ │ -00020ef0: 643a 2043 6f6d 704d 6574 686f 642c 3d3e d: CompMethod,=> │ │ │ │ -00020f00: 5072 6f6a 6563 7469 7665 4465 6772 6565 ProjectiveDegree │ │ │ │ -00020f10: 2e20 5468 6520 4d65 7468 6f64 2049 6e63 . The Method Inc │ │ │ │ -00020f20: 6c75 7369 6f6e 4578 636c 7573 696f 6e0a lusionExclusion. │ │ │ │ -00020f30: 7769 6c6c 2061 6c77 6179 7320 6265 2075 will always be u │ │ │ │ -00020f40: 7365 6420 7769 7468 202a 6e6f 7465 2043 sed with *note C │ │ │ │ -00020f50: 6f6d 704d 6574 686f 643a 2043 6f6d 704d ompMethod: CompM │ │ │ │ -00020f60: 6574 686f 642c 2050 6e52 6573 6964 7561 ethod, PnResidua │ │ │ │ -00020f70: 6c20 6f72 2062 6572 7469 6e69 2e0a 0a2b l or bertini...+ │ │ │ │ -00020f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020fb0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -00020fc0: 5220 3d20 5a5a 2f33 3237 3439 5b78 5f30 R = ZZ/32749[x_0 │ │ │ │ -00020fd0: 2e2e 785f 365d 2020 2020 2020 2020 2020 ..x_6] │ │ │ │ +000208b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000208c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000208d0: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ +000208e0: 203d 2066 616c 7365 2020 2020 2020 2020 = false │ │ │ │ +000208f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020910: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00020920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00020960: 0a0a 4e6f 7465 2074 6861 7420 666f 7220 ..Note that for │ │ │ │ +00020970: 616e 2069 6465 616c 2074 6f20 6265 206d an ideal to be m │ │ │ │ +00020980: 756c 7469 2d68 6f6d 6f67 656e 656f 7573 ulti-homogeneous │ │ │ │ +00020990: 2074 6865 2064 6567 7265 6520 7665 6374 the degree vect │ │ │ │ +000209a0: 6f72 206f 6620 616c 6c0a 6d6f 6e6f 6d69 or of all.monomi │ │ │ │ +000209b0: 616c 7320 696e 2061 2067 6976 656e 2067 als in a given g │ │ │ │ +000209c0: 656e 6572 6174 6f72 206d 7573 7420 6265 enerator must be │ │ │ │ +000209d0: 2074 6865 2073 616d 652e 0a0a 5761 7973 the same...Ways │ │ │ │ +000209e0: 2074 6f20 7573 6520 6973 4d75 6c74 6948 to use isMultiH │ │ │ │ +000209f0: 6f6d 6f67 656e 656f 7573 3a0a 3d3d 3d3d omogeneous:.==== │ │ │ │ +00020a00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00020a10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00020a20: 2022 6973 4d75 6c74 6948 6f6d 6f67 656e "isMultiHomogen │ │ │ │ +00020a30: 656f 7573 2849 6465 616c 2922 0a20 202a eous(Ideal)". * │ │ │ │ +00020a40: 2022 6973 4d75 6c74 6948 6f6d 6f67 656e "isMultiHomogen │ │ │ │ +00020a50: 656f 7573 2852 696e 6745 6c65 6d65 6e74 eous(RingElement │ │ │ │ +00020a60: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +00020a70: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00020a80: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00020a90: 626a 6563 7420 2a6e 6f74 6520 6973 4d75 bject *note isMu │ │ │ │ +00020aa0: 6c74 6948 6f6d 6f67 656e 656f 7573 3a20 ltiHomogeneous: │ │ │ │ +00020ab0: 6973 4d75 6c74 6948 6f6d 6f67 656e 656f isMultiHomogeneo │ │ │ │ +00020ac0: 7573 2c20 6973 2061 202a 6e6f 7465 206d us, is a *note m │ │ │ │ +00020ad0: 6574 686f 640a 6675 6e63 7469 6f6e 3a20 ethod.function: │ │ │ │ +00020ae0: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +00020af0: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +00020b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00020b50: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00020b60: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00020b70: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00020b80: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00020b90: 756c 6179 322d 312e 3235 2e31 312b 6473 ulay2-1.25.11+ds │ │ │ │ +00020ba0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00020bb0: 636b 6167 6573 2f0a 4368 6172 6163 7465 ckages/.Characte │ │ │ │ +00020bc0: 7269 7374 6963 436c 6173 7365 732e 6d32 risticClasses.m2 │ │ │ │ +00020bd0: 3a32 3031 323a 302e 0a1f 0a46 696c 653a :2012:0....File: │ │ │ │ +00020be0: 2043 6861 7261 6374 6572 6973 7469 6343 CharacteristicC │ │ │ │ +00020bf0: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64 lasses.info, Nod │ │ │ │ +00020c00: 653a 204d 6574 686f 642c 204e 6578 743a e: Method, Next: │ │ │ │ +00020c10: 204d 756c 7469 5072 6f6a 436f 6f72 6452 MultiProjCoordR │ │ │ │ +00020c20: 696e 672c 2050 7265 763a 2069 734d 756c ing, Prev: isMul │ │ │ │ +00020c30: 7469 486f 6d6f 6765 6e65 6f75 732c 2055 tiHomogeneous, U │ │ │ │ +00020c40: 703a 2054 6f70 0a0a 4d65 7468 6f64 0a2a p: Top..Method.* │ │ │ │ +00020c50: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ +00020c60: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +00020c70: 5468 6520 6f70 7469 6f6e 204d 6574 686f The option Metho │ │ │ │ +00020c80: 6420 6973 206f 6e6c 7920 7573 6564 2062 d is only used b │ │ │ │ +00020c90: 7920 7468 6520 636f 6d6d 616e 6473 202a y the commands * │ │ │ │ +00020ca0: 6e6f 7465 2043 534d 3a20 4353 4d2c 2061 note CSM: CSM, a │ │ │ │ +00020cb0: 6e64 202a 6e6f 7465 2045 756c 6572 3a0a nd *note Euler:. │ │ │ │ +00020cc0: 4575 6c65 722c 2061 6e64 206f 6e6c 7920 Euler, and only │ │ │ │ +00020cd0: 696e 2063 6f6d 6269 6e61 7469 6f6e 2077 in combination w │ │ │ │ +00020ce0: 6974 6820 2a6e 6f74 6520 436f 6d70 4d65 ith *note CompMe │ │ │ │ +00020cf0: 7468 6f64 3a0a 436f 6d70 4d65 7468 6f64 thod:.CompMethod │ │ │ │ +00020d00: 2c3d 3e50 726f 6a65 6374 6976 6544 6567 ,=>ProjectiveDeg │ │ │ │ +00020d10: 7265 652e 2054 6865 204d 6574 686f 6420 ree. The Method │ │ │ │ +00020d20: 496e 636c 7573 696f 6e45 7863 6c75 7369 InclusionExclusi │ │ │ │ +00020d30: 6f6e 2077 696c 6c20 616c 7761 7973 2062 on will always b │ │ │ │ +00020d40: 650a 7573 6564 2077 6974 6820 2a6e 6f74 e.used with *not │ │ │ │ +00020d50: 6520 436f 6d70 4d65 7468 6f64 3a20 436f e CompMethod: Co │ │ │ │ +00020d60: 6d70 4d65 7468 6f64 2c20 506e 5265 7369 mpMethod, PnResi │ │ │ │ +00020d70: 6475 616c 206f 7220 6265 7274 696e 692e dual or bertini. │ │ │ │ +00020d80: 2057 6865 6e20 7468 6520 696e 7075 740a When the input. │ │ │ │ +00020d90: 6964 6561 6c20 6973 2061 2063 6f6d 706c ideal is a compl │ │ │ │ +00020da0: 6574 6520 696e 7465 7273 6563 7469 6f6e ete intersection │ │ │ │ +00020db0: 206f 6e65 206d 6179 2c20 706f 7465 6e74 one may, potent │ │ │ │ +00020dc0: 6961 6c6c 792c 2073 7065 6564 2075 7020 ially, speed up │ │ │ │ +00020dd0: 7468 6520 636f 6d70 7574 6174 696f 6e0a the computation. │ │ │ │ +00020de0: 6279 2073 6574 7469 6e67 204d 6574 686f by setting Metho │ │ │ │ +00020df0: 643d 3e20 4469 7265 6374 436f 6d70 6c65 d=> DirectComple │ │ │ │ +00020e00: 7465 496e 742e 2054 6865 206f 7074 696f teInt. The optio │ │ │ │ +00020e10: 6e20 4d65 7468 6f64 2069 7320 6f6e 6c79 n Method is only │ │ │ │ +00020e20: 2075 7365 6420 6279 2074 6865 0a63 6f6d used by the.com │ │ │ │ +00020e30: 6d61 6e64 7320 2a6e 6f74 6520 4353 4d3a mands *note CSM: │ │ │ │ +00020e40: 2043 534d 2c20 616e 6420 2a6e 6f74 6520 CSM, and *note │ │ │ │ +00020e50: 4575 6c65 723a 2045 756c 6572 2c20 616e Euler: Euler, an │ │ │ │ +00020e60: 6420 6f6e 6c79 2069 6e20 636f 6d62 696e d only in combin │ │ │ │ +00020e70: 6174 696f 6e20 7769 7468 0a2a 6e6f 7465 ation with.*note │ │ │ │ +00020e80: 2043 6f6d 704d 6574 686f 643a 2043 6f6d CompMethod: Com │ │ │ │ +00020e90: 704d 6574 686f 642c 3d3e 5072 6f6a 6563 pMethod,=>Projec │ │ │ │ +00020ea0: 7469 7665 4465 6772 6565 2e20 5468 6520 tiveDegree. The │ │ │ │ +00020eb0: 4d65 7468 6f64 2049 6e63 6c75 7369 6f6e Method Inclusion │ │ │ │ +00020ec0: 4578 636c 7573 696f 6e0a 7769 6c6c 2061 Exclusion.will a │ │ │ │ +00020ed0: 6c77 6179 7320 6265 2075 7365 6420 7769 lways be used wi │ │ │ │ +00020ee0: 7468 202a 6e6f 7465 2043 6f6d 704d 6574 th *note CompMet │ │ │ │ +00020ef0: 686f 643a 2043 6f6d 704d 6574 686f 642c hod: CompMethod, │ │ │ │ +00020f00: 2050 6e52 6573 6964 7561 6c20 6f72 2062 PnResidual or b │ │ │ │ +00020f10: 6572 7469 6e69 2e0a 0a2b 2d2d 2d2d 2d2d ertini...+------ │ │ │ │ +00020f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020f50: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 5a5a --+.|i1 : R = ZZ │ │ │ │ +00020f60: 2f33 3237 3439 5b78 5f30 2e2e 785f 365d /32749[x_0..x_6] │ │ │ │ +00020f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020f80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00020f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020fc0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ +00020fd0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00020fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ff0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021000: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00021010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021020: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00021030: 7c6f 3120 3d20 5220 2020 2020 2020 2020 |o1 = R │ │ │ │ -00021040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021060: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00021070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000210a0: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79 |.|o1 : Poly │ │ │ │ -000210b0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ -000210c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000210d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000210e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -000210f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021110: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ -00021120: 3a20 493d 6964 6561 6c28 7261 6e64 6f6d : I=ideal(random │ │ │ │ -00021130: 2832 2c52 292c 7261 6e64 6f6d 2831 2c52 (2,R),random(1,R │ │ │ │ -00021140: 292c 525f 302a 525f 312a 525f 362d 525f ),R_0*R_1*R_6-R_ │ │ │ │ -00021150: 305e 3329 3b7c 0a7c 2020 2020 2020 2020 0^3);|.| │ │ │ │ -00021160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021190: 7c0a 7c6f 3220 3a20 4964 6561 6c20 6f66 |.|o2 : Ideal of │ │ │ │ -000211a0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -000211b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000211c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -000211d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000211e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000211f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021200: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7469 ------+.|i3 : ti │ │ │ │ -00021210: 6d65 2043 534d 2049 2020 2020 2020 2020 me CSM I │ │ │ │ +00021020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021040: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c |o1 : Polynomial │ │ │ │ +00021050: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +00021060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021070: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00021080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000210a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000210b0: 2d2d 2d2d 2b0a 7c69 3220 3a20 493d 6964 ----+.|i2 : I=id │ │ │ │ +000210c0: 6561 6c28 7261 6e64 6f6d 2832 2c52 292c eal(random(2,R), │ │ │ │ +000210d0: 7261 6e64 6f6d 2831 2c52 292c 525f 302a random(1,R),R_0* │ │ │ │ +000210e0: 525f 312a 525f 362d 525f 305e 3329 3b7c R_1*R_6-R_0^3);| │ │ │ │ +000210f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00021100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021120: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +00021130: 3a20 4964 6561 6c20 6f66 2052 2020 2020 : Ideal of R │ │ │ │ +00021140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021160: 2020 2020 207c 0a2b 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: 2b0a 7c69 3320 3a20 7469 6d65 2043 534d +.|i3 : time CSM │ │ │ │ +000211b0: 2049 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ +000211c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000211d0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ +000211e0: 2d20 7573 6564 2033 2e32 3532 3335 7320 - used 3.25235s │ │ │ │ +000211f0: 2863 7075 293b 2031 2e32 3737 3534 7320 (cpu); 1.27754s │ │ │ │ +00021200: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +00021210: 2920 2020 2020 7c0a 7c20 2020 2020 2020 ) |.| │ │ │ │ 00021220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021240: 207c 0a7c 202d 2d20 7573 6564 2031 2e31 |.| -- used 1.1 │ │ │ │ -00021250: 3435 3035 7320 2863 7075 293b 2030 2e38 4505s (cpu); 0.8 │ │ │ │ -00021260: 3839 3338 3373 2028 7468 7265 6164 293b 89383s (thread); │ │ │ │ -00021270: 2030 7320 2867 6329 2020 2020 7c0a 7c20 0s (gc) |.| │ │ │ │ -00021280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000212a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000212b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000212c0: 2020 3520 2020 2020 2034 2020 2020 2033 5 4 3 │ │ │ │ -000212d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021250: 207c 0a7c 2020 2020 2020 2020 3520 2020 |.| 5 │ │ │ │ +00021260: 2020 2034 2020 2020 2033 2020 2020 2020 4 3 │ │ │ │ +00021270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021280: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00021290: 3320 3d20 3132 6820 202b 2031 3068 2020 3 = 12h + 10h │ │ │ │ +000212a0: 2b20 3668 2020 2020 2020 2020 2020 2020 + 6h │ │ │ │ +000212b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000212c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000212d0: 2020 3120 2020 2020 2031 2020 2020 2031 1 1 1 │ │ │ │ 000212e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000212f0: 2020 7c0a 7c6f 3320 3d20 3132 6820 202b |.|o3 = 12h + │ │ │ │ -00021300: 2031 3068 2020 2b20 3668 2020 2020 2020 10h + 6h │ │ │ │ +000212f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021300: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00021310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021320: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00021330: 2020 2020 2020 2020 3120 2020 2020 2031 1 1 │ │ │ │ -00021340: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +00021320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021330: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00021340: 2020 2020 205a 5a5b 6820 5d20 2020 2020 ZZ[h ] │ │ │ │ 00021350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021360: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00021370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021370: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00021380: 2020 2020 3120 2020 2020 2020 2020 2020 1 │ │ │ │ 00021390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000213a0: 2020 207c 0a7c 2020 2020 205a 5a5b 6820 |.| ZZ[h │ │ │ │ -000213b0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -000213c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000213d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000213e0: 7c20 2020 2020 2020 2020 3120 2020 2020 | 1 │ │ │ │ -000213f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000213a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000213b0: 2020 207c 0a7c 6f33 203a 202d 2d2d 2d2d |.|o3 : ----- │ │ │ │ +000213c0: 2d20 2020 2020 2020 2020 2020 2020 2020 - │ │ │ │ +000213d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000213e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000213f0: 7c20 2020 2020 2020 2037 2020 2020 2020 | 7 │ │ │ │ 00021400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021410: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ -00021420: 202d 2d2d 2d2d 2d20 2020 2020 2020 2020 ------ │ │ │ │ -00021430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021420: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00021430: 2020 2068 2020 2020 2020 2020 2020 2020 h │ │ │ │ 00021440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021450: 2020 2020 7c0a 7c20 2020 2020 2020 2037 |.| 7 │ │ │ │ -00021460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021460: 2020 2020 7c0a 7c20 2020 2020 2020 2031 |.| 1 │ │ │ │ 00021470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021480: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021490: 0a7c 2020 2020 2020 2068 2020 2020 2020 .| h │ │ │ │ -000214a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000214d0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ -000214e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021500: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00021510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021540: 2b0a 7c69 3420 3a20 7469 6d65 2043 534d +.|i4 : time CSM │ │ │ │ -00021550: 2849 2c4d 6574 686f 643d 3e44 6972 6563 (I,Method=>Direc │ │ │ │ -00021560: 7443 6f6d 706c 6574 6549 6e74 2920 2020 tCompleteInt) │ │ │ │ -00021570: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00021580: 2d20 7573 6564 2030 2e32 3838 3434 3673 - used 0.288446s │ │ │ │ -00021590: 2028 6370 7529 3b20 302e 3232 3432 3238 (cpu); 0.224228 │ │ │ │ -000215a0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -000215b0: 6763 2920 2020 7c0a 7c20 2020 2020 2020 gc) |.| │ │ │ │ -000215c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000214a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000214b0: 2d2d 2d2d 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2020 2034 2020 5 4 │ │ │ │ +000215a0: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +000215b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000215c0: 2020 2020 2020 7c0a 7c6f 3420 3d20 3132 |.|o4 = 12 │ │ │ │ +000215d0: 6820 202b 2031 3068 2020 2b20 3668 2020 h + 10h + 6h │ │ │ │ 000215e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215f0: 207c 0a7c 2020 2020 2020 2020 3520 2020 |.| 5 │ │ │ │ -00021600: 2020 2034 2020 2020 2033 2020 2020 2020 4 3 │ │ │ │ -00021610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021620: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00021630: 3420 3d20 3132 6820 202b 2031 3068 2020 4 = 12h + 10h │ │ │ │ -00021640: 2b20 3668 2020 2020 2020 2020 2020 2020 + 6h │ │ │ │ +000215f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021600: 207c 0a7c 2020 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ +00021610: 2020 2031 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ +00021620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021630: 2020 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-00021840: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00021850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021870: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6865 ----------+..Whe │ │ │ │ -00021880: 6e20 7573 696e 6720 7468 6520 4469 7265 n using the Dire │ │ │ │ -00021890: 6374 436f 6d70 6c65 7465 496e 7420 6d65 ctCompleteInt me │ │ │ │ -000218a0: 7468 6f64 206f 6e65 206d 6179 2070 6f74 thod one may pot │ │ │ │ -000218b0: 656e 7469 616c 6c79 2066 7572 7468 6572 entially further │ │ │ │ -000218c0: 2073 7065 6564 2075 700a 636f 6d70 7574 speed up.comput │ │ │ │ -000218d0: 6174 696f 6e20 7469 6d65 2062 7920 7370 ation time by sp │ │ │ │ -000218e0: 6563 6966 7969 6e67 2077 6861 7420 7375 ecifying what su │ │ │ │ -000218f0: 6273 6574 206f 6620 7468 6520 6765 6e65 bset of the gene │ │ │ │ -00021900: 7261 746f 7273 206f 6620 7468 6520 696e rators of the in │ │ │ │ -00021910: 7075 7420 6964 6561 6c0a 6465 6669 6e65 put ideal.define │ │ │ │ -00021920: 2061 2073 6d6f 6f74 6820 7375 6273 6368 a smooth subsch │ │ │ │ -00021930: 656d 6520 2869 6620 7468 6973 2069 7320 eme (if this is │ │ │ │ -00021940: 6b6e 6f77 6e29 2c20 7365 6520 2a6e 6f74 known), see *not │ │ │ │ -00021950: 6520 496e 6473 4f66 536d 6f6f 7468 3a0a e IndsOfSmooth:. │ │ │ │ -00021960: 496e 6473 4f66 536d 6f6f 7468 2c2e 0a0a IndsOfSmooth,... │ │ │ │ -00021970: 4675 6e63 7469 6f6e 7320 7769 7468 206f Functions with o │ │ │ │ -00021980: 7074 696f 6e61 6c20 6172 6775 6d65 6e74 ptional argument │ │ │ │ -00021990: 206e 616d 6564 204d 6574 686f 643a 0a3d named Method:.= │ │ │ │ -000219a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000219b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000219c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -000219d0: 202a 2022 4353 4d28 2e2e 2e2c 4d65 7468 * "CSM(...,Meth │ │ │ │ -000219e0: 6f64 3d3e 2e2e 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(missing │ │ │ │ -00021a50: 646f 6375 6d65 6e74 6174 696f 6e29 0a0a documentation).. │ │ │ │ -00021a60: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -00021a70: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -00021a80: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -00021a90: 7420 2a6e 6f74 6520 4d65 7468 6f64 3a20 t *note Method: │ │ │ │ -00021aa0: 4d65 7468 6f64 2c20 6973 2061 202a 6e6f Method, is a *no │ │ │ │ -00021ab0: 7465 2073 796d 626f 6c3a 2028 4d61 6361 te symbol: (Maca │ │ │ │ -00021ac0: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ -00021ad0: 2e0a 0a2d 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 2d2d 2d2d ---------------- │ │ │ │ -00021b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021b20: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -00021b30: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -00021b40: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -00021b50: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -00021b60: 6163 6175 6c61 7932 2d31 2e32 352e 3131 acaulay2-1.25.11 │ │ │ │ -00021b70: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -00021b80: 2f70 6163 6b61 6765 732f 0a43 6861 7261 /packages/.Chara │ │ │ │ -00021b90: 6374 6572 6973 7469 6343 6c61 7373 6573 cteristicClasses │ │ │ │ -00021ba0: 2e6d 323a 3234 3332 3a30 2e0a 1f0a 4669 .m2:2432:0....Fi │ │ │ │ -00021bb0: 6c65 3a20 4368 6172 6163 7465 7269 7374 le: Characterist │ │ │ │ -00021bc0: 6963 436c 6173 7365 732e 696e 666f 2c20 icClasses.info, │ │ │ │ -00021bd0: 4e6f 6465 3a20 4d75 6c74 6950 726f 6a43 Node: MultiProjC │ │ │ │ -00021be0: 6f6f 7264 5269 6e67 2c20 4e65 7874 3a20 oordRing, Next: │ │ │ │ -00021bf0: 4f75 7470 7574 2c20 5072 6576 3a20 4d65 Output, Prev: Me │ │ │ │ -00021c00: 7468 6f64 2c20 5570 3a20 546f 700a 0a4d thod, Up: Top..M │ │ │ │ -00021c10: 756c 7469 5072 6f6a 436f 6f72 6452 696e ultiProjCoordRin │ │ │ │ -00021c20: 6720 2d2d 2041 2071 7569 636b 2077 6179 g -- A quick way │ │ │ │ -00021c30: 2074 6f20 6275 696c 6420 7468 6520 636f to build the co │ │ │ │ -00021c40: 6f72 6469 6e61 7465 2072 696e 6720 6f66 ordinate ring of │ │ │ │ -00021c50: 2061 2070 726f 6475 6374 206f 6620 7072 a product of pr │ │ │ │ -00021c60: 6f6a 6563 7469 7665 2073 7061 6365 730a ojective spaces. │ │ │ │ -00021c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021cd0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -00021ce0: 2020 2020 2020 4d75 6c74 6950 726f 6a43 MultiProjC │ │ │ │ -00021cf0: 6f6f 7264 5269 6e67 2044 696d 730a 2020 oordRing Dims. │ │ │ │ -00021d00: 2020 2020 2020 4d75 6c74 6950 726f 6a43 MultiProjC │ │ │ │ -00021d10: 6f6f 7264 5269 6e67 2028 436f 6566 6652 oordRing (CoeffR │ │ │ │ -00021d20: 696e 672c 4469 6d73 290a 2020 2020 2020 ing,Dims). │ │ │ │ -00021d30: 2020 4d75 6c74 6950 726f 6a43 6f6f 7264 MultiProjCoord │ │ │ │ -00021d40: 5269 6e67 2028 7661 722c 4469 6d73 290a Ring (var,Dims). │ │ │ │ -00021d50: 2020 2020 2020 2020 4d75 6c74 6950 726f MultiPro │ │ │ │ -00021d60: 6a43 6f6f 7264 5269 6e67 2028 436f 6566 jCoordRing (Coef │ │ │ │ -00021d70: 6652 696e 672c 7661 722c 4469 6d73 290a fRing,var,Dims). │ │ │ │ -00021d80: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -00021d90: 2020 2a20 4469 6d73 2c20 6120 2a6e 6f74 * Dims, a *not │ │ │ │ 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6865 2063 6f65 6666 Ring,, the coeff │ │ │ │ -00021e80: 6963 6965 6e74 2072 696e 6720 6f66 0a20 icient ring of. │ │ │ │ -00021e90: 2020 2020 2020 2074 6865 2067 7261 6465 the grade │ │ │ │ -00021ea0: 6420 706f 6c79 6e6f 6d69 616c 2072 696e d polynomial rin │ │ │ │ -00021eb0: 6720 746f 2062 6520 6275 696c 7420 6279 g to be built by │ │ │ │ -00021ec0: 2074 6865 206d 6574 686f 642c 2062 7920 the method, by │ │ │ │ -00021ed0: 6465 6661 756c 7420 7468 6973 0a20 2020 default this. │ │ │ │ -00021ee0: 2020 2020 2069 7320 5c5a 5a2f 3332 3734 is \ZZ/3274 │ │ │ │ -00021ef0: 390a 2020 2020 2020 2a20 7661 722c 2061 9. * var, a │ │ │ │ -00021f00: 202a 6e6f 7465 2073 796d 626f 6c3a 2028 *note symbol: ( │ │ │ │ -00021f10: 4d61 6361 756c 6179 3244 6f63 2953 796d Macaulay2Doc)Sym │ │ │ │ -00021f20: 626f 6c2c 2c20 746f 2062 6520 7573 6564 bol,, to be used │ │ │ │ -00021f30: 2066 6f72 2074 6865 0a20 2020 2020 2020 for the. │ │ │ │ -00021f40: 2069 6e74 6572 6d65 6469 6174 6573 206f intermediates o │ │ │ │ -00021f50: 6620 7468 6520 6772 6164 6564 2070 6f6c f the graded pol │ │ │ │ -00021f60: 796e 6f6d 6961 6c20 7269 6e67 2074 6f20 ynomial ring to │ │ │ │ -00021f70: 6265 2062 7569 6c74 2062 7920 7468 6520 be built by the │ │ │ │ -00021f80: 6d65 7468 6f64 0a20 202a 204f 7574 7075 method. * Outpu │ │ │ │ -00021f90: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e ts:. * a *n │ │ │ │ -00021fa0: 6f74 6520 7269 6e67 3a20 284d 6163 6175 ote ring: (Macau │ │ │ │ -00021fb0: 6c61 7932 446f 6329 5269 6e67 2c2c 2074 lay2Doc)Ring,, t │ │ │ │ -00021fc0: 6865 2067 7261 6465 6420 636f 6f72 6469 he graded coordi │ │ │ │ -00021fd0: 6e61 7465 2072 696e 6720 6f66 2074 6865 nate ring of the │ │ │ │ -00021fe0: 0a20 2020 2020 2020 205c 5050 5e7b 6e5f . \PP^{n_ │ │ │ │ -00021ff0: 317d 2078 2e2e 2e2e 2078 205c 5050 5e7b 1} x.... x \PP^{ │ │ │ │ -00022000: 6e5f 6d7d 2077 6865 7265 207b 6e5f 312c n_m} where {n_1, │ │ │ │ -00022010: 2e2e 2e2c 6e5f 6d7d 2069 7320 7468 6520 ...,n_m} is the │ │ │ │ -00022020: 696e 7075 7420 6c69 7374 206f 660a 2020 input list of. │ │ │ │ -00022030: 2020 2020 2020 6469 6d65 6e73 696f 6e73 dimensions │ │ │ │ -00022040: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -00022050: 3d3d 3d3d 3d3d 3d3d 3d0a 0a43 6f6d 7075 =========..Compu │ │ │ │ -00022060: 7465 7320 7468 6520 6772 6164 6564 2063 tes the graded c │ │ │ │ -00022070: 6f6f 7264 696e 6174 6520 7269 6e67 206f oordinate ring o │ │ │ │ -00022080: 6620 7468 6520 5c50 505e 7b6e 5f31 7d20 f the \PP^{n_1} │ │ │ │ -00022090: 782e 2e2e 2e20 7820 5c50 505e 7b6e 5f6d x.... x \PP^{n_m │ │ │ │ -000220a0: 7d20 7768 6572 650a 7b6e 5f31 2c2e 2e2e } where.{n_1,... │ │ │ │ -000220b0: 2c6e 5f6d 7d20 6973 2074 6865 2069 6e70 ,n_m} is the inp │ │ │ │ -000220c0: 7574 206c 6973 7420 6f66 2064 696d 656e ut list of dimen │ │ │ │ -000220d0: 7369 6f6e 732e 2054 6869 7320 6d65 7468 sions. This meth │ │ │ │ -000220e0: 6f64 2069 7320 7573 6564 2074 6f20 7175 od is used to qu │ │ │ │ -000220f0: 6963 6b6c 790a 6275 696c 6420 7468 6520 ickly.build the │ │ │ │ -00022100: 636f 6f72 6469 6e61 7465 2072 696e 6720 coordinate ring │ │ │ │ -00022110: 6f66 2061 2070 726f 6475 6374 206f 6620 of a product of │ │ │ │ -00022120: 7072 6f6a 6563 7469 7665 2073 7061 6365 projective space │ │ │ │ -00022130: 7320 666f 7220 7573 6520 696e 0a63 6f6d s for use in.com │ │ │ │ -00022140: 7075 7461 7469 6f6e 732e 0a0a 2b2d 2d2d putations...+--- │ │ │ │ -00022150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022190: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ -000221a0: 3a20 533d 4d75 6c74 6950 726f 6a43 6f6f : S=MultiProjCoo │ │ │ │ -000221b0: 7264 5269 6e67 2851 512c 7379 6d62 6f6c rdRing(QQ,symbol │ │ │ │ -000221c0: 207a 2c7b 312c 332c 337d 2920 2020 2020 z,{1,3,3}) │ │ │ │ -000221d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000221e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000217c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000217d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000217e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000217f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021810: 2d2d 2d2d 2b0a 0a57 6865 6e20 7573 696e ----+..When usin │ │ │ │ +00021820: 6720 7468 6520 4469 7265 6374 436f 6d70 g the DirectComp │ │ │ │ +00021830: 6c65 7465 496e 7420 6d65 7468 6f64 206f leteInt method o │ │ │ │ +00021840: 6e65 206d 6179 2070 6f74 656e 7469 616c ne may potential │ │ │ │ +00021850: 6c79 2066 7572 7468 6572 2073 7065 6564 ly further speed │ │ │ │ +00021860: 2075 700a 636f 6d70 7574 6174 696f 6e20 up.computation │ │ │ │ +00021870: 7469 6d65 2062 7920 7370 6563 6966 7969 time by specifyi │ │ │ │ +00021880: 6e67 2077 6861 7420 7375 6273 6574 206f ng what subset o │ │ │ │ +00021890: 6620 7468 6520 6765 6e65 7261 746f 7273 f the generators │ │ │ │ +000218a0: 206f 6620 7468 6520 696e 7075 7420 6964 of the input id │ │ │ │ +000218b0: 6561 6c0a 6465 6669 6e65 2061 2073 6d6f eal.define a smo │ │ │ │ +000218c0: 6f74 6820 7375 6273 6368 656d 6520 2869 oth subscheme (i │ │ │ │ +000218d0: 6620 7468 6973 2069 7320 6b6e 6f77 6e29 f this is known) │ │ │ │ +000218e0: 2c20 7365 6520 2a6e 6f74 6520 496e 6473 , see *note Inds │ │ │ │ +000218f0: 4f66 536d 6f6f 7468 3a0a 496e 6473 4f66 OfSmooth:.IndsOf │ │ │ │ +00021900: 536d 6f6f 7468 2c2e 0a0a 4675 6e63 7469 Smooth,...Functi │ │ │ │ +00021910: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ +00021920: 6c20 6172 6775 6d65 6e74 206e 616d 6564 l argument named │ │ │ │ +00021930: 204d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d Method:.======= │ │ │ │ +00021940: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00021950: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00021960: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4353 =======.. * "CS │ │ │ │ +00021970: 4d28 2e2e 2e2c 4d65 7468 6f64 3d3e 2e2e M(...,Method=>.. │ │ │ │ +00021980: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465 .)" -- see *note │ │ │ │ +00021990: 2043 534d 3a20 4353 4d2c 202d 2d20 5468 CSM: CSM, -- Th │ │ │ │ +000219a0: 650a 2020 2020 4368 6572 6e2d 5363 6877 e. Chern-Schw │ │ │ │ +000219b0: 6172 747a 2d4d 6163 5068 6572 736f 6e20 artz-MacPherson │ │ │ │ +000219c0: 636c 6173 730a 2020 2a20 4575 6c65 7228 class. * Euler( │ │ │ │ +000219d0: 2e2e 2e2c 4d65 7468 6f64 3d3e 2e2e 2e29 ...,Method=>...) │ │ │ │ +000219e0: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +000219f0: 6e74 6174 696f 6e29 0a0a 466f 7220 7468 ntation)..For th │ │ │ │ +00021a00: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +00021a10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00021a20: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +00021a30: 6520 4d65 7468 6f64 3a20 4d65 7468 6f64 e Method: Method │ │ │ │ +00021a40: 2c20 6973 2061 202a 6e6f 7465 2073 796d , is a *note sym │ │ │ │ +00021a50: 626f 6c3a 2028 4d61 6361 756c 6179 3244 bol: (Macaulay2D │ │ │ │ +00021a60: 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2d2d oc)Symbol,...--- │ │ │ │ +00021a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00021ac0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00021ad0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00021ae0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00021af0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00021b00: 7932 2d31 2e32 352e 3131 2b64 732f 4d32 y2-1.25.11+ds/M2 │ │ │ │ +00021b10: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00021b20: 6765 732f 0a43 6861 7261 6374 6572 6973 ges/.Characteris │ │ │ │ +00021b30: 7469 6343 6c61 7373 6573 2e6d 323a 3234 ticClasses.m2:24 │ │ │ │ +00021b40: 3332 3a30 2e0a 1f0a 4669 6c65 3a20 4368 32:0....File: Ch │ │ │ │ +00021b50: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ +00021b60: 7365 732e 696e 666f 2c20 4e6f 6465 3a20 ses.info, Node: │ │ │ │ +00021b70: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ +00021b80: 6e67 2c20 4e65 7874 3a20 4f75 7470 7574 ng, Next: Output │ │ │ │ +00021b90: 2c20 5072 6576 3a20 4d65 7468 6f64 2c20 , Prev: Method, │ │ │ │ +00021ba0: 5570 3a20 546f 700a 0a4d 756c 7469 5072 Up: Top..MultiPr │ │ │ │ +00021bb0: 6f6a 436f 6f72 6452 696e 6720 2d2d 2041 ojCoordRing -- A │ │ │ │ +00021bc0: 2071 7569 636b 2077 6179 2074 6f20 6275 quick way to bu │ │ │ │ +00021bd0: 696c 6420 7468 6520 636f 6f72 6469 6e61 ild the coordina │ │ │ │ +00021be0: 7465 2072 696e 6720 6f66 2061 2070 726f te ring of a pro │ │ │ │ +00021bf0: 6475 6374 206f 6620 7072 6f6a 6563 7469 duct of projecti │ │ │ │ +00021c00: 7665 2073 7061 6365 730a 2a2a 2a2a 2a2a ve spaces.****** │ │ │ │ +00021c10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021c20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021c30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021c40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021c60: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +00021c70: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +00021c80: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ +00021c90: 6e67 2044 696d 730a 2020 2020 2020 2020 ng Dims. │ │ │ │ +00021ca0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ +00021cb0: 6e67 2028 436f 6566 6652 696e 672c 4469 ng (CoeffRing,Di │ │ │ │ +00021cc0: 6d73 290a 2020 2020 2020 2020 4d75 6c74 ms). Mult │ │ │ │ +00021cd0: 6950 726f 6a43 6f6f 7264 5269 6e67 2028 iProjCoordRing ( │ │ │ │ +00021ce0: 7661 722c 4469 6d73 290a 2020 2020 2020 var,Dims). │ │ │ │ +00021cf0: 2020 4d75 6c74 6950 726f 6a43 6f6f 7264 MultiProjCoord │ │ │ │ +00021d00: 5269 6e67 2028 436f 6566 6652 696e 672c Ring (CoeffRing, │ │ │ │ +00021d10: 7661 722c 4469 6d73 290a 2020 2a20 496e var,Dims). * In │ │ │ │ +00021d20: 7075 7473 3a0a 2020 2020 2020 2a20 4469 puts:. * Di │ │ │ │ +00021d30: 6d73 2c20 6120 2a6e 6f74 6520 6c69 7374 ms, a *note list │ │ │ │ +00021d40: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00021d50: 4c69 7374 2c2c 2072 6570 7265 7365 6e74 List,, represent │ │ │ │ +00021d60: 696e 6720 7468 6520 6469 6d65 6e73 696f ing the dimensio │ │ │ │ +00021d70: 6e73 206f 660a 2020 2020 2020 2020 7468 ns of. th │ │ │ │ +00021d80: 6520 7072 6f6a 6563 7469 7665 2073 7061 e projective spa │ │ │ │ +00021d90: 6365 732c 2069 2e65 2e20 7b6e 5f31 2c2e ces, i.e. {n_1,. │ │ │ │ +00021da0: 2e2e 2c6e 5f6d 7d20 636f 7272 6573 706f ..,n_m} correspo │ │ │ │ +00021db0: 6e64 7320 746f 205c 5050 5e7b 6e5f 317d nds to \PP^{n_1} │ │ │ │ +00021dc0: 0a20 2020 2020 2020 2078 2e2e 2e2e 2078 . x.... x │ │ │ │ +00021dd0: 205c 5050 5e7b 6e5f 6d7d 0a20 2020 2020 \PP^{n_m}. │ │ │ │ +00021de0: 202a 2043 6f65 6666 5269 6e67 2c20 6120 * CoeffRing, a │ │ │ │ +00021df0: 2a6e 6f74 6520 7269 6e67 3a20 284d 6163 *note ring: (Mac │ │ │ │ +00021e00: 6175 6c61 7932 446f 6329 5269 6e67 2c2c aulay2Doc)Ring,, │ │ │ │ +00021e10: 2074 6865 2063 6f65 6666 6963 6965 6e74 the coefficient │ │ │ │ +00021e20: 2072 696e 6720 6f66 0a20 2020 2020 2020 ring of. │ │ │ │ +00021e30: 2074 6865 2067 7261 6465 6420 706f 6c79 the graded poly │ │ │ │ +00021e40: 6e6f 6d69 616c 2072 696e 6720 746f 2062 nomial ring to b │ │ │ │ +00021e50: 6520 6275 696c 7420 6279 2074 6865 206d e built by the m │ │ │ │ +00021e60: 6574 686f 642c 2062 7920 6465 6661 756c ethod, by defaul │ │ │ │ +00021e70: 7420 7468 6973 0a20 2020 2020 2020 2069 t this. i │ │ │ │ +00021e80: 7320 5c5a 5a2f 3332 3734 390a 2020 2020 s \ZZ/32749. │ │ │ │ +00021e90: 2020 2a20 7661 722c 2061 202a 6e6f 7465 * var, a *note │ │ │ │ +00021ea0: 2073 796d 626f 6c3a 2028 4d61 6361 756c symbol: (Macaul │ │ │ │ +00021eb0: 6179 3244 6f63 2953 796d 626f 6c2c 2c20 ay2Doc)Symbol,, │ │ │ │ +00021ec0: 746f 2062 6520 7573 6564 2066 6f72 2074 to be used for t │ │ │ │ +00021ed0: 6865 0a20 2020 2020 2020 2069 6e74 6572 he. inter │ │ │ │ +00021ee0: 6d65 6469 6174 6573 206f 6620 7468 6520 mediates of the │ │ │ │ +00021ef0: 6772 6164 6564 2070 6f6c 796e 6f6d 6961 graded polynomia │ │ │ │ +00021f00: 6c20 7269 6e67 2074 6f20 6265 2062 7569 l ring to be bui │ │ │ │ +00021f10: 6c74 2062 7920 7468 6520 6d65 7468 6f64 lt by the method │ │ │ │ +00021f20: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00021f30: 2020 2020 2a20 6120 2a6e 6f74 6520 7269 * a *note ri │ │ │ │ +00021f40: 6e67 3a20 284d 6163 6175 6c61 7932 446f ng: (Macaulay2Do │ │ │ │ +00021f50: 6329 5269 6e67 2c2c 2074 6865 2067 7261 c)Ring,, the gra │ │ │ │ +00021f60: 6465 6420 636f 6f72 6469 6e61 7465 2072 ded coordinate r │ │ │ │ +00021f70: 696e 6720 6f66 2074 6865 0a20 2020 2020 ing of the. │ │ │ │ +00021f80: 2020 205c 5050 5e7b 6e5f 317d 2078 2e2e \PP^{n_1} x.. │ │ │ │ +00021f90: 2e2e 2078 205c 5050 5e7b 6e5f 6d7d 2077 .. x \PP^{n_m} w │ │ │ │ +00021fa0: 6865 7265 207b 6e5f 312c 2e2e 2e2c 6e5f here {n_1,...,n_ │ │ │ │ +00021fb0: 6d7d 2069 7320 7468 6520 696e 7075 7420 m} is the input │ │ │ │ +00021fc0: 6c69 7374 206f 660a 2020 2020 2020 2020 list of. │ │ │ │ +00021fd0: 6469 6d65 6e73 696f 6e73 0a0a 4465 7363 dimensions..Desc │ │ │ │ +00021fe0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00021ff0: 3d3d 3d0a 0a43 6f6d 7075 7465 7320 7468 ===..Computes th │ │ │ │ +00022000: 6520 6772 6164 6564 2063 6f6f 7264 696e e graded coordin │ │ │ │ +00022010: 6174 6520 7269 6e67 206f 6620 7468 6520 ate ring of the │ │ │ │ +00022020: 5c50 505e 7b6e 5f31 7d20 782e 2e2e 2e20 \PP^{n_1} x.... │ │ │ │ +00022030: 7820 5c50 505e 7b6e 5f6d 7d20 7768 6572 x \PP^{n_m} wher │ │ │ │ +00022040: 650a 7b6e 5f31 2c2e 2e2e 2c6e 5f6d 7d20 e.{n_1,...,n_m} │ │ │ │ +00022050: 6973 2074 6865 2069 6e70 7574 206c 6973 is the input lis │ │ │ │ +00022060: 7420 6f66 2064 696d 656e 7369 6f6e 732e t of dimensions. │ │ │ │ +00022070: 2054 6869 7320 6d65 7468 6f64 2069 7320 This method is │ │ │ │ +00022080: 7573 6564 2074 6f20 7175 6963 6b6c 790a used to quickly. │ │ │ │ +00022090: 6275 696c 6420 7468 6520 636f 6f72 6469 build the coordi │ │ │ │ +000220a0: 6e61 7465 2072 696e 6720 6f66 2061 2070 nate ring of a p │ │ │ │ +000220b0: 726f 6475 6374 206f 6620 7072 6f6a 6563 roduct of projec │ │ │ │ +000220c0: 7469 7665 2073 7061 6365 7320 666f 7220 tive spaces for │ │ │ │ +000220d0: 7573 6520 696e 0a63 6f6d 7075 7461 7469 use in.computati │ │ │ │ +000220e0: 6f6e 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ons...+--------- │ │ │ │ +000220f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022130: 2d2d 2d2d 2b0a 7c69 3120 3a20 533d 4d75 ----+.|i1 : S=Mu │ │ │ │ +00022140: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ +00022150: 2851 512c 7379 6d62 6f6c 207a 2c7b 312c (QQ,symbol z,{1, │ │ │ │ +00022160: 332c 337d 2920 2020 2020 2020 2020 2020 3,3}) │ │ │ │ +00022170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022180: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00022190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000221a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000221b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000221c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000221d0: 2020 2020 7c0a 7c6f 3120 3d20 5320 2020 |.|o1 = S │ │ │ │ +000221e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000221f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022230: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -00022240: 3d20 5320 2020 2020 2020 2020 2020 2020 = S │ │ │ │ +00022220: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022280: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022270: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79 |.|o1 : Poly │ │ │ │ +00022280: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ 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 2020 2020 │ │ │ │ -000222c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -000222e0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ -000222f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022320: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00022330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022370: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ -00022380: 3a20 6465 6772 6565 7320 5320 2020 2020 : degrees S 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----------+.|i3 │ │ │ │ -00022600: 3a20 523d 4d75 6c74 6950 726f 6a43 6f6f : R=MultiProjCoo │ │ │ │ -00022610: 7264 5269 6e67 207b 322c 337d 2020 2020 rdRing {2,3} │ │ │ │ +00022540: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00022550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022590: 2d2d 2d2d 2b0a 7c69 3320 3a20 523d 4d75 ----+.|i3 : R=Mu │ │ │ │ +000225a0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ +000225b0: 207b 322c 337d 2020 2020 2020 2020 2020 {2,3} │ │ │ │ +000225c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000225d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000225e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000225f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022600: 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020 |.| │ │ │ │ +00022630: 2020 2020 7c0a 7c6f 3320 3d20 5220 2020 |.|o3 = R │ │ │ │ +00022640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022690: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -000226a0: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +00022680: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -000226e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000226d0: 2020 2020 7c0a 7c6f 3320 3a20 506f 6c79 |.|o3 : Poly │ │ │ │ +000226e0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ 000226f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022730: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -00022740: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ -00022750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022780: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00022790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000227a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000227b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000227c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000227d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ -000227e0: 3a20 636f 6566 6669 6369 656e 7452 696e : coefficientRin │ │ │ │ -000227f0: 6720 5220 2020 2020 2020 2020 2020 2020 g R │ │ │ │ +00022720: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00022730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022770: 2d2d 2d2d 2b0a 7c69 3420 3a20 636f 6566 ----+.|i4 : coef │ │ │ │ +00022780: 6669 6369 656e 7452 696e 6720 5220 2020 ficientRing R │ │ │ │ +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 2020 │ │ │ │ +000227c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000227d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000227e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000227f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022820: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022810: 2020 2020 7c0a 7c20 2020 2020 2020 5a5a |.| ZZ │ │ │ │ +00022820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022870: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00022880: 2020 2020 5a5a 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00022860: 2020 2020 7c0a 7c6f 3420 3d20 2d2d 2d2d |.|o4 = ---- │ │ │ │ +00022870: 2d20 2020 2020 2020 2020 2020 2020 2020 - │ │ │ │ +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 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000228c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -000228d0: 3d20 2d2d 2d2d 2d20 2020 2020 2020 2020 = ----- │ │ │ │ +000228b0: 2020 2020 7c0a 7c20 2020 2020 3332 3734 |.| 3274 │ │ │ │ +000228c0: 3920 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ +000228d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000228e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000228f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022910: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00022920: 2020 3332 3734 3920 2020 2020 2020 2020 32749 │ │ │ │ +00022900: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00022910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022960: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022950: 2020 2020 7c0a 7c6f 3420 3a20 5175 6f74 |.|o4 : Quot │ │ │ │ +00022960: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ 00022970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -000229c0: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ -000229d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a00: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00022a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -00022a60: 3a20 6465 7363 7269 6265 2052 2020 2020 : describe R │ │ │ │ +000229a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000229b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000229c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000229d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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2020 2020 2020 │ │ │ │ 00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022af0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00022b00: 2020 2020 5a5a 2020 2020 2020 2020 2020 ZZ │ │ │ │ -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 7c0a 7c6f 3520 |.|o5 │ │ │ │ -00022b50: 3d20 2d2d 2d2d 2d5b 7820 2e2e 7820 2c20 = -----[x ..x , │ │ │ │ -00022b60: 4465 6772 6565 7320 3d3e 207b 333a 7b31 Degrees => {3:{1 │ │ │ │ -00022b70: 7d2c 2034 3a7b 307d 7d2c 2048 6566 7420 }, 4:{0}}, Heft │ │ │ │ -00022b80: 3d3e 207b 323a 317d 5d20 2020 2020 2020 => {2:1}] │ │ │ │ -00022b90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00022ba0: 2020 3332 3734 3920 2030 2020 2036 2020 32749 0 6 │ │ │ │ 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│ │ +00022b20: 317d 5d20 2020 2020 2020 2020 2020 2020 1}] │ │ │ │ +00022b30: 2020 2020 7c0a 7c20 2020 2020 3332 3734 |.| 3274 │ │ │ │ +00022b40: 3920 2030 2020 2036 2020 2020 2020 2020 9 0 6 │ │ │ │ +00022b50: 2020 2020 2020 2020 7b30 7d20 2020 207b {0} { │ │ │ │ +00022b60: 317d 2020 2020 2020 2020 2020 2020 2020 1} │ │ │ │ +00022b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022b80: 2020 2020 7c0a 2b2d 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 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022bd0: 2d2d 2d2d 2b0a 7c69 3620 3a20 413d 4368 ----+.|i6 : A=Ch │ │ │ │ +00022be0: 6f77 5269 6e67 2052 2020 2020 2020 2020 owRing R │ │ │ │ +00022bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022c20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00022c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022c70: 2020 2020 7c0a 7c6f 3620 3d20 4120 2020 |.|o6 = A │ │ │ │ +00022c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022cd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -00022ce0: 3d20 4120 2020 2020 2020 2020 2020 2020 = A │ │ │ │ +00022cc0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022d10: 2020 2020 7c0a 7c6f 3620 3a20 5175 6f74 |.|o6 : Quot │ │ │ │ +00022d20: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ 00022d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d70: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -00022d80: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ -00022d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022dc0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00022dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ -00022e20: 3a20 6465 7363 7269 6265 2041 2020 2020 : describe A │ │ │ │ +00022d60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00022d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022db0: 2d2d 2d2d 2b0a 7c69 3720 3a20 6465 7363 ----+.|i7 : desc │ │ │ │ +00022dc0: 7269 6265 2041 2020 2020 2020 2020 2020 ribe A │ │ │ │ +00022dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022e00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00022e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e60: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022e50: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b68 |.| ZZ[h │ │ │ │ +00022e60: 202e 2e68 205d 2020 2020 2020 2020 2020 ..h ] │ │ │ │ 00022e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022eb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00022ec0: 2020 5a5a 5b68 202e 2e68 205d 2020 2020 ZZ[h ..h ] │ │ │ │ +00022ea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00022eb0: 3120 2020 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00022f10: 2020 2020 2020 3120 2020 3220 2020 2020 1 2 │ │ │ │ +00022ef0: 2020 2020 7c0a 7c6f 3720 3d20 2d2d 2d2d |.|o7 = ---- │ │ │ │ +00022f00: 2d2d 2d2d 2d2d 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f50: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -00022f60: 3d20 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020 = ---------- │ │ │ │ +00022f40: 2020 2020 7c0a 7c20 2020 2020 2020 2033 |.| 3 │ │ │ │ +00022f50: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022fa0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00022fb0: 2020 2020 2033 2020 2034 2020 2020 2020 3 4 │ │ │ │ +00022f90: 2020 2020 7c0a 7c20 2020 2020 2028 6820 |.| (h │ │ │ │ +00022fa0: 2c20 6820 2920 2020 2020 2020 2020 2020 , h ) │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ff0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00023000: 2020 2028 6820 2c20 6820 2920 2020 2020 (h , h ) │ │ │ │ +00022fe0: 2020 2020 7c0a 7c20 2020 2020 2020 2031 |.| 1 │ │ │ │ +00022ff0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023040: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00023050: 2020 2020 2031 2020 2032 2020 2020 2020 1 2 │ │ │ │ -00023060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023090: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -000230a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000230b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000230c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000230d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000230e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 ----------+.|i8 │ │ │ │ -000230f0: 3a20 5365 6772 6528 412c 6964 6561 6c20 : Segre(A,ideal │ │ │ │ -00023100: 7261 6e64 6f6d 287b 312c 317d 2c52 2929 random({1,1},R)) │ │ │ │ +00023030: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00023040: 2d2d 2d2d 2d2d 2d2d 2d2d 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 2b0a 7c69 3820 3a20 5365 6772 ----+.|i8 : Segr │ │ │ │ +00023090: 6528 412c 6964 6561 6c20 7261 6e64 6f6d e(A,ideal random │ │ │ │ +000230a0: 287b 312c 317d 2c52 2929 2020 2020 2020 ({1,1},R)) │ │ │ │ +000230b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000230c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000230d0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 │ │ │ │ 00023110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023130: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00023140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023180: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00023190: 2020 2020 2032 2033 2020 2020 2032 2032 2 3 2 2 │ │ │ │ -000231a0: 2020 2020 2020 2033 2020 2020 2032 2020 3 2 │ │ │ │ -000231b0: 2020 2020 2020 2032 2020 2020 3320 2020 2 3 │ │ │ │ -000231c0: 2032 2020 2020 2020 2020 2020 2020 3220 2 2 │ │ │ │ -000231d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820 |.|o8 │ │ │ │ -000231e0: 3d20 3130 6820 6820 202d 2036 6820 6820 = 10h h - 6h h │ │ │ │ -000231f0: 202d 2034 6820 6820 202b 2033 6820 6820 - 4h h + 3h h │ │ │ │ -00023200: 202b 2033 6820 6820 202b 2068 2020 2d20 + 3h h + h - │ │ │ │ -00023210: 6820 202d 2032 6820 6820 202d 2068 2020 h - 2h h - h │ │ │ │ -00023220: 2b20 6820 202b 2068 2020 7c0a 7c20 2020 + h + h |.| │ │ │ │ -00023230: 2020 2020 2031 2032 2020 2020 2031 2032 1 2 1 2 │ │ │ │ -00023240: 2020 2020 2031 2032 2020 2020 2031 2032 1 2 1 2 │ │ │ │ -00023250: 2020 2020 2031 2032 2020 2020 3220 2020 1 2 2 │ │ │ │ -00023260: 2031 2020 2020 2031 2032 2020 2020 3220 1 1 2 2 │ │ │ │ -00023270: 2020 2031 2020 2020 3220 7c0a 7c20 2020 1 2 |.| │ │ │ │ +00023120: 2020 2020 7c0a 7c20 2020 2020 2020 2032 |.| 2 │ │ │ │ +00023130: 2033 2020 2020 2032 2032 2020 2020 2020 3 2 2 │ │ │ │ +00023140: 2033 2020 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ +00023150: 2032 2020 2020 3320 2020 2032 2020 2020 2 3 2 │ │ │ │ +00023160: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00023170: 2020 2020 7c0a 7c6f 3820 3d20 3130 6820 |.|o8 = 10h │ │ │ │ +00023180: 6820 202d 2036 6820 6820 202d 2034 6820 h - 6h h - 4h │ │ │ │ +00023190: 6820 202b 2033 6820 6820 202b 2033 6820 h + 3h h + 3h │ │ │ │ +000231a0: 6820 202b 2068 2020 2d20 6820 202d 2032 h + h - h - 2 │ │ │ │ +000231b0: 6820 6820 202d 2068 2020 2b20 6820 202b h h - h + h + │ │ │ │ +000231c0: 2068 2020 7c0a 7c20 2020 2020 2020 2031 h |.| 1 │ │ │ │ +000231d0: 2032 2020 2020 2031 2032 2020 2020 2031 2 1 2 1 │ │ │ │ +000231e0: 2032 2020 2020 2031 2032 2020 2020 2031 2 1 2 1 │ │ │ │ +000231f0: 2032 2020 2020 3220 2020 2031 2020 2020 2 2 1 │ │ │ │ +00023200: 2031 2032 2020 2020 3220 2020 2031 2020 1 2 2 1 │ │ │ │ +00023210: 2020 3220 7c0a 7c20 2020 2020 2020 2020 2 |.| │ │ │ │ +00023220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023260: 2020 2020 7c0a 7c6f 3820 3a20 4120 2020 |.|o8 : A │ │ │ │ +00023270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820 |.|o8 │ │ │ │ -000232d0: 3a20 4120 2020 2020 2020 2020 2020 2020 : A │ │ │ │ -000232e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023310: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00023320: 2d2d 2d2d 2d2d 2d2d 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6a43 . * "MultiProjC │ │ │ │ -00023400: 6f6f 7264 5269 6e67 2852 696e 672c 5379 oordRing(Ring,Sy │ │ │ │ -00023410: 6d62 6f6c 2c4c 6973 7429 220a 2020 2a20 mbol,List)". * │ │ │ │ -00023420: 224d 756c 7469 5072 6f6a 436f 6f72 6452 "MultiProjCoordR │ │ │ │ -00023430: 696e 6728 5379 6d62 6f6c 2c4c 6973 7429 ing(Symbol,List) │ │ │ │ -00023440: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ -00023450: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -00023460: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -00023470: 6a65 6374 202a 6e6f 7465 204d 756c 7469 ject *note Multi │ │ │ │ -00023480: 5072 6f6a 436f 6f72 6452 696e 673a 204d ProjCoordRing: M │ │ │ │ -00023490: 756c 7469 5072 6f6a 436f 6f72 6452 696e ultiProjCoordRin │ │ │ │ -000234a0: 672c 2069 7320 6120 2a6e 6f74 6520 6d65 g, is a *note me │ │ │ │ -000234b0: 7468 6f64 0a66 756e 6374 696f 6e3a 2028 thod.function: ( │ │ │ │ -000234c0: 4d61 6361 756c 6179 3244 6f63 294d 6574 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kages/.Character │ │ │ │ -000235a0: 6973 7469 6343 6c61 7373 6573 2e6d 323a isticClasses.m2: │ │ │ │ -000235b0: 3230 3530 3a30 2e0a 1f0a 4669 6c65 3a20 2050:0....File: │ │ │ │ -000235c0: 4368 6172 6163 7465 7269 7374 6963 436c CharacteristicCl │ │ │ │ -000235d0: 6173 7365 732e 696e 666f 2c20 4e6f 6465 asses.info, Node │ │ │ │ -000235e0: 3a20 4f75 7470 7574 2c20 4e65 7874 3a20 : Output, Next: │ │ │ │ -000235f0: 7072 6f62 6162 696c 6973 7469 6320 616c probabilistic al │ │ │ │ -00023600: 676f 7269 7468 6d2c 2050 7265 763a 204d gorithm, Prev: M │ │ │ │ -00023610: 756c 7469 5072 6f6a 436f 6f72 6452 696e ultiProjCoordRin │ │ │ │ -00023620: 672c 2055 703a 2054 6f70 0a0a 4f75 7470 g, Up: Top..Outp │ │ │ │ -00023630: 7574 0a2a 2a2a 2a2a 2a0a 0a44 6573 6372 ut.******..Descr │ │ │ │ -00023640: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00023650: 3d3d 0a0a 5468 6520 6f70 7469 6f6e 204f ==..The option O │ │ │ │ -00023660: 7574 7075 7420 6973 206f 6e6c 7920 7573 utput is only us │ │ │ │ -00023670: 6564 2062 7920 7468 6520 636f 6d6d 616e ed by the comman │ │ │ │ -00023680: 6473 202a 6e6f 7465 2043 534d 3a20 4353 ds *note CSM: CS │ │ │ │ -00023690: 4d2c 2c20 2a6e 6f74 6520 5365 6772 653a M,, *note Segre: │ │ │ │ -000236a0: 0a53 6567 7265 2c2c 202a 6e6f 7465 2043 .Segre,, *note C │ │ │ │ -000236b0: 6865 726e 3a20 4368 6572 6e2c 2061 6e64 hern: Chern, and │ │ │ │ -000236c0: 202a 6e6f 7465 2045 756c 6572 3a20 4575 *note Euler: Eu │ │ │ │ -000236d0: 6c65 722c 2074 6f20 7370 6563 6966 7920 ler, to specify │ │ │ │ -000236e0: 7468 6520 7479 7065 206f 660a 6f75 7470 the type of.outp │ │ │ │ -000236f0: 7574 2074 6f20 6265 2072 6574 7572 6e65 ut to be returne │ │ │ │ -00023700: 6420 746f 2074 6865 2075 7365 642e 2054 d to the used. T │ │ │ │ -00023710: 6869 7320 6f70 7469 6f6e 2077 696c 6c20 his option will │ │ │ │ -00023720: 6265 2069 676e 6f72 6564 2077 6865 6e20 be ignored when │ │ │ │ -00023730: 7573 6564 2077 6974 680a 2a6e 6f74 6520 used with.*note │ │ │ │ -00023740: 436f 6d70 4d65 7468 6f64 3a20 436f 6d70 CompMethod: Comp │ │ │ │ -00023750: 4d65 7468 6f64 2c20 506e 5265 7369 6475 Method, PnResidu │ │ │ │ -00023760: 616c 206f 7220 6265 7274 696e 692e 2054 al or bertini. T │ │ │ │ -00023770: 6865 206f 7074 696f 6e20 7769 6c6c 2061 he option will a │ │ │ │ -00023780: 6c73 6f20 6265 0a69 676e 6f72 6520 7768 lso be.ignore wh │ │ │ │ -00023790: 656e 202a 6e6f 7465 204d 6574 686f 643a en *note Method: │ │ │ │ -000237a0: 204d 6574 686f 642c 3d3e 4469 7265 6374 Method,=>Direct │ │ │ │ -000237b0: 436f 6d70 6c65 7465 496e 7420 6973 2075 CompleteInt is u │ │ │ │ -000237c0: 7365 642e 2054 6865 2064 6566 6175 6c74 sed. The default │ │ │ │ -000237d0: 0a6f 7574 7075 7420 666f 7220 616c 6c20 .output for all │ │ │ │ -000237e0: 7468 6573 6520 6d65 7468 6f64 7320 6973 these methods is │ │ │ │ -000237f0: 2043 686f 7752 696e 6745 6c65 6c6d 656e ChowRingElelmen │ │ │ │ -00023800: 7420 7768 6963 6820 7769 6c6c 2072 6574 t which will ret │ │ │ │ -00023810: 7572 6e20 616e 2065 6c65 6d65 6e74 0a6f urn an element.o │ │ │ │ -00023820: 6620 7468 6520 6170 7072 6f70 7269 6174 f the appropriat │ │ │ │ -00023830: 6520 4368 6f77 2072 696e 672e 2041 6c6c e Chow ring. All │ │ │ │ -00023840: 206d 6574 686f 6473 2061 6c73 6f20 6861 methods also ha │ │ │ │ -00023850: 7665 2061 6e20 6f70 7469 6f6e 2048 6173 ve an option Has │ │ │ │ -00023860: 6846 6f72 6d20 7768 6963 680a 7265 7475 hForm which.retu │ │ │ │ -00023870: 726e 7320 6164 6469 7469 6f6e 616c 2069 rns additional i │ │ │ │ -00023880: 6e66 6f72 6d61 7469 6f6e 2063 6f6d 7075 nformation compu │ │ │ │ -00023890: 7465 6420 6279 2074 6865 206d 6574 686f ted by the metho │ │ │ │ -000238a0: 6473 2064 7572 696e 6720 7468 6569 7220 ds during their │ │ │ │ -000238b0: 7374 616e 6461 7264 0a6f 7065 7261 7469 standard.operati │ │ │ │ -000238c0: 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d on...+---------- │ │ │ │ -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 2d2d 2d2d ---------------- │ │ │ │ -00023900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023910: 2d2d 2d2b 0a7c 6931 203a 2052 203d 205a ---+.|i1 : R = Z │ │ │ │ -00023920: 5a2f 3332 3734 395b 785f 302e 2e78 5f36 Z/32749[x_0..x_6 │ │ │ │ -00023930: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -00023940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023960: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000232b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000232c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000232d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000232e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000232f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023300: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075 ----+..Ways to u │ │ │ │ +00023310: 7365 204d 756c 7469 5072 6f6a 436f 6f72 se MultiProjCoor │ │ │ │ +00023320: 6452 696e 673a 0a3d 3d3d 3d3d 3d3d 3d3d dRing:.========= │ │ │ │ +00023330: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00023340: 3d3d 3d3d 3d3d 0a0a 2020 2a20 224d 756c ======.. * "Mul │ │ │ │ +00023350: 7469 5072 6f6a 436f 6f72 6452 696e 6728 tiProjCoordRing( │ │ │ │ +00023360: 4c69 7374 2922 0a20 202a 2022 4d75 6c74 List)". * "Mult │ │ │ │ +00023370: 6950 726f 6a43 6f6f 7264 5269 6e67 2852 iProjCoordRing(R │ │ │ │ +00023380: 696e 672c 4c69 7374 2922 0a20 202a 2022 ing,List)". * " │ │ │ │ +00023390: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ +000233a0: 6e67 2852 696e 672c 5379 6d62 6f6c 2c4c ng(Ring,Symbol,L │ │ │ │ +000233b0: 6973 7429 220a 2020 2a20 224d 756c 7469 ist)". * "Multi │ │ │ │ +000233c0: 5072 6f6a 436f 6f72 6452 696e 6728 5379 ProjCoordRing(Sy │ │ │ │ +000233d0: 6d62 6f6c 2c4c 6973 7429 220a 0a46 6f72 mbol,List)"..For │ │ │ │ +000233e0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +000233f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00023400: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00023410: 6e6f 7465 204d 756c 7469 5072 6f6a 436f note MultiProjCo │ │ │ │ +00023420: 6f72 6452 696e 673a 204d 756c 7469 5072 ordRing: MultiPr │ │ │ │ +00023430: 6f6a 436f 6f72 6452 696e 672c 2069 7320 ojCoordRing, is │ │ │ │ +00023440: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66 a *note method.f │ │ │ │ +00023450: 756e 6374 696f 6e3a 2028 4d61 6361 756c unction: (Macaul │ │ │ │ +00023460: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +00023470: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +00023480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000234a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000234b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000234c0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +000234d0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +000234e0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +000234f0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00023500: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00023510: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163 .25.11+ds/M2/Mac │ │ │ │ +00023520: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00023530: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ +00023540: 6c61 7373 6573 2e6d 323a 3230 3530 3a30 lasses.m2:2050:0 │ │ │ │ +00023550: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ +00023560: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +00023570: 696e 666f 2c20 4e6f 6465 3a20 4f75 7470 info, Node: Outp │ │ │ │ +00023580: 7574 2c20 4e65 7874 3a20 7072 6f62 6162 ut, Next: probab │ │ │ │ +00023590: 696c 6973 7469 6320 616c 676f 7269 7468 ilistic algorith │ │ │ │ +000235a0: 6d2c 2050 7265 763a 204d 756c 7469 5072 m, Prev: MultiPr │ │ │ │ +000235b0: 6f6a 436f 6f72 6452 696e 672c 2055 703a ojCoordRing, Up: │ │ │ │ +000235c0: 2054 6f70 0a0a 4f75 7470 7574 0a2a 2a2a Top..Output.*** │ │ │ │ +000235d0: 2a2a 2a0a 0a44 6573 6372 6970 7469 6f6e ***..Description │ │ │ │ +000235e0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ +000235f0: 6520 6f70 7469 6f6e 204f 7574 7075 7420 e option Output │ │ │ │ +00023600: 6973 206f 6e6c 7920 7573 6564 2062 7920 is only used by │ │ │ │ +00023610: 7468 6520 636f 6d6d 616e 6473 202a 6e6f the commands *no │ │ │ │ +00023620: 7465 2043 534d 3a20 4353 4d2c 2c20 2a6e te CSM: CSM,, *n │ │ │ │ +00023630: 6f74 6520 5365 6772 653a 0a53 6567 7265 ote Segre:.Segre │ │ │ │ +00023640: 2c2c 202a 6e6f 7465 2043 6865 726e 3a20 ,, *note Chern: │ │ │ │ +00023650: 4368 6572 6e2c 2061 6e64 202a 6e6f 7465 Chern, and *note │ │ │ │ +00023660: 2045 756c 6572 3a20 4575 6c65 722c 2074 Euler: Euler, t │ │ │ │ +00023670: 6f20 7370 6563 6966 7920 7468 6520 7479 o specify the ty │ │ │ │ +00023680: 7065 206f 660a 6f75 7470 7574 2074 6f20 pe of.output to │ │ │ │ +00023690: 6265 2072 6574 7572 6e65 6420 746f 2074 be returned to t │ │ │ │ +000236a0: 6865 2075 7365 642e 2054 6869 7320 6f70 he used. This op │ │ │ │ +000236b0: 7469 6f6e 2077 696c 6c20 6265 2069 676e tion will be ign │ │ │ │ +000236c0: 6f72 6564 2077 6865 6e20 7573 6564 2077 ored when used w │ │ │ │ +000236d0: 6974 680a 2a6e 6f74 6520 436f 6d70 4d65 ith.*note CompMe │ │ │ │ +000236e0: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64 thod: CompMethod │ │ │ │ +000236f0: 2c20 506e 5265 7369 6475 616c 206f 7220 , PnResidual or │ │ │ │ +00023700: 6265 7274 696e 692e 2054 6865 206f 7074 bertini. The opt │ │ │ │ +00023710: 696f 6e20 7769 6c6c 2061 6c73 6f20 6265 ion will also be │ │ │ │ +00023720: 0a69 676e 6f72 6520 7768 656e 202a 6e6f .ignore when *no │ │ │ │ +00023730: 7465 204d 6574 686f 643a 204d 6574 686f te Method: Metho │ │ │ │ +00023740: 642c 3d3e 4469 7265 6374 436f 6d70 6c65 d,=>DirectComple │ │ │ │ +00023750: 7465 496e 7420 6973 2075 7365 642e 2054 teInt is used. T │ │ │ │ +00023760: 6865 2064 6566 6175 6c74 0a6f 7574 7075 he default.outpu │ │ │ │ +00023770: 7420 666f 7220 616c 6c20 7468 6573 6520 t for all these │ │ │ │ +00023780: 6d65 7468 6f64 7320 6973 2043 686f 7752 methods is ChowR │ │ │ │ +00023790: 696e 6745 6c65 6c6d 656e 7420 7768 6963 ingElelment whic │ │ │ │ +000237a0: 6820 7769 6c6c 2072 6574 7572 6e20 616e h will return an │ │ │ │ +000237b0: 2065 6c65 6d65 6e74 0a6f 6620 7468 6520 element.of the │ │ │ │ +000237c0: 6170 7072 6f70 7269 6174 6520 4368 6f77 appropriate Chow │ │ │ │ +000237d0: 2072 696e 672e 2041 6c6c 206d 6574 686f ring. All metho │ │ │ │ +000237e0: 6473 2061 6c73 6f20 6861 7665 2061 6e20 ds also have an │ │ │ │ +000237f0: 6f70 7469 6f6e 2048 6173 6846 6f72 6d20 option HashForm │ │ │ │ +00023800: 7768 6963 680a 7265 7475 726e 7320 6164 which.returns ad │ │ │ │ +00023810: 6469 7469 6f6e 616c 2069 6e66 6f72 6d61 ditional informa │ │ │ │ +00023820: 7469 6f6e 2063 6f6d 7075 7465 6420 6279 tion computed by │ │ │ │ +00023830: 2074 6865 206d 6574 686f 6473 2064 7572 the methods dur │ │ │ │ +00023840: 696e 6720 7468 6569 7220 7374 616e 6461 ing their standa │ │ │ │ +00023850: 7264 0a6f 7065 7261 7469 6f6e 2e0a 0a2b rd.operation...+ │ │ │ │ +00023860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000238a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000238b0: 6931 203a 2052 203d 205a 5a2f 3332 3734 i1 : R = ZZ/3274 │ │ │ │ +000238c0: 395b 785f 302e 2e78 5f36 5d20 2020 2020 9[x_0..x_6] │ │ │ │ +000238d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000238e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000238f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023940: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023950: 6f31 203d 2052 2020 2020 2020 2020 2020 o1 = R │ │ │ │ +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 2020 2020 │ │ │ │ +00023990: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000239a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000239b0: 2020 207c 0a7c 6f31 203d 2052 2020 2020 |.|o1 = R │ │ │ │ +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 2020 2020 │ │ │ │ -000239f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000239e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000239f0: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ +00023a00: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ 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 207c 0a7c 6f31 203a 2050 6f6c 796e |.|o1 : Polyn │ │ │ │ -00023a60: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ -00023a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023aa0: 2020 207c 0a2b 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 2d2d 2d2d ---------------- │ │ │ │ -00023ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023af0: 2d2d 2d2b 0a7c 6932 203a 2041 3d43 686f ---+.|i2 : A=Cho │ │ │ │ -00023b00: 7752 696e 6728 5229 2020 2020 2020 2020 wRing(R) │ │ │ │ +00023a30: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00023a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00023a90: 6932 203a 2041 3d43 686f 7752 696e 6728 i2 : A=ChowRing( │ │ │ │ +00023aa0: 5229 2020 2020 2020 2020 2020 2020 2020 R) │ │ │ │ +00023ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ad0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00023b20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023b30: 6f32 203d 2041 2020 2020 2020 2020 2020 o2 = A │ │ │ │ +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 2020 2020 │ │ │ │ +00023b70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00023b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b90: 2020 207c 0a7c 6f32 203d 2041 2020 2020 |.|o2 = A │ │ │ │ +00023b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023be0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00023bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023bd0: 6f32 203a 2051 756f 7469 656e 7452 696e o2 : QuotientRin │ │ │ │ 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2033 2020 2020 4 3 │ │ │ │ -000241c0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000241d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -000241e0: 2020 2020 2020 2020 2020 2020 4353 4d20 CSM │ │ │ │ -000241f0: 3d3e 2031 3068 2020 2b20 3132 6820 202b => 10h + 12h + │ │ │ │ -00024200: 2032 3268 2020 2b20 3136 6820 202b 2036 22h + 16h + 6 │ │ │ │ -00024210: 6820 2020 2020 2020 2020 2020 2020 2020 h │ │ │ │ -00024220: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00024230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024240: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ -00024250: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00024260: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00024270: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00024280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024290: 2020 2020 2036 2020 2020 2020 3520 2020 6 5 │ │ │ │ -000242a0: 2020 2034 2020 2020 2020 3320 2020 2020 4 3 │ │ │ │ -000242b0: 2032 2020 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2d2d ---------------- │ │ │ │ -000244a0: 2d2d 2d2b 0a7c 6936 203a 2043 534d 2841 ---+.|i6 : CSM(A │ │ │ │ -000244b0: 2c69 6465 616c 2049 5f30 293d 3d63 736d ,ideal I_0)==csm │ │ │ │ -000244c0: 237b 307d 2020 2020 2020 2020 2020 2020 #{0} │ │ │ │ -000244d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000244e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000244f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024050: 2020 3620 2020 2020 2035 2020 2020 2020 6 5 │ │ │ │ +00024060: 3420 2020 2020 2033 2020 2020 2020 3220 4 3 2 │ │ │ │ +00024070: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00024080: 6f35 203d 204d 7574 6162 6c65 4861 7368 o5 = MutableHash │ │ │ │ +00024090: 5461 626c 657b 7b30 2c20 317d 203d 3e20 Table{{0, 1} => │ │ │ │ +000240a0: 3268 2020 2b20 3233 6820 202b 2033 3268 2h + 23h + 32h │ │ │ │ +000240b0: 2020 2b20 3333 6820 202b 2031 3868 2020 + 33h + 18h │ │ │ │ +000240c0: 2b20 3568 207d 2020 2020 2020 207c 0a7c + 5h } |.| │ │ │ │ +000240d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000240e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000240f0: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ +00024100: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ +00024110: 2020 2020 3120 2020 2020 2020 207c 0a7c 1 |.| │ │ │ │ +00024120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024140: 3620 2020 2020 2035 2020 2020 2020 3420 6 5 4 │ │ │ │ +00024150: 2020 2020 2033 2020 2020 2032 2020 2020 3 2 │ │ │ │ +00024160: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00024170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024180: 2020 2020 2020 4353 4d20 3d3e 2031 3068 CSM => 10h │ │ │ │ +00024190: 2020 2b20 3132 6820 202b 2032 3268 2020 + 12h + 22h │ │ │ │ +000241a0: 2b20 3136 6820 202b 2036 6820 2020 2020 + 16h + 6h │ │ │ │ +000241b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 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+000256e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000256f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00025700: 6931 3220 3a20 6575 3d45 756c 6572 2820 i12 : eu=Euler( │ │ │ │ +00025710: 492c 204f 7574 7075 743d 3e48 6173 6846 I, Output=>HashF │ │ │ │ +00025720: 6f72 6d29 2020 2020 2020 2020 2020 2020 orm) │ │ │ │ +00025730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025740: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00025750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025790: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000257a0: 6f31 3220 3d20 4d75 7461 626c 6548 6173 o12 = MutableHas │ │ │ │ +000257b0: 6854 6162 6c65 7b2e 2e2e 352e 2e2e 7d20 hTable{...5...} │ │ │ │ 000257c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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hTable │ │ │ │ 00025860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258a0: 2020 207c 0a7c 6f31 3220 3a20 4d75 7461 |.|o12 : Muta │ │ │ │ -000258b0: 626c 6548 6173 6854 6162 6c65 2020 2020 bleHashTable │ │ │ │ -000258c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00025900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025940: 2d2d 2d2b 0a7c 6931 3320 3a20 7065 656b ---+.|i13 : peek │ │ │ │ -00025950: 2065 7520 2020 2020 2020 2020 2020 2020 eu │ │ │ │ +00025880: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00025890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000258a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000258b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000258c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000258d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000258e0: 6931 3320 3a20 7065 656b 2065 7520 2020 i13 : peek eu │ │ │ │ +000258f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025920: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00025930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025990: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -000259a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025970: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00025980: 6f31 3320 3d20 4d75 7461 626c 6548 6173 o13 = MutableHas │ │ │ │ +00025990: 6854 6162 6c65 7b45 756c 6572 203d 3e20 hTable{Euler => │ │ │ │ +000259a0: 3130 2020 2020 2020 2020 2020 2020 2020 10 │ │ │ │ 000259b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000259c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000259c0: 2020 2020 2020 7d20 2020 2020 207c 0a7c } |.| │ │ │ │ 000259d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000259e0: 2020 207c 0a7c 6f31 3320 3d20 4d75 7461 |.|o13 = Muta │ │ │ │ -000259f0: 626c 6548 6173 6854 6162 6c65 7b45 756c bleHashTable{Eul │ │ │ │ -00025a00: 6572 203d 3e20 3130 2020 2020 2020 2020 er => 10 │ │ │ │ -00025a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025a20: 2020 2020 2020 2020 2020 2020 7d20 2020 } │ │ │ │ -00025a30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025a50: 2020 2020 2020 2020 2036 2020 2020 2020 6 │ │ │ │ -00025a60: 3520 2020 2020 2034 2020 2020 2020 3320 5 4 3 │ │ │ │ -00025a70: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00025a80: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025a90: 2020 2020 2020 2020 2020 2020 207b 302c {0, │ │ │ │ -00025aa0: 2031 7d20 3d3e 2032 6820 202b 2032 3368 1} => 2h + 23h │ │ │ │ -00025ab0: 2020 2b20 3332 6820 202b 2033 3368 2020 + 32h + 33h │ │ │ │ -00025ac0: 2b20 3138 6820 202b 2035 6820 2020 2020 + 18h + 5h │ │ │ │ -00025ad0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025af0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ -00025b00: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ -00025b10: 2020 2020 2031 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00025b20: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025b40: 2020 2020 2020 2036 2020 2020 2020 3520 6 5 │ │ │ │ -00025b50: 2020 2020 2034 2020 2020 2020 3320 2020 4 3 │ │ │ │ -00025b60: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00025b70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025b80: 2020 2020 2020 2020 2020 2020 2043 534d CSM │ │ │ │ -00025b90: 203d 3e20 3130 6820 202b 2031 3268 2020 => 10h + 12h │ │ │ │ -00025ba0: 2b20 3232 6820 202b 2031 3668 2020 2b20 + 22h + 16h + │ │ │ │ -00025bb0: 3668 2020 2020 2020 2020 2020 2020 2020 6h │ │ │ │ -00025bc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025be0: 2020 2020 2020 2031 2020 2020 2020 3120 1 1 │ │ │ │ -00025bf0: 2020 2020 2031 2020 2020 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3120 2020 2020 3120 2020 2020 2020 1 1 │ │ │ │ -00025d00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d20: 2020 2020 2020 3620 2020 2020 2035 2020 6 5 │ │ │ │ -00025d30: 2020 2020 3420 2020 2020 2033 2020 2020 4 3 │ │ │ │ -00025d40: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00025d50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025d60: 2020 2020 2020 2020 2020 2020 207b 317d {1} │ │ │ │ -00025d70: 203d 3e20 3668 2020 2b20 3137 6820 202b => 6h + 17h + │ │ │ │ -00025d80: 2032 3868 2020 2b20 3237 6820 202b 2031 28h + 27h + 1 │ │ │ │ -00025d90: 3468 2020 2b20 3368 2020 2020 2020 2020 4h + 3h │ │ │ │ -00025da0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025dc0: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ -00025dd0: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00025de0: 2020 3120 2020 2020 3120 2020 2020 2020 1 1 │ │ │ │ -00025df0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00025e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e40: 2d2d 2d2b 0a0a 5468 6520 4d75 7461 626c ---+..The Mutabl │ │ │ │ -00025e50: 6548 6173 6854 6162 6c65 2072 6574 7572 eHashTable retur │ │ │ │ -00025e60: 6e65 6420 7769 7468 2074 6865 206f 7074 ned with the opt │ │ │ │ -00025e70: 696f 6e20 4f75 7470 7574 3d3e 4861 7368 ion Output=>Hash │ │ │ │ -00025e80: 466f 726d 2063 6f6e 7461 696e 730a 6469 Form contains.di │ │ │ │ -00025e90: 6666 6572 656e 7420 696e 666f 726d 6174 fferent informat │ │ │ │ -00025ea0: 696f 6e20 6465 7065 6e64 696e 6720 6f6e ion depending on │ │ │ │ -00025eb0: 2074 6865 206d 6574 686f 6420 7769 7468 the method with │ │ │ │ -00025ec0: 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534d 3a20 4353 4d2c 2020 note CSM: CSM, │ │ │ │ -00025fa0: 636f 6d6d 616e 6420 696e 2074 6865 2064 command in the d │ │ │ │ -00025fb0: 6566 6175 6c74 0a63 6f6e 6669 6775 7261 efault.configura │ │ │ │ -00025fc0: 7469 6f6e 7320 2874 6861 7420 6973 202a tions (that is * │ │ │ │ -00025fd0: 6e6f 7465 204d 6574 686f 643a 204d 6574 note Method: Met │ │ │ │ -00025fe0: 686f 642c 3d3e 496e 636c 7573 696f 6e45 hod,=>InclusionE │ │ │ │ -00025ff0: 7863 6c75 7369 6f6e 2c20 2a6e 6f74 650a xclusion, *note. │ │ │ │ -00026000: 436f 6d70 4d65 7468 6f64 3a20 436f 6d70 CompMethod: Comp │ │ │ │ -00026010: 4d65 7468 6f64 2c3d 3e50 726f 6a65 6374 Method,=>Project │ │ │ │ -00026020: 6976 6544 6567 7265 6529 2074 6865 7265 iveDegree) there │ │ │ │ -00026030: 2069 7320 7468 6520 6164 6469 7469 6f6e is the addition │ │ │ │ -00026040: 616c 206f 7074 696f 6e20 746f 0a73 6574 al option to.set │ │ │ │ -00026050: 204f 7574 7075 743d 3e48 6173 6846 6f72 Output=>HashFor │ │ │ │ -00026060: 6d58 4c2e 2054 6869 7320 7265 7475 726e mXL. This return │ │ │ │ -00026070: 7320 616c 6c20 7468 6520 7573 7561 6c20 s all the usual │ │ │ │ -00026080: 696e 666f 726d 6174 696f 6e20 7468 6174 information that │ │ │ │ -00026090: 0a4f 7574 7075 743d 3e48 6173 6846 6f72 .Output=>HashFor │ │ │ │ -000260a0: 6d20 776f 756c 6420 666f 7220 7468 6973 m would for this │ │ │ │ -000260b0: 2063 6f6e 6669 6775 7261 7469 6f6e 2077 configuration w │ │ │ │ -000260c0: 6974 6820 7468 6520 6164 6469 7469 6f6e ith the addition │ │ │ │ -000260d0: 206f 6620 7468 650a 7072 6f6a 6563 7469 of the.projecti │ │ │ │ -000260e0: 7665 2064 6567 7265 6573 2061 6e64 2053 ve degrees and S │ │ │ │ -000260f0: 6567 7265 2063 6c61 7373 6573 206f 6620 egre classes of │ │ │ │ -00026100: 7369 6e67 756c 6172 6974 7920 7375 6273 singularity subs │ │ │ │ -00026110: 6368 656d 6573 2067 656e 6572 6174 6564 chemes generated │ │ │ │ -00026120: 2062 7920 7468 650a 6879 7065 7273 7572 by the.hypersur │ │ │ │ -00026130: 6661 6365 7320 636f 6e73 6964 6572 6564 faces considered │ │ │ │ -00026140: 2069 6e20 7468 6520 696e 636c 7573 696f in the inclusio │ │ │ │ -00026150: 6e2f 6578 636c 7573 696f 6e20 7072 6f63 n/exclusion proc │ │ │ │ -00026160: 6564 7572 652c 2074 6861 7420 6973 2069 edure, that is i │ │ │ │ -00026170: 6e0a 6669 6e64 696e 6720 7468 6520 4353 n.finding the CS │ │ │ │ -00026180: 4d20 636c 6173 7320 6f66 2061 6c6c 2068 M class of all h │ │ │ │ -00026190: 7970 6572 7375 7266 6163 6573 2067 656e ypersurfaces gen │ │ │ │ -000261a0: 6572 6174 6564 2062 7920 7461 6b69 6e67 erated by taking │ │ │ │ -000261b0: 2061 2070 726f 6475 6374 206f 660a 736f a product of.so │ │ │ │ -000261c0: 6d65 2073 7562 7365 7473 206f 6620 6765 me subsets of ge │ │ │ │ -000261d0: 6e65 7261 746f 7273 206f 6620 7468 6520 nerators of the │ │ │ │ -000261e0: 696e 7075 7420 6964 6561 6c2e 204e 6f74 input ideal. Not │ │ │ │ -000261f0: 6520 7468 6174 2c20 7369 6e63 6520 7468 e that, since th │ │ │ │ -00026200: 6520 4353 4d20 636c 6173 730a 6f66 2061 e CSM class.of a │ │ │ │ -00026210: 2073 7562 7363 6865 6d65 2065 7175 616c subscheme equal │ │ │ │ -00026220: 7320 7468 6520 4353 4d20 636c 6173 7320 s the CSM class │ │ │ │ -00026230: 6f66 2069 7473 2072 6564 7563 6564 2073 of its reduced s │ │ │ │ -00026240: 6368 656d 652c 206f 7220 6571 7569 7661 cheme, or equiva │ │ │ │ -00026250: 6c65 6e74 6c79 2066 6f72 0a75 7320 7468 lently for.us th │ │ │ │ -00026260: 6520 4353 4d20 636c 6173 7320 636f 7272 e CSM class corr │ │ │ │ -00026270: 6573 706f 6e64 696e 6720 746f 2061 6e20 esponding to an │ │ │ │ -00026280: 6964 6561 6c20 4920 6571 7561 6c73 2074 ideal I equals t │ │ │ │ -00026290: 6865 2043 534d 2063 6c61 7373 206f 6620 he CSM class of │ │ │ │ -000262a0: 7468 650a 7261 6469 6361 6c20 6f66 2049 the.radical of I │ │ │ │ -000262b0: 2c20 7468 656e 2069 6e74 6572 6e61 6c6c , then internall │ │ │ │ -000262c0: 7920 7765 2061 6c77 6179 7320 776f 726b y we always work │ │ │ │ -000262d0: 2077 6974 6820 7261 6469 6361 6c20 6964 with radical id │ │ │ │ -000262e0: 6561 6c73 2028 666f 720a 6566 6669 6369 eals (for.effici │ │ │ │ -000262f0: 656e 6379 2072 6561 736f 6e73 292e 2048 ency reasons). H │ │ │ │ -00026300: 656e 6365 2074 6865 2070 726f 6a65 6374 ence the project │ │ │ │ -00026310: 6976 6520 6465 6772 6565 7320 616e 6420 ive degrees and │ │ │ │ -00026320: 5365 6772 6520 636c 6173 7365 7320 636f Segre classes co │ │ │ │ -00026330: 6d70 7574 6564 0a69 6e74 6572 6e61 6c6c mputed.internall │ │ │ │ -00026340: 7920 7769 6c6c 2062 6520 7468 6f73 6520 y will be those │ │ │ │ -00026350: 6f66 2074 6865 2072 6164 6963 616c 206f of the radical o │ │ │ │ -00026360: 6620 616e 2069 6465 616c 2064 6566 696e f an ideal defin │ │ │ │ -00026370: 6564 2062 7920 6120 706f 6c79 6e6f 6d69 ed by a polynomi │ │ │ │ -00026380: 616c 0a77 6869 6368 2069 7320 6120 7072 al.which is a pr │ │ │ │ -00026390: 6f64 7563 7420 6f66 2073 6f6d 6520 7375 oduct of some su │ │ │ │ -000263a0: 6273 6574 206f 6620 7468 6520 6765 6e65 bset of the gene │ │ │ │ -000263b0: 7261 746f 7273 2e20 5765 2069 6c6c 7573 rators. We illus │ │ │ │ -000263c0: 7472 6174 6520 7468 6973 2077 6974 6820 trate this with │ │ │ │ -000263d0: 616e 0a65 7861 6d70 6c65 2062 656c 6f77 an.example below │ │ │ │ -000263e0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ -000263f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026430: 2d2b 0a7c 6931 3420 3a20 6373 6d58 4c68 -+.|i14 : csmXLh │ │ │ │ -00026440: 6173 683d 4353 4d28 412c 492c 4f75 7470 ash=CSM(A,I,Outp │ │ │ │ -00026450: 7574 3d3e 4861 7368 466f 726d 584c 2920 ut=>HashFormXL) │ │ │ │ -00026460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026480: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000259e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000259f0: 2020 2036 2020 2020 2020 3520 2020 2020 6 5 │ │ │ │ +00025a00: 2034 2020 2020 2020 3320 2020 2020 2032 4 3 2 │ │ │ │ +00025a10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00025a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a30: 2020 2020 2020 207b 302c 2031 7d20 3d3e {0, 1} => │ │ │ │ +00025a40: 2032 6820 202b 2032 3368 2020 2b20 3332 2h + 23h + 32 │ │ │ │ +00025a50: 6820 202b 2033 3368 2020 2b20 3138 6820 h + 33h + 18h │ │ │ │ +00025a60: 202b 2035 6820 2020 2020 2020 207c 0a7c + 5h |.| │ │ │ │ +00025a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a90: 2020 2031 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ +00025aa0: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ +00025ab0: 2020 2020 2031 2020 2020 2020 207c 0a7c 1 |.| │ │ │ │ +00025ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ae0: 2036 2020 2020 2020 3520 2020 2020 2034 6 5 4 │ │ │ │ +00025af0: 2020 2020 2020 3320 2020 2020 3220 2020 3 2 │ │ │ │ +00025b00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00025b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b20: 2020 2020 2020 2043 534d 203d 3e20 3130 CSM => 10 │ │ │ │ +00025b30: 6820 202b 2031 3268 2020 2b20 3232 6820 h + 12h + 22h │ │ │ │ +00025b40: 202b 2031 3668 2020 2b20 3668 2020 2020 + 16h + 6h │ │ │ │ +00025b50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00025b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b80: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ +00025b90: 2020 2020 2020 3120 2020 2020 3120 2020 1 1 │ │ │ │ +00025ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00025bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025bd0: 3620 2020 2020 2035 2020 2020 2020 3420 6 5 4 │ │ │ │ +00025be0: 2020 2020 2033 2020 2020 2020 3220 2020 3 2 │ │ │ │ +00025bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00025c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c10: 2020 2020 2020 207b 307d 203d 3e20 3668 {0} => 6h │ │ │ │ +00025c20: 2020 2b20 3138 6820 202b 2032 3668 2020 + 18h + 26h │ │ │ │ +00025c30: 2b20 3232 6820 202b 2031 3068 2020 2b20 + 22h + 10h + │ │ │ │ +00025c40: 3268 2020 2020 2020 2020 2020 207c 0a7c 2h |.| │ │ │ │ +00025c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c70: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ +00025c80: 2020 2020 2031 2020 2020 2020 3120 2020 1 1 │ │ │ │ +00025c90: 2020 3120 2020 2020 2020 2020 207c 0a7c 1 |.| │ │ │ │ +00025ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025cc0: 3620 2020 2020 2035 2020 2020 2020 3420 6 5 4 │ │ │ │ +00025cd0: 2020 2020 2033 2020 2020 2020 3220 2020 3 2 │ │ │ │ +00025ce0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00025cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025d00: 2020 2020 2020 207b 317d 203d 3e20 3668 {1} => 6h │ │ │ │ +00025d10: 2020 2b20 3137 6820 202b 2032 3868 2020 + 17h + 28h │ │ │ │ +00025d20: 2b20 3237 6820 202b 2031 3468 2020 2b20 + 27h + 14h + │ │ │ │ +00025d30: 3368 2020 2020 2020 2020 2020 207c 0a7c 3h |.| │ │ │ │ +00025d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025d60: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ +00025d70: 2020 2020 2031 2020 2020 2020 3120 2020 1 1 │ │ │ │ +00025d80: 2020 3120 2020 2020 2020 2020 207c 0a2b 1 |.+ │ │ │ │ +00025d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00025de0: 5468 6520 4d75 7461 626c 6548 6173 6854 The MutableHashT │ │ │ │ +00025df0: 6162 6c65 2072 6574 7572 6e65 6420 7769 able returned wi │ │ │ │ +00025e00: 7468 2074 6865 206f 7074 696f 6e20 4f75 th the option Ou │ │ │ │ +00025e10: 7470 7574 3d3e 4861 7368 466f 726d 2063 tput=>HashForm c │ │ │ │ +00025e20: 6f6e 7461 696e 730a 6469 6666 6572 656e ontains.differen │ │ │ │ +00025e30: 7420 696e 666f 726d 6174 696f 6e20 6465 t information de │ │ │ │ +00025e40: 7065 6e64 696e 6720 6f6e 2074 6865 206d pending on the m │ │ │ │ +00025e50: 6574 686f 6420 7769 7468 2077 6869 6368 ethod with which │ │ │ │ +00025e60: 2069 7420 6973 2075 7365 642e 0a41 6464 it is used..Add │ │ │ │ +00025e70: 6974 696f 6e61 6c6c 7920 6966 2074 6865 itionally if the │ │ │ │ +00025e80: 206f 7074 696f 6e20 2a6e 6f74 6520 496e option *note In │ │ │ │ +00025e90: 7075 7449 7353 6d6f 6f74 683a 2049 6e70 putIsSmooth: Inp │ │ │ │ +00025ea0: 7574 4973 536d 6f6f 7468 2c20 6973 2075 utIsSmooth, is u │ │ │ │ +00025eb0: 7365 6420 7468 656e 2074 6865 0a68 6173 sed then the.has │ │ │ │ +00025ec0: 6820 7461 626c 6520 7265 7475 726e 6564 h table returned │ │ │ │ +00025ed0: 2062 7920 7468 6520 6d65 7468 6f64 7320 by the methods │ │ │ │ +00025ee0: 4575 6c65 7220 616e 6420 4353 4d20 7769 Euler and CSM wi │ │ │ │ +00025ef0: 6c6c 2062 6520 7468 6520 7361 6d65 2061 ll be the same a │ │ │ │ +00025f00: 7320 7468 6174 0a72 6574 7572 6e65 6420 s that.returned │ │ │ │ +00025f10: 6279 2043 6865 726e 2e20 5768 656e 2075 by Chern. When u │ │ │ │ +00025f20: 7369 6e67 2074 6865 202a 6e6f 7465 2043 sing the *note C │ │ │ │ +00025f30: 534d 3a20 4353 4d2c 2020 636f 6d6d 616e SM: CSM, comman │ │ │ │ +00025f40: 6420 696e 2074 6865 2064 6566 6175 6c74 d in the default │ │ │ │ +00025f50: 0a63 6f6e 6669 6775 7261 7469 6f6e 7320 .configurations │ │ │ │ +00025f60: 2874 6861 7420 6973 202a 6e6f 7465 204d (that is *note M │ │ │ │ +00025f70: 6574 686f 643a 204d 6574 686f 642c 3d3e ethod: Method,=> │ │ │ │ +00025f80: 496e 636c 7573 696f 6e45 7863 6c75 7369 InclusionExclusi │ │ │ │ +00025f90: 6f6e 2c20 2a6e 6f74 650a 436f 6d70 4d65 on, *note.CompMe │ │ │ │ +00025fa0: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64 thod: CompMethod │ │ │ │ +00025fb0: 2c3d 3e50 726f 6a65 6374 6976 6544 6567 ,=>ProjectiveDeg │ │ │ │ +00025fc0: 7265 6529 2074 6865 7265 2069 7320 7468 ree) there is th │ │ │ │ +00025fd0: 6520 6164 6469 7469 6f6e 616c 206f 7074 e additional opt │ │ │ │ +00025fe0: 696f 6e20 746f 0a73 6574 204f 7574 7075 ion to.set Outpu │ │ │ │ +00025ff0: 743d 3e48 6173 6846 6f72 6d58 4c2e 2054 t=>HashFormXL. T │ │ │ │ +00026000: 6869 7320 7265 7475 726e 7320 616c 6c20 his returns all │ │ │ │ +00026010: 7468 6520 7573 7561 6c20 696e 666f 726d the usual inform │ │ │ │ +00026020: 6174 696f 6e20 7468 6174 0a4f 7574 7075 ation that.Outpu │ │ │ │ +00026030: 743d 3e48 6173 6846 6f72 6d20 776f 756c t=>HashForm woul │ │ │ │ +00026040: 6420 666f 7220 7468 6973 2063 6f6e 6669 d for this confi │ │ │ │ +00026050: 6775 7261 7469 6f6e 2077 6974 6820 7468 guration with th │ │ │ │ +00026060: 6520 6164 6469 7469 6f6e 206f 6620 7468 e addition of th │ │ │ │ +00026070: 650a 7072 6f6a 6563 7469 7665 2064 6567 e.projective deg │ │ │ │ +00026080: 7265 6573 2061 6e64 2053 6567 7265 2063 rees and Segre c │ │ │ │ +00026090: 6c61 7373 6573 206f 6620 7369 6e67 756c lasses of singul │ │ │ │ +000260a0: 6172 6974 7920 7375 6273 6368 656d 6573 arity subschemes │ │ │ │ +000260b0: 2067 656e 6572 6174 6564 2062 7920 7468 generated by th │ │ │ │ +000260c0: 650a 6879 7065 7273 7572 6661 6365 7320 e.hypersurfaces │ │ │ │ +000260d0: 636f 6e73 6964 6572 6564 2069 6e20 7468 considered in th │ │ │ │ +000260e0: 6520 696e 636c 7573 696f 6e2f 6578 636c e inclusion/excl │ │ │ │ +000260f0: 7573 696f 6e20 7072 6f63 6564 7572 652c usion procedure, │ │ │ │ +00026100: 2074 6861 7420 6973 2069 6e0a 6669 6e64 that is in.find │ │ │ │ +00026110: 696e 6720 7468 6520 4353 4d20 636c 6173 ing the CSM clas │ │ │ │ +00026120: 7320 6f66 2061 6c6c 2068 7970 6572 7375 s of all hypersu │ │ │ │ +00026130: 7266 6163 6573 2067 656e 6572 6174 6564 rfaces generated │ │ │ │ +00026140: 2062 7920 7461 6b69 6e67 2061 2070 726f by taking a pro │ │ │ │ +00026150: 6475 6374 206f 660a 736f 6d65 2073 7562 duct of.some sub │ │ │ │ +00026160: 7365 7473 206f 6620 6765 6e65 7261 746f sets of generato │ │ │ │ +00026170: 7273 206f 6620 7468 6520 696e 7075 7420 rs of the input │ │ │ │ +00026180: 6964 6561 6c2e 204e 6f74 6520 7468 6174 ideal. Note that │ │ │ │ +00026190: 2c20 7369 6e63 6520 7468 6520 4353 4d20 , since the CSM │ │ │ │ +000261a0: 636c 6173 730a 6f66 2061 2073 7562 7363 class.of a subsc │ │ │ │ +000261b0: 6865 6d65 2065 7175 616c 7320 7468 6520 heme equals the │ │ │ │ +000261c0: 4353 4d20 636c 6173 7320 6f66 2069 7473 CSM class of its │ │ │ │ +000261d0: 2072 6564 7563 6564 2073 6368 656d 652c reduced scheme, │ │ │ │ +000261e0: 206f 7220 6571 7569 7661 6c65 6e74 6c79 or equivalently │ │ │ │ +000261f0: 2066 6f72 0a75 7320 7468 6520 4353 4d20 for.us the CSM │ │ │ │ +00026200: 636c 6173 7320 636f 7272 6573 706f 6e64 class correspond │ │ │ │ +00026210: 696e 6720 746f 2061 6e20 6964 6561 6c20 ing to an ideal │ │ │ │ +00026220: 4920 6571 7561 6c73 2074 6865 2043 534d I equals the CSM │ │ │ │ +00026230: 2063 6c61 7373 206f 6620 7468 650a 7261 class of the.ra │ │ │ │ +00026240: 6469 6361 6c20 6f66 2049 2c20 7468 656e dical of I, then │ │ │ │ +00026250: 2069 6e74 6572 6e61 6c6c 7920 7765 2061 internally we a │ │ │ │ +00026260: 6c77 6179 7320 776f 726b 2077 6974 6820 lways work with │ │ │ │ +00026270: 7261 6469 6361 6c20 6964 6561 6c73 2028 radical ideals ( │ │ │ │ +00026280: 666f 720a 6566 6669 6369 656e 6379 2072 for.efficiency r │ │ │ │ +00026290: 6561 736f 6e73 292e 2048 656e 6365 2074 easons). Hence t │ │ │ │ +000262a0: 6865 2070 726f 6a65 6374 6976 6520 6465 he projective de │ │ │ │ +000262b0: 6772 6565 7320 616e 6420 5365 6772 6520 grees and Segre │ │ │ │ +000262c0: 636c 6173 7365 7320 636f 6d70 7574 6564 classes computed │ │ │ │ +000262d0: 0a69 6e74 6572 6e61 6c6c 7920 7769 6c6c .internally will │ │ │ │ +000262e0: 2062 6520 7468 6f73 6520 6f66 2074 6865 be those of the │ │ │ │ +000262f0: 2072 6164 6963 616c 206f 6620 616e 2069 radical of an i │ │ │ │ +00026300: 6465 616c 2064 6566 696e 6564 2062 7920 deal defined by │ │ │ │ +00026310: 6120 706f 6c79 6e6f 6d69 616c 0a77 6869 a polynomial.whi │ │ │ │ +00026320: 6368 2069 7320 6120 7072 6f64 7563 7420 ch is a product │ │ │ │ +00026330: 6f66 2073 6f6d 6520 7375 6273 6574 206f of some subset o │ │ │ │ +00026340: 6620 7468 6520 6765 6e65 7261 746f 7273 f the generators │ │ │ │ +00026350: 2e20 5765 2069 6c6c 7573 7472 6174 6520 . We illustrate │ │ │ │ +00026360: 7468 6973 2077 6974 6820 616e 0a65 7861 this with an.exa │ │ │ │ +00026370: 6d70 6c65 2062 656c 6f77 2e0a 0a2b 2d2d mple below...+-- │ │ │ │ +00026380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000263a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000263b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000263c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +000263d0: 3420 3a20 6373 6d58 4c68 6173 683d 4353 4 : csmXLhash=CS │ │ │ │ +000263e0: 4d28 412c 492c 4f75 7470 7574 3d3e 4861 M(A,I,Output=>Ha │ │ │ │ +000263f0: 7368 466f 726d 584c 2920 2020 2020 2020 shFormXL) │ │ │ │ +00026400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026410: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00026420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00026470: 3420 3d20 4d75 7461 626c 6548 6173 6854 4 = MutableHashT │ │ │ │ +00026480: 6162 6c65 7b2e 2e2e 3130 2e2e 2e7d 2020 able{...10...} │ │ │ │ 00026490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000264a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000264b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000264b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000264c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000264d0: 207c 0a7c 6f31 3420 3d20 4d75 7461 626c |.|o14 = Mutabl │ │ │ │ -000264e0: 6548 6173 6854 6162 6c65 7b2e 2e2e 3130 eHashTable{...10 │ │ │ │ -000264f0: 2e2e 2e7d 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: 207c 0a7c 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 6f31 |.|o1 │ │ │ │ +00026510: 3420 3a20 4d75 7461 626c 6548 6173 6854 4 : MutableHashT │ │ │ │ +00026520: 6162 6c65 2020 2020 2020 2020 2020 2020 able │ │ │ │ 00026530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026570: 207c 0a7c 6f31 3420 3a20 4d75 7461 626c |.|o14 : Mutabl │ │ │ │ -00026580: 6548 6173 6854 6162 6c65 2020 2020 2020 eHashTable │ │ │ │ -00026590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000265a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000265b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000265c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -000265d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026610: 2d2b 0a7c 6931 3520 3a20 7065 656b 2063 -+.|i15 : peek c │ │ │ │ -00026620: 736d 584c 6861 7368 2020 2020 2020 2020 smXLhash │ │ │ │ +00026550: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00026560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000265a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +000265b0: 3520 3a20 7065 656b 2063 736d 584c 6861 5 : peek csmXLha │ │ │ │ +000265c0: 7368 2020 2020 2020 2020 2020 2020 2020 sh │ │ │ │ +000265d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000265e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000265f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00026600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026660: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00026670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026640: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00026650: 3520 3d20 4d75 7461 626c 6548 6173 6854 5 = MutableHashT │ │ │ │ +00026660: 6162 6c65 7b47 284a 6163 6f62 6961 6e29 able{G(Jacobian) │ │ │ │ +00026670: 7b30 7d20 3d3e 2030 2020 2020 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2020 2020 2020 2020 2020 │ │ │ │ 000271b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000271c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000271d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000271e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000271f0: 207c 0a7c 3868 2020 2b20 3468 2020 2b20 |.|8h + 4h + │ │ │ │ -00027200: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +000271d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000271e0: 3120 2020 2020 3120 2020 2020 2020 2020 1 1 │ │ │ │ +000271f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027240: 207c 0a7c 2020 3120 2020 2020 3120 2020 |.| 1 1 │ │ │ │ -00027250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027290: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -000272a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000272b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000272c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000272d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000272e0: 2d2b 0a7c 6931 3620 3a20 4b3d 6964 6561 -+.|i16 : K=idea │ │ │ │ -000272f0: 6c20 495f 302a 495f 313b 2020 2020 2020 l I_0*I_1; │ │ │ │ +00027220: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00027230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00027280: 3620 3a20 4b3d 6964 6561 6c20 495f 302a 6 : K=ideal I_0* │ │ │ │ +00027290: 495f 313b 2020 2020 2020 2020 2020 2020 I_1; │ │ │ │ +000272a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000272b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000272c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000272d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000272e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000272f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027330: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027310: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00027320: 3620 3a20 4964 6561 6c20 6f66 2052 2020 6 : Ideal of R │ │ │ │ +00027330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027380: 207c 0a7c 6f31 3620 3a20 4964 6561 6c20 |.|o16 : Ideal │ │ │ │ -00027390: 6f66 2052 2020 2020 2020 2020 2020 2020 of R │ │ │ │ -000273a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000273b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000273c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000273d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -000273e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000273f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027420: 2d2b 0a7c 6931 3720 3a20 4353 4d28 412c -+.|i17 : CSM(A, │ │ │ │ -00027430: 7261 6469 6361 6c20 4b29 3d3d 4353 4d28 radical K)==CSM( │ │ │ │ -00027440: 412c 4b29 2020 2020 2020 2020 2020 2020 A,K) │ │ │ │ -00027450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027470: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027360: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00027370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000273a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000273b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +000273c0: 3720 3a20 4353 4d28 412c 7261 6469 6361 7 : CSM(A,radica │ │ │ │ +000273d0: 6c20 4b29 3d3d 4353 4d28 412c 4b29 2020 l K)==CSM(A,K) │ │ │ │ +000273e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000273f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027400: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00027410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027450: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00027460: 3720 3d20 7472 7565 2020 2020 2020 2020 7 = true │ │ │ │ +00027470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274c0: 207c 0a7c 6f31 3720 3d20 7472 7565 2020 |.|o17 = true │ │ │ │ -000274d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2d2d 2d2d ---------------- │ │ │ │ +000274c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000274d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000274e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000274f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00027500: 3820 3a20 4a3d 6964 6561 6c20 6a61 636f 8 : J=ideal jaco │ │ │ │ +00027510: 6269 616e 2072 6164 6963 616c 204b 3b20 bian radical K; │ │ │ │ +00027520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027540: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00027550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 6f31 |.|o1 │ │ │ │ +000275a0: 3820 3a20 4964 6561 6c20 6f66 2052 2020 8 : Ideal of R │ │ │ │ +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 2020 2020 2020 │ │ │ │ -000275f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027600: 207c 0a7c 6f31 3820 3a20 4964 6561 6c20 |.|o18 : Ideal │ │ │ │ -00027610: 6f66 2052 2020 2020 2020 2020 2020 2020 of R │ │ │ │ -00027620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027650: 207c 0a2b 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 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000276a0: 2d2b 0a7c 6931 3920 3a20 7365 674a 3d53 -+.|i19 : segJ=S │ │ │ │ -000276b0: 6567 7265 2841 2c4a 2c4f 7574 7075 743d egre(A,J,Output= │ │ │ │ -000276c0: 3e48 6173 6846 6f72 6d29 2020 2020 2020 >HashForm) │ │ │ │ -000276d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000276e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000276f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000275e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +000275f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00027640: 3920 3a20 7365 674a 3d53 6567 7265 2841 9 : segJ=Segre(A │ │ │ │ +00027650: 2c4a 2c4f 7574 7075 743d 3e48 6173 6846 ,J,Output=>HashF │ │ │ │ +00027660: 6f72 6d29 2020 2020 2020 2020 2020 2020 orm) │ │ │ │ +00027670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027680: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00027690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000276a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000276b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000276c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000276d0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +000276e0: 3920 3d20 4d75 7461 626c 6548 6173 6854 9 = MutableHashT │ │ │ │ +000276f0: 6162 6c65 7b2e 2e2e 342e 2e2e 7d20 2020 able{...4...} │ │ │ │ 00027700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027720: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00027730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027740: 207c 0a7c 6f31 3920 3d20 4d75 7461 626c |.|o19 = Mutabl │ │ │ │ -00027750: 6548 6173 6854 6162 6c65 7b2e 2e2e 342e eHashTable{...4. │ │ │ │ -00027760: 2e2e 7d20 2020 2020 2020 2020 2020 2020 ..} │ │ │ │ -00027770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027790: 207c 0a7c 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 6f31 |.|o1 │ │ │ │ +00027780: 3920 3a20 4d75 7461 626c 6548 6173 6854 9 : MutableHashT │ │ │ │ +00027790: 6162 6c65 2020 2020 2020 2020 2020 2020 able │ │ │ │ 000277a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000277b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277e0: 207c 0a7c 6f31 3920 3a20 4d75 7461 626c |.|o19 : Mutabl │ │ │ │ -000277f0: 6548 6173 6854 6162 6c65 2020 2020 2020 eHashTable │ │ │ │ -00027800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027830: 207c 0a2b 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 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027880: 2d2b 0a7c 6932 3020 3a20 6373 6d58 4c68 -+.|i20 : csmXLh │ │ │ │ -00027890: 6173 6823 2822 4728 4a61 636f 6269 616e ash#("G(Jacobian │ │ │ │ -000278a0: 2922 7c74 6f53 7472 696e 6728 7b30 2c31 )"|toString({0,1 │ │ │ │ -000278b0: 7d29 293d 3d73 6567 4a23 2247 2220 2020 }))==segJ#"G" │ │ │ │ -000278c0: 2020 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2020 2020 2020 2020 │ │ │ │ +00027890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000278a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000278b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +000278c0: 3020 3d20 7472 7565 2020 2020 2020 2020 0 = true │ │ │ │ +000278d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000278e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000278f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027920: 207c 0a7c 6f32 3020 3d20 7472 7565 2020 |.|o20 = true │ │ │ │ -00027930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027970: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000279b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000279c0: 2d2b 0a7c 6932 3120 3a20 6373 6d58 4c68 -+.|i21 : csmXLh │ │ │ │ -000279d0: 6173 6823 2822 5365 6772 6528 4a61 636f ash#("Segre(Jaco │ │ │ │ -000279e0: 6269 616e 2922 7c74 6f53 7472 696e 6728 bian)"|toString( │ │ │ │ -000279f0: 7b30 2c31 7d29 293d 3d73 6567 4a23 2253 {0,1}))==segJ#"S │ │ │ │ -00027a00: 6567 7265 2220 2020 2020 2020 2020 2020 egre" │ │ │ │ -00027a10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027900: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00027910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +00027960: 3120 3a20 6373 6d58 4c68 6173 6823 2822 1 : csmXLhash#(" │ │ │ │ +00027970: 5365 6772 6528 4a61 636f 6269 616e 2922 Segre(Jacobian)" │ │ │ │ +00027980: 7c74 6f53 7472 696e 6728 7b30 2c31 7d29 |toString({0,1}) │ │ │ │ +00027990: 293d 3d73 6567 4a23 2253 6567 7265 2220 )==segJ#"Segre" │ │ │ │ +000279a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000279b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000279c0: 2020 2020 2020 2020 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 207c 0a7c 6f32 |.|o2 │ │ │ │ +00027a00: 3120 3d20 7472 7565 2020 2020 2020 2020 1 = true │ │ │ │ +00027a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a60: 207c 0a7c 6f32 3120 3d20 7472 7565 2020 |.|o21 = true │ │ │ │ -00027a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027ab0: 207c 0a2b 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 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027b00: 2d2b 0a0a 4675 6e63 7469 6f6e 7320 7769 -+..Functions wi │ │ │ │ -00027b10: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ -00027b20: 6d65 6e74 206e 616d 6564 204f 7574 7075 ment named Outpu │ │ │ │ -00027b30: 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d t:.============= │ │ │ │ -00027b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027b60: 3d0a 0a20 202a 2022 4368 6572 6e28 2e2e =.. * "Chern(.. │ │ │ │ -00027b70: 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 2220 .,Output=>...)" │ │ │ │ -00027b80: 2d2d 2073 6565 202a 6e6f 7465 2043 6865 -- see *note Che │ │ │ │ -00027b90: 726e 3a20 4368 6572 6e2c 202d 2d20 5468 rn: Chern, -- Th │ │ │ │ -00027ba0: 6520 4368 6572 6e20 636c 6173 730a 2020 e Chern class. │ │ │ │ -00027bb0: 2a20 2243 534d 282e 2e2e 2c4f 7574 7075 * "CSM(...,Outpu │ │ │ │ -00027bc0: 743d 3e2e 2e2e 2922 202d 2d20 7365 6520 t=>...)" -- see │ │ │ │ -00027bd0: 2a6e 6f74 6520 4353 4d3a 2043 534d 2c20 *note CSM: CSM, │ │ │ │ -00027be0: 2d2d 2054 6865 0a20 2020 2043 6865 726e -- The. Chern │ │ │ │ -00027bf0: 2d53 6368 7761 7274 7a2d 4d61 6350 6865 -Schwartz-MacPhe │ │ │ │ -00027c00: 7273 6f6e 2063 6c61 7373 0a20 202a 2022 rson class. * " │ │ │ │ -00027c10: 4575 6c65 7228 2e2e 2e2c 4f75 7470 7574 Euler(...,Output │ │ │ │ -00027c20: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ -00027c30: 6e6f 7465 2045 756c 6572 3a20 4575 6c65 note Euler: Eule │ │ │ │ -00027c40: 722c 202d 2d20 5468 6520 4575 6c65 720a r, -- The Euler. │ │ │ │ -00027c50: 2020 2020 4368 6172 6163 7465 7269 7374 Characterist │ │ │ │ -00027c60: 6963 0a20 202a 2022 5365 6772 6528 2e2e ic. * "Segre(.. │ │ │ │ -00027c70: 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 2220 .,Output=>...)" │ │ │ │ -00027c80: 2d2d 2073 6565 202a 6e6f 7465 2053 6567 -- see *note Seg │ │ │ │ -00027c90: 7265 3a20 5365 6772 652c 202d 2d20 5468 re: Segre, -- Th │ │ │ │ -00027ca0: 6520 5365 6772 6520 636c 6173 7320 6f66 e Segre class of │ │ │ │ -00027cb0: 2061 0a20 2020 2073 7562 7363 6865 6d65 a. subscheme │ │ │ │ -00027cc0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -00027cd0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -00027ce0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -00027cf0: 6563 7420 2a6e 6f74 6520 4f75 7470 7574 ect *note Output │ │ │ │ -00027d00: 3a20 4f75 7470 7574 2c20 6973 2061 202a : Output, is a * │ │ │ │ -00027d10: 6e6f 7465 2073 796d 626f 6c3a 2028 4d61 note symbol: (Ma │ │ │ │ -00027d20: 6361 756c 6179 3244 6f63 2953 796d 626f caulay2Doc)Symbo │ │ │ │ -00027d30: 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d l,...----------- │ │ │ │ -00027d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d80: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00027d90: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00027da0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00027db0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00027dc0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ -00027dd0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61 11+ds/M2/Macaula │ │ │ │ -00027de0: 7932 2f70 6163 6b61 6765 732f 0a43 6861 y2/packages/.Cha │ │ │ │ -00027df0: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ -00027e00: 6573 2e6d 323a 3234 3639 3a30 2e0a 1f0a es.m2:2469:0.... │ │ │ │ -00027e10: 4669 6c65 3a20 4368 6172 6163 7465 7269 File: Characteri │ │ │ │ -00027e20: 7374 6963 436c 6173 7365 732e 696e 666f sticClasses.info │ │ │ │ -00027e30: 2c20 4e6f 6465 3a20 7072 6f62 6162 696c , Node: probabil │ │ │ │ -00027e40: 6973 7469 6320 616c 676f 7269 7468 6d2c istic algorithm, │ │ │ │ -00027e50: 204e 6578 743a 2053 6567 7265 2c20 5072 Next: Segre, Pr │ │ │ │ -00027e60: 6576 3a20 4f75 7470 7574 2c20 5570 3a20 ev: Output, Up: │ │ │ │ -00027e70: 546f 700a 0a70 726f 6261 6269 6c69 7374 Top..probabilist │ │ │ │ -00027e80: 6963 2061 6c67 6f72 6974 686d 0a2a 2a2a ic algorithm.*** │ │ │ │ -00027e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00027ea0: 2a2a 2a2a 0a0a 5468 6520 616c 676f 7269 ****..The algori │ │ │ │ -00027eb0: 7468 6d73 2075 7365 6420 666f 7220 7468 thms used for th │ │ │ │ -00027ec0: 6520 636f 6d70 7574 6174 696f 6e20 6f66 e computation of │ │ │ │ -00027ed0: 2063 6861 7261 6374 6572 6973 7469 6320 characteristic │ │ │ │ -00027ee0: 636c 6173 7365 7320 6172 650a 7072 6f62 classes are.prob │ │ │ │ -00027ef0: 6162 696c 6973 7469 632e 2054 6865 6f72 abilistic. Theor │ │ │ │ -00027f00: 6574 6963 616c 6c79 2c20 7468 6579 2063 etically, they c │ │ │ │ -00027f10: 616c 6375 6c61 7465 2074 6865 2063 6c61 alculate the cla │ │ │ │ -00027f20: 7373 6573 2063 6f72 7265 6374 6c79 2066 sses correctly f │ │ │ │ -00027f30: 6f72 2061 0a67 656e 6572 616c 2063 686f or a.general cho │ │ │ │ -00027f40: 6963 6520 6f66 2063 6572 7461 696e 2070 ice of certain p │ │ │ │ -00027f50: 6f6c 796e 6f6d 6961 6c73 2e20 5468 6174 olynomials. That │ │ │ │ -00027f60: 2069 732c 2074 6865 7265 2069 7320 616e is, there is an │ │ │ │ -00027f70: 206f 7065 6e20 6465 6e73 6520 5a61 7269 open dense Zari │ │ │ │ -00027f80: 736b 690a 7365 7420 666f 7220 7768 6963 ski.set for whic │ │ │ │ -00027f90: 6820 7468 6520 616c 676f 7269 7468 6d20 h the algorithm │ │ │ │ -00027fa0: 7969 656c 6473 2074 6865 2063 6f72 7265 yields the corre │ │ │ │ -00027fb0: 6374 2063 6c61 7373 2c20 692e 652e 2c20 ct class, i.e., │ │ │ │ -00027fc0: 7468 6520 636f 7272 6563 7420 636c 6173 the correct clas │ │ │ │ -00027fd0: 730a 6973 2063 616c 6375 6c61 7465 6420 s.is calculated │ │ │ │ -00027fe0: 7769 7468 2070 726f 6261 6269 6c69 7479 with probability │ │ │ │ -00027ff0: 2031 2e20 486f 7765 7665 722c 2073 696e 1. However, sin │ │ │ │ -00028000: 6365 2074 6865 2069 6d70 6c65 6d65 6e74 ce the implement │ │ │ │ -00028010: 6174 696f 6e20 776f 726b 7320 6f76 6572 ation works over │ │ │ │ -00028020: 0a61 2064 6973 6372 6574 6520 7072 6f62 .a discrete prob │ │ │ │ -00028030: 6162 696c 6974 7920 7370 6163 6520 7468 ability space th │ │ │ │ -00028040: 6572 6520 6973 2061 2076 6572 7920 736d ere is a very sm │ │ │ │ -00028050: 616c 6c2c 2062 7574 206e 6f6e 2d7a 6572 all, but non-zer │ │ │ │ -00028060: 6f2c 2070 726f 6261 6269 6c69 7479 0a6f o, probability.o │ │ │ │ -00028070: 6620 6e6f 7420 636f 6d70 7574 696e 6720 f not computing │ │ │ │ -00028080: 7468 6520 636f 7272 6563 7420 636c 6173 the correct clas │ │ │ │ -00028090: 732e 2053 6b65 7074 6963 616c 2075 7365 s. Skeptical use │ │ │ │ -000280a0: 7273 2073 686f 756c 6420 7265 7065 6174 rs should repeat │ │ │ │ -000280b0: 2063 616c 6375 6c61 7469 6f6e 730a 7365 calculations.se │ │ │ │ -000280c0: 7665 7261 6c20 7469 6d65 7320 746f 2069 veral times to i │ │ │ │ -000280d0: 6e63 7265 6173 6520 7468 6520 7072 6f62 ncrease the prob │ │ │ │ -000280e0: 6162 696c 6974 7920 6f66 2063 6f6d 7075 ability of compu │ │ │ │ -000280f0: 7469 6e67 2074 6865 2063 6f72 7265 6374 ting the correct │ │ │ │ -00028100: 2063 6c61 7373 2e0a 0a49 6e20 7468 6520 class...In the │ │ │ │ -00028110: 6361 7365 206f 6620 7468 6520 7379 6d62 case of the symb │ │ │ │ -00028120: 6f6c 6963 2069 6d70 6c65 6d65 6e74 6174 olic implementat │ │ │ │ -00028130: 696f 6e20 6f66 2074 6865 2050 726f 6a65 ion of the Proje │ │ │ │ -00028140: 6374 6976 6544 6567 7265 6520 6d65 7468 ctiveDegree meth │ │ │ │ -00028150: 6f64 0a70 7261 6374 6963 616c 2065 7870 od.practical exp │ │ │ │ -00028160: 6572 6965 6e63 6520 616e 6420 616c 676f erience and algo │ │ │ │ -00028170: 7269 7468 6d20 7465 7374 696e 6720 696e rithm testing in │ │ │ │ -00028180: 6469 6361 7465 2074 6861 7420 6120 6669 dicate that a fi │ │ │ │ -00028190: 6e69 7465 2066 6965 6c64 2077 6974 680a nite field with. │ │ │ │ -000281a0: 6f76 6572 2032 3530 3030 2065 6c65 6d65 over 25000 eleme │ │ │ │ -000281b0: 6e74 7320 6973 206d 6f72 6520 7468 616e nts is more than │ │ │ │ -000281c0: 2073 7566 6669 6369 656e 7420 746f 2065 sufficient to e │ │ │ │ -000281d0: 7870 6563 7420 6120 636f 7272 6563 7420 xpect a correct │ │ │ │ -000281e0: 7265 7375 6c74 2077 6974 680a 6869 6768 result with.high │ │ │ │ -000281f0: 2070 726f 6261 6269 6c69 7479 2c20 692e probability, i. │ │ │ │ -00028200: 652e 2075 7369 6e67 2074 6865 2066 696e e. using the fin │ │ │ │ -00028210: 6974 6520 6669 656c 6420 6b6b 3d5a 5a2f ite field kk=ZZ/ │ │ │ │ -00028220: 3235 3037 3320 7468 6520 6578 7065 7269 25073 the experi │ │ │ │ -00028230: 6d65 6e74 616c 0a63 6861 6e63 6520 6f66 mental.chance of │ │ │ │ -00028240: 2066 6169 6c75 7265 2077 6974 6820 7468 failure with th │ │ │ │ -00028250: 6520 5072 6f6a 6563 7469 7665 4465 6772 e ProjectiveDegr │ │ │ │ -00028260: 6565 2061 6c67 6f72 6974 686d 206f 6e20 ee algorithm on │ │ │ │ -00028270: 6120 7661 7269 6574 7920 6f66 2065 7861 a variety of exa │ │ │ │ -00028280: 6d70 6c65 730a 7761 7320 6c65 7373 2074 mples.was less t │ │ │ │ -00028290: 6861 6e20 312f 3230 3030 2e20 5573 696e han 1/2000. Usin │ │ │ │ -000282a0: 6720 7468 6520 6669 6e69 7465 2066 6965 g the finite fie │ │ │ │ -000282b0: 6c64 206b 6b3d 5a5a 2f33 3237 3439 2072 ld kk=ZZ/32749 r │ │ │ │ -000282c0: 6573 756c 7465 6420 696e 206e 6f0a 6661 esulted in no.fa │ │ │ │ -000282d0: 696c 7572 6573 2069 6e20 6f76 6572 2031 ilures in over 1 │ │ │ │ -000282e0: 3030 3030 2061 7474 656d 7074 7320 6f66 0000 attempts of │ │ │ │ -000282f0: 2073 6576 6572 616c 2064 6966 6665 7265 several differe │ │ │ │ -00028300: 6e74 2065 7861 6d70 6c65 732e 0a0a 5765 nt examples...We │ │ │ │ -00028310: 2069 6c6c 7573 7472 6174 6520 7468 6520 illustrate the │ │ │ │ -00028320: 7072 6f62 6162 696c 6973 7469 6320 6265 probabilistic be │ │ │ │ -00028330: 6861 7669 6f75 7220 7769 7468 2061 6e20 haviour with an │ │ │ │ -00028340: 6578 616d 706c 6520 7768 6572 6520 7468 example where th │ │ │ │ -00028350: 6520 6368 6f73 656e 0a72 616e 646f 6d20 e chosen.random │ │ │ │ -00028360: 7365 6564 206c 6561 6473 2074 6f20 6120 seed leads to a │ │ │ │ -00028370: 7772 6f6e 6720 7265 7375 6c74 2069 6e20 wrong result in │ │ │ │ -00028380: 7468 6520 6669 7273 7420 6361 6c63 756c the first calcul │ │ │ │ -00028390: 6174 696f 6e2e 0a0a 2b2d 2d2d 2d2d 2d2d ation...+------- │ │ │ │ -000283a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000283b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000283c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ -000283d0: 2073 6574 5261 6e64 6f6d 5365 6564 2031 setRandomSeed 1 │ │ │ │ -000283e0: 3231 3b20 2020 2020 2020 2020 2020 2020 21; │ │ │ │ -000283f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00028400: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ -00028410: 6d20 7365 6564 2074 6f20 3132 3120 2020 m seed to 121 │ │ │ │ +00027a40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00027a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4675 -----------+..Fu │ │ │ │ +00027aa0: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +00027ab0: 696f 6e61 6c20 6172 6775 6d65 6e74 206e ional argument n │ │ │ │ +00027ac0: 616d 6564 204f 7574 7075 743a 0a3d 3d3d amed Output:.=== │ │ │ │ +00027ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00027ae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00027af0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00027b00: 2022 4368 6572 6e28 2e2e 2e2c 4f75 7470 "Chern(...,Outp │ │ │ │ +00027b10: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565 ut=>...)" -- see │ │ │ │ +00027b20: 202a 6e6f 7465 2043 6865 726e 3a20 4368 *note Chern: Ch │ │ │ │ +00027b30: 6572 6e2c 202d 2d20 5468 6520 4368 6572 ern, -- The Cher │ │ │ │ +00027b40: 6e20 636c 6173 730a 2020 2a20 2243 534d n class. * "CSM │ │ │ │ +00027b50: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e (...,Output=>... │ │ │ │ +00027b60: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ +00027b70: 4353 4d3a 2043 534d 2c20 2d2d 2054 6865 CSM: CSM, -- The │ │ │ │ +00027b80: 0a20 2020 2043 6865 726e 2d53 6368 7761 . Chern-Schwa │ │ │ │ +00027b90: 7274 7a2d 4d61 6350 6865 7273 6f6e 2063 rtz-MacPherson c │ │ │ │ +00027ba0: 6c61 7373 0a20 202a 2022 4575 6c65 7228 lass. * "Euler( │ │ │ │ +00027bb0: 2e2e 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 ...,Output=>...) │ │ │ │ +00027bc0: 2220 2d2d 2073 6565 202a 6e6f 7465 2045 " -- see *note E │ │ │ │ +00027bd0: 756c 6572 3a20 4575 6c65 722c 202d 2d20 uler: Euler, -- │ │ │ │ +00027be0: 5468 6520 4575 6c65 720a 2020 2020 4368 The Euler. Ch │ │ │ │ +00027bf0: 6172 6163 7465 7269 7374 6963 0a20 202a aracteristic. * │ │ │ │ +00027c00: 2022 5365 6772 6528 2e2e 2e2c 4f75 7470 "Segre(...,Outp │ │ │ │ +00027c10: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565 ut=>...)" -- see │ │ │ │ +00027c20: 202a 6e6f 7465 2053 6567 7265 3a20 5365 *note Segre: Se │ │ │ │ +00027c30: 6772 652c 202d 2d20 5468 6520 5365 6772 gre, -- The Segr │ │ │ │ +00027c40: 6520 636c 6173 7320 6f66 2061 0a20 2020 e class of a. │ │ │ │ +00027c50: 2073 7562 7363 6865 6d65 0a0a 466f 7220 subscheme..For │ │ │ │ +00027c60: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +00027c70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00027c80: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +00027c90: 6f74 6520 4f75 7470 7574 3a20 4f75 7470 ote Output: Outp │ │ │ │ +00027ca0: 7574 2c20 6973 2061 202a 6e6f 7465 2073 ut, is a *note s │ │ │ │ +00027cb0: 796d 626f 6c3a 2028 4d61 6361 756c 6179 ymbol: (Macaulay │ │ │ │ +00027cc0: 3244 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2Doc)Symbol,...- │ │ │ │ +00027cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00027d20: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00027d30: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00027d40: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +00027d50: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +00027d60: 6c61 7932 2d31 2e32 352e 3131 2b64 732f lay2-1.25.11+ds/ │ │ │ │ +00027d70: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +00027d80: 6b61 6765 732f 0a43 6861 7261 6374 6572 kages/.Character │ │ │ │ +00027d90: 6973 7469 6343 6c61 7373 6573 2e6d 323a isticClasses.m2: │ │ │ │ +00027da0: 3234 3639 3a30 2e0a 1f0a 4669 6c65 3a20 2469:0....File: │ │ │ │ +00027db0: 4368 6172 6163 7465 7269 7374 6963 436c CharacteristicCl │ │ │ │ +00027dc0: 6173 7365 732e 696e 666f 2c20 4e6f 6465 asses.info, Node │ │ │ │ +00027dd0: 3a20 7072 6f62 6162 696c 6973 7469 6320 : probabilistic │ │ │ │ +00027de0: 616c 676f 7269 7468 6d2c 204e 6578 743a algorithm, Next: │ │ │ │ +00027df0: 2053 6567 7265 2c20 5072 6576 3a20 4f75 Segre, Prev: Ou │ │ │ │ +00027e00: 7470 7574 2c20 5570 3a20 546f 700a 0a70 tput, Up: Top..p │ │ │ │ +00027e10: 726f 6261 6269 6c69 7374 6963 2061 6c67 robabilistic alg │ │ │ │ +00027e20: 6f72 6974 686d 0a2a 2a2a 2a2a 2a2a 2a2a orithm.********* │ │ │ │ +00027e30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00027e40: 5468 6520 616c 676f 7269 7468 6d73 2075 The algorithms u │ │ │ │ +00027e50: 7365 6420 666f 7220 7468 6520 636f 6d70 sed for the comp │ │ │ │ +00027e60: 7574 6174 696f 6e20 6f66 2063 6861 7261 utation of chara │ │ │ │ +00027e70: 6374 6572 6973 7469 6320 636c 6173 7365 cteristic classe │ │ │ │ +00027e80: 7320 6172 650a 7072 6f62 6162 696c 6973 s are.probabilis │ │ │ │ +00027e90: 7469 632e 2054 6865 6f72 6574 6963 616c tic. Theoretical │ │ │ │ +00027ea0: 6c79 2c20 7468 6579 2063 616c 6375 6c61 ly, they calcula │ │ │ │ +00027eb0: 7465 2074 6865 2063 6c61 7373 6573 2063 te the classes c │ │ │ │ +00027ec0: 6f72 7265 6374 6c79 2066 6f72 2061 0a67 orrectly for a.g │ │ │ │ +00027ed0: 656e 6572 616c 2063 686f 6963 6520 6f66 eneral choice of │ │ │ │ +00027ee0: 2063 6572 7461 696e 2070 6f6c 796e 6f6d certain polynom │ │ │ │ +00027ef0: 6961 6c73 2e20 5468 6174 2069 732c 2074 ials. That is, t │ │ │ │ +00027f00: 6865 7265 2069 7320 616e 206f 7065 6e20 here is an open │ │ │ │ +00027f10: 6465 6e73 6520 5a61 7269 736b 690a 7365 dense Zariski.se │ │ │ │ +00027f20: 7420 666f 7220 7768 6963 6820 7468 6520 t for which the │ │ │ │ +00027f30: 616c 676f 7269 7468 6d20 7969 656c 6473 algorithm yields │ │ │ │ +00027f40: 2074 6865 2063 6f72 7265 6374 2063 6c61 the correct cla │ │ │ │ +00027f50: 7373 2c20 692e 652e 2c20 7468 6520 636f ss, i.e., the co │ │ │ │ +00027f60: 7272 6563 7420 636c 6173 730a 6973 2063 rrect class.is c │ │ │ │ +00027f70: 616c 6375 6c61 7465 6420 7769 7468 2070 alculated with p │ │ │ │ +00027f80: 726f 6261 6269 6c69 7479 2031 2e20 486f robability 1. Ho │ │ │ │ +00027f90: 7765 7665 722c 2073 696e 6365 2074 6865 wever, since the │ │ │ │ +00027fa0: 2069 6d70 6c65 6d65 6e74 6174 696f 6e20 implementation │ │ │ │ +00027fb0: 776f 726b 7320 6f76 6572 0a61 2064 6973 works over.a dis │ │ │ │ +00027fc0: 6372 6574 6520 7072 6f62 6162 696c 6974 crete probabilit │ │ │ │ +00027fd0: 7920 7370 6163 6520 7468 6572 6520 6973 y space there is │ │ │ │ +00027fe0: 2061 2076 6572 7920 736d 616c 6c2c 2062 a very small, b │ │ │ │ +00027ff0: 7574 206e 6f6e 2d7a 6572 6f2c 2070 726f ut non-zero, pro │ │ │ │ +00028000: 6261 6269 6c69 7479 0a6f 6620 6e6f 7420 bability.of not │ │ │ │ +00028010: 636f 6d70 7574 696e 6720 7468 6520 636f computing the co │ │ │ │ +00028020: 7272 6563 7420 636c 6173 732e 2053 6b65 rrect class. Ske │ │ │ │ +00028030: 7074 6963 616c 2075 7365 7273 2073 686f ptical users sho │ │ │ │ +00028040: 756c 6420 7265 7065 6174 2063 616c 6375 uld repeat calcu │ │ │ │ +00028050: 6c61 7469 6f6e 730a 7365 7665 7261 6c20 lations.several │ │ │ │ +00028060: 7469 6d65 7320 746f 2069 6e63 7265 6173 times to increas │ │ │ │ +00028070: 6520 7468 6520 7072 6f62 6162 696c 6974 e the probabilit │ │ │ │ +00028080: 7920 6f66 2063 6f6d 7075 7469 6e67 2074 y of computing t │ │ │ │ +00028090: 6865 2063 6f72 7265 6374 2063 6c61 7373 he correct class │ │ │ │ +000280a0: 2e0a 0a49 6e20 7468 6520 6361 7365 206f ...In the case o │ │ │ │ +000280b0: 6620 7468 6520 7379 6d62 6f6c 6963 2069 f the symbolic i │ │ │ │ +000280c0: 6d70 6c65 6d65 6e74 6174 696f 6e20 6f66 mplementation of │ │ │ │ +000280d0: 2074 6865 2050 726f 6a65 6374 6976 6544 the ProjectiveD │ │ │ │ +000280e0: 6567 7265 6520 6d65 7468 6f64 0a70 7261 egree method.pra │ │ │ │ +000280f0: 6374 6963 616c 2065 7870 6572 6965 6e63 ctical experienc │ │ │ │ +00028100: 6520 616e 6420 616c 676f 7269 7468 6d20 e and algorithm │ │ │ │ +00028110: 7465 7374 696e 6720 696e 6469 6361 7465 testing indicate │ │ │ │ +00028120: 2074 6861 7420 6120 6669 6e69 7465 2066 that a finite f │ │ │ │ +00028130: 6965 6c64 2077 6974 680a 6f76 6572 2032 ield with.over 2 │ │ │ │ +00028140: 3530 3030 2065 6c65 6d65 6e74 7320 6973 5000 elements is │ │ │ │ +00028150: 206d 6f72 6520 7468 616e 2073 7566 6669 more than suffi │ │ │ │ +00028160: 6369 656e 7420 746f 2065 7870 6563 7420 cient to expect │ │ │ │ +00028170: 6120 636f 7272 6563 7420 7265 7375 6c74 a correct result │ │ │ │ +00028180: 2077 6974 680a 6869 6768 2070 726f 6261 with.high proba │ │ │ │ +00028190: 6269 6c69 7479 2c20 692e 652e 2075 7369 bility, i.e. usi │ │ │ │ +000281a0: 6e67 2074 6865 2066 696e 6974 6520 6669 ng the finite fi │ │ │ │ +000281b0: 656c 6420 6b6b 3d5a 5a2f 3235 3037 3320 eld kk=ZZ/25073 │ │ │ │ +000281c0: 7468 6520 6578 7065 7269 6d65 6e74 616c the experimental │ │ │ │ +000281d0: 0a63 6861 6e63 6520 6f66 2066 6169 6c75 .chance of failu │ │ │ │ +000281e0: 7265 2077 6974 6820 7468 6520 5072 6f6a re with the Proj │ │ │ │ +000281f0: 6563 7469 7665 4465 6772 6565 2061 6c67 ectiveDegree alg │ │ │ │ +00028200: 6f72 6974 686d 206f 6e20 6120 7661 7269 orithm on a vari │ │ │ │ +00028210: 6574 7920 6f66 2065 7861 6d70 6c65 730a ety of examples. │ │ │ │ +00028220: 7761 7320 6c65 7373 2074 6861 6e20 312f was less than 1/ │ │ │ │ +00028230: 3230 3030 2e20 5573 696e 6720 7468 6520 2000. Using the │ │ │ │ +00028240: 6669 6e69 7465 2066 6965 6c64 206b 6b3d finite field kk= │ │ │ │ +00028250: 5a5a 2f33 3237 3439 2072 6573 756c 7465 ZZ/32749 resulte │ │ │ │ +00028260: 6420 696e 206e 6f0a 6661 696c 7572 6573 d in no.failures │ │ │ │ +00028270: 2069 6e20 6f76 6572 2031 3030 3030 2061 in over 10000 a │ │ │ │ +00028280: 7474 656d 7074 7320 6f66 2073 6576 6572 ttempts of sever │ │ │ │ +00028290: 616c 2064 6966 6665 7265 6e74 2065 7861 al different exa │ │ │ │ +000282a0: 6d70 6c65 732e 0a0a 5765 2069 6c6c 7573 mples...We illus │ │ │ │ +000282b0: 7472 6174 6520 7468 6520 7072 6f62 6162 trate the probab │ │ │ │ +000282c0: 696c 6973 7469 6320 6265 6861 7669 6f75 ilistic behaviou │ │ │ │ +000282d0: 7220 7769 7468 2061 6e20 6578 616d 706c r with an exampl │ │ │ │ +000282e0: 6520 7768 6572 6520 7468 6520 6368 6f73 e where the chos │ │ │ │ +000282f0: 656e 0a72 616e 646f 6d20 7365 6564 206c en.random seed l │ │ │ │ +00028300: 6561 6473 2074 6f20 6120 7772 6f6e 6720 eads to a wrong │ │ │ │ +00028310: 7265 7375 6c74 2069 6e20 7468 6520 6669 result in the fi │ │ │ │ +00028320: 7273 7420 6361 6c63 756c 6174 696f 6e2e rst calculation. │ │ │ │ +00028330: 0a0a 2b2d 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 2d2b 0a7c 6931 203a 2073 6574 5261 ---+.|i1 : setRa │ │ │ │ +00028370: 6e64 6f6d 5365 6564 2031 3231 3b20 2020 ndomSeed 121; │ │ │ │ +00028380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028390: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574 |.| -- set │ │ │ │ +000283a0: 7469 6e67 2072 616e 646f 6d20 7365 6564 ting random seed │ │ │ │ +000283b0: 2074 6f20 3132 3120 2020 2020 2020 2020 to 121 │ │ │ │ +000283c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +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 2b0a 7c69 ------------+.|i │ │ │ │ +00028400: 3220 3a20 5220 3d20 5151 5b78 2c79 2c7a 2 : R = QQ[x,y,z │ │ │ │ +00028410: 2c77 5d20 2020 2020 2020 2020 2020 2020 ,w] │ │ │ │ 00028420: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028430: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00028440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028460: 2d2d 2b0a 7c69 3220 3a20 5220 3d20 5151 --+.|i2 : R = QQ │ │ │ │ -00028470: 5b78 2c79 2c7a 2c77 5d20 2020 2020 2020 [x,y,z,w] │ │ │ │ +00028430: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00028440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028460: 2020 7c0a 7c6f 3220 3d20 5220 2020 2020 |.|o2 = R │ │ │ │ +00028470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028490: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000284a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000284b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000284c0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -000284d0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +000284c0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ +000284d0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ 000284e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000284f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00028500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028520: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028530: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c |o2 : Polynomial │ │ │ │ -00028540: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ -00028550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028560: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00028570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028590: 2d2d 2d2d 2b0a 7c69 3320 3a20 4920 3d20 ----+.|i3 : I = │ │ │ │ -000285a0: 6d69 6e6f 7273 2832 2c6d 6174 7269 787b minors(2,matrix{ │ │ │ │ -000285b0: 7b78 2c79 2c7a 7d2c 7b79 2c7a 2c77 7d7d {x,y,z},{y,z,w}} │ │ │ │ -000285c0: 2920 2020 2020 207c 0a7c 2020 2020 2020 ) |.| │ │ │ │ -000285d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000285e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000285f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00028600: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +000284f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00028500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00028530: 7c69 3320 3a20 4920 3d20 6d69 6e6f 7273 |i3 : I = minors │ │ │ │ +00028540: 2832 2c6d 6174 7269 787b 7b78 2c79 2c7a (2,matrix{{x,y,z │ │ │ │ +00028550: 7d2c 7b79 2c7a 2c77 7d7d 2920 2020 2020 },{y,z,w}}) │ │ │ │ +00028560: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028590: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000285a0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +000285b0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +000285c0: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ +000285d0: 6465 616c 2028 2d20 7920 202b 2078 2a7a deal (- y + x*z │ │ │ │ +000285e0: 2c20 2d20 792a 7a20 2b20 782a 772c 202d , - y*z + x*w, - │ │ │ │ +000285f0: 207a 2020 2b20 792a 7729 7c0a 7c20 2020 z + y*w)|.| │ │ │ │ +00028600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028620: 2020 2020 2032 2020 2020 2020 207c 0a7c 2 |.| │ │ │ │ -00028630: 6f33 203d 2069 6465 616c 2028 2d20 7920 o3 = ideal (- y │ │ │ │ -00028640: 202b 2078 2a7a 2c20 2d20 792a 7a20 2b20 + x*z, - y*z + │ │ │ │ -00028650: 782a 772c 202d 207a 2020 2b20 792a 7729 x*w, - z + y*w) │ │ │ │ -00028660: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00028670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028690: 2020 207c 0a7c 6f33 203a 2049 6465 616c |.|o3 : Ideal │ │ │ │ -000286a0: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ -000286b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000286c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -000286d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000286e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000286f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -00028700: 2043 6865 726e 2028 492c 436f 6d70 4d65 Chern (I,CompMe │ │ │ │ -00028710: 7468 6f64 3d3e 506e 5265 7369 6475 616c thod=>PnResidual │ │ │ │ -00028720: 2920 2020 2020 2020 2020 2020 7c0a 7c20 ) |.| │ │ │ │ -00028730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028620: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00028630: 6f33 203a 2049 6465 616c 206f 6620 5220 o3 : Ideal of R │ │ │ │ +00028640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028660: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00028670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028690: 2d2d 2d2b 0a7c 6934 203a 2043 6865 726e ---+.|i4 : Chern │ │ │ │ +000286a0: 2028 492c 436f 6d70 4d65 7468 6f64 3d3e (I,CompMethod=> │ │ │ │ +000286b0: 506e 5265 7369 6475 616c 2920 2020 2020 PnResidual) │ │ │ │ +000286c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000286d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000286e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000286f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00028700: 2020 2033 2020 2020 2032 2020 2020 2020 3 2 │ │ │ │ +00028710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028720: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00028730: 3420 3d20 3248 2020 2b20 3348 2020 2020 4 = 2H + 3H │ │ │ │ 00028740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028750: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028760: 0a7c 2020 2020 2020 2033 2020 2020 2032 .| 3 2 │ │ │ │ +00028760: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028790: 2020 7c0a 7c6f 3420 3d20 3248 2020 2b20 |.|o4 = 2H + │ │ │ │ -000287a0: 3348 2020 2020 2020 2020 2020 2020 2020 3H │ │ │ │ +00028790: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20 |.| ZZ[H] │ │ │ │ +000287a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000287b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000287c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000287d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000287c0: 2020 2020 207c 0a7c 6f34 203a 202d 2d2d |.|o4 : --- │ │ │ │ +000287d0: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ 000287e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000287f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00028800: 5a5a 5b48 5d20 2020 2020 2020 2020 2020 ZZ[H] │ │ │ │ +00028800: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 00028810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028820: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ -00028830: 203a 202d 2d2d 2d2d 2020 2020 2020 2020 : ----- │ │ │ │ +00028820: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00028830: 2020 2020 2048 2020 2020 2020 2020 2020 H │ │ │ │ 00028840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028850: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028860: 7c20 2020 2020 2020 2034 2020 2020 2020 | 4 │ │ │ │ -00028870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028890: 207c 0a7c 2020 2020 2020 2048 2020 2020 |.| H │ │ │ │ -000288a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000288b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000288c0: 2020 2020 7c0a 2b2d 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 2d2b 0a7c 6935 203a 2043 -------+.|i5 : C │ │ │ │ -00028900: 6865 726e 2028 492c 436f 6d70 4d65 7468 hern (I,CompMeth │ │ │ │ -00028910: 6f64 3d3e 506e 5265 7369 6475 616c 2920 od=>PnResidual) │ │ │ │ -00028920: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00028930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028860: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00028870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028890: 2d2b 0a7c 6935 203a 2043 6865 726e 2028 -+.|i5 : Chern ( │ │ │ │ +000288a0: 492c 436f 6d70 4d65 7468 6f64 3d3e 506e I,CompMethod=>Pn │ │ │ │ +000288b0: 5265 7369 6475 616c 2920 2020 2020 2020 Residual) │ │ │ │ +000288c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000288d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00028900: 2033 2020 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ +00028910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028920: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ +00028930: 3d20 3248 2020 2b20 3348 2020 2020 2020 = 2H + 3H │ │ │ │ 00028940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028950: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028960: 2020 2020 2020 2033 2020 2020 2032 2020 3 2 │ │ │ │ +00028960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028990: 7c0a 7c6f 3520 3d20 3248 2020 2b20 3348 |.|o5 = 2H + 3H │ │ │ │ +00028990: 7c0a 7c20 2020 2020 5a5a 5b48 5d20 2020 |.| ZZ[H] │ │ │ │ 000289a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000289b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000289c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000289c0: 2020 207c 0a7c 6f35 203a 202d 2d2d 2d2d |.|o5 : ----- │ │ │ │ 000289d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000289e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000289f0: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a |.| ZZ │ │ │ │ -00028a00: 5b48 5d20 2020 2020 2020 2020 2020 2020 [H] │ │ │ │ +000289f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00028a00: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 00028a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a20: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ -00028a30: 202d 2d2d 2d2d 2020 2020 2020 2020 2020 ----- │ │ │ │ +00028a20: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00028a30: 2020 2048 2020 2020 2020 2020 2020 2020 H │ │ │ │ 00028a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00028a60: 2020 2020 2020 2034 2020 2020 2020 2020 4 │ │ │ │ -00028a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028a90: 0a7c 2020 2020 2020 2048 2020 2020 2020 .| H │ │ │ │ -00028aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ac0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00028ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028af0: 2d2d 2d2d 2d2b 0a7c 6936 203a 2043 6865 -----+.|i6 : Che │ │ │ │ -00028b00: 726e 2028 492c 436f 6d70 4d65 7468 6f64 rn (I,CompMethod │ │ │ │ -00028b10: 3d3e 506e 5265 7369 6475 616c 2920 2020 =>PnResidual) │ │ │ │ -00028b20: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00028b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028a50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00028a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00028a90: 0a7c 6936 203a 2043 6865 726e 2028 492c .|i6 : Chern (I, │ │ │ │ +00028aa0: 436f 6d70 4d65 7468 6f64 3d3e 506e 5265 CompMethod=>PnRe │ │ │ │ +00028ab0: 7369 6475 616c 2920 2020 2020 2020 2020 sidual) │ │ │ │ +00028ac0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028af0: 2020 2020 207c 0a7c 2020 2020 2020 2033 |.| 3 │ │ │ │ +00028b00: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00028b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028b20: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ +00028b30: 3248 2020 2b20 3348 2020 2020 2020 2020 2H + 3H │ │ │ │ 00028b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00028b60: 2020 2020 2033 2020 2020 2032 2020 2020 3 2 │ │ │ │ +00028b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028b90: 7c6f 3620 3d20 3248 2020 2b20 3348 2020 |o6 = 2H + 3H │ │ │ │ +00028b90: 7c20 2020 2020 5a5a 5b48 5d20 2020 2020 | ZZ[H] │ │ │ │ 00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028bc0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028bc0: 207c 0a7c 6f36 203a 202d 2d2d 2d2d 2020 |.|o6 : ----- │ │ │ │ 00028bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028bf0: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b48 |.| ZZ[H │ │ │ │ -00028c00: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +00028bf0: 2020 2020 7c0a 7c20 2020 2020 2020 2034 |.| 4 │ │ │ │ +00028c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c20: 2020 2020 2020 207c 0a7c 6f36 203a 202d |.|o6 : - │ │ │ │ -00028c30: 2d2d 2d2d 2020 2020 2020 2020 2020 2020 ---- │ │ │ │ +00028c20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00028c30: 2048 2020 2020 2020 2020 2020 2020 2020 H │ │ │ │ 00028c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00028c60: 2020 2020 2034 2020 2020 2020 2020 2020 4 │ │ │ │ -00028c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028c90: 2020 2020 2020 2048 2020 2020 2020 2020 H │ │ │ │ -00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028cc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00028cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028cf0: 2d2d 2d2b 0a7c 6937 203a 2043 6865 726e ---+.|i7 : Chern │ │ │ │ -00028d00: 2849 2c43 6f6d 704d 6574 686f 643d 3e50 (I,CompMethod=>P │ │ │ │ -00028d10: 726f 6a65 6374 6976 6544 6567 7265 6529 rojectiveDegree) │ │ │ │ -00028d20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00028d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c50: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00028c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00028c90: 6937 203a 2043 6865 726e 2849 2c43 6f6d i7 : Chern(I,Com │ │ │ │ +00028ca0: 704d 6574 686f 643d 3e50 726f 6a65 6374 pMethod=>Project │ │ │ │ +00028cb0: 6976 6544 6567 7265 6529 2020 2020 2020 iveDegree) │ │ │ │ +00028cc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028cf0: 2020 207c 0a7c 2020 2020 2020 2033 2020 |.| 3 │ │ │ │ +00028d00: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00028d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028d20: 2020 2020 2020 7c0a 7c6f 3720 3d20 3268 |.|o7 = 2h │ │ │ │ +00028d30: 2020 2b20 3368 2020 2020 2020 2020 2020 + 3h │ │ │ │ 00028d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00028d60: 2020 2033 2020 2020 2032 2020 2020 2020 3 2 │ │ │ │ +00028d60: 2020 2031 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ 00028d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00028d90: 3720 3d20 3268 2020 2b20 3368 2020 2020 7 = 2h + 3h │ │ │ │ +00028d80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00028d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028db0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028dc0: 0a7c 2020 2020 2020 2031 2020 2020 2031 .| 1 1 │ │ │ │ +00028dc0: 0a7c 2020 2020 205a 5a5b 6820 5d20 2020 .| ZZ[h ] │ │ │ │ 00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028df0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028df0: 2020 7c0a 7c20 2020 2020 2020 2020 3120 |.| 1 │ │ │ │ 00028e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e20: 2020 2020 207c 0a7c 2020 2020 205a 5a5b |.| ZZ[ │ │ │ │ -00028e30: 6820 5d20 2020 2020 2020 2020 2020 2020 h ] │ │ │ │ +00028e20: 2020 2020 207c 0a7c 6f37 203a 202d 2d2d |.|o7 : --- │ │ │ │ +00028e30: 2d2d 2d20 2020 2020 2020 2020 2020 2020 --- │ │ │ │ 00028e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e50: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00028e60: 2020 2020 3120 2020 2020 2020 2020 2020 1 │ │ │ │ +00028e60: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 00028e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e80: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -00028e90: 203a 202d 2d2d 2d2d 2d20 2020 2020 2020 : ------ │ │ │ │ +00028e80: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00028e90: 2020 2020 2068 2020 2020 2020 2020 2020 h │ │ │ │ 00028ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028eb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028ec0: 7c20 2020 2020 2020 2034 2020 2020 2020 | 4 │ │ │ │ +00028ec0: 7c20 2020 2020 2020 2031 2020 2020 2020 | 1 │ │ │ │ 00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ef0: 207c 0a7c 2020 2020 2020 2068 2020 2020 |.| h │ │ │ │ -00028f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f20: 2020 2020 7c0a 7c20 2020 2020 2020 2031 |.| 1 │ │ │ │ -00028f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00028ef0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00028f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f20: 2d2d 2d2d 2b0a 2d2d 2d2d 2d2d 2d2d 2d2d ----+.---------- │ │ │ │ +00028f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f40: 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 2d2d 2d2d ----------+.---- │ │ │ │ -00028f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -00028fe0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -00028ff0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -00029000: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -00029010: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -00029020: 322d 312e 3235 2e31 312b 6473 2f4d 322f 2-1.25.11+ds/M2/ │ │ │ │ -00029030: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -00029040: 6573 2f0a 4368 6172 6163 7465 7269 7374 es/.Characterist │ │ │ │ -00029050: 6963 436c 6173 7365 732e 6d32 3a32 3337 icClasses.m2:237 │ │ │ │ -00029060: 383a 302e 0a1f 0a46 696c 653a 2043 6861 8:0....File: Cha │ │ │ │ -00029070: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ -00029080: 6573 2e69 6e66 6f2c 204e 6f64 653a 2053 es.info, Node: S │ │ │ │ -00029090: 6567 7265 2c20 4e65 7874 3a20 546f 7269 egre, Next: Tori │ │ │ │ -000290a0: 6343 686f 7752 696e 672c 2050 7265 763a cChowRing, Prev: │ │ │ │ -000290b0: 2070 726f 6261 6269 6c69 7374 6963 2061 probabilistic a │ │ │ │ -000290c0: 6c67 6f72 6974 686d 2c20 5570 3a20 546f lgorithm, Up: To │ │ │ │ -000290d0: 700a 0a53 6567 7265 202d 2d20 5468 6520 p..Segre -- The │ │ │ │ -000290e0: 5365 6772 6520 636c 6173 7320 6f66 2061 Segre class of a │ │ │ │ -000290f0: 2073 7562 7363 6865 6d65 0a2a 2a2a 2a2a subscheme.***** │ │ │ │ -00029100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00029110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00029120: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ -00029130: 2020 2020 2020 2020 5365 6772 6520 490a Segre I. │ │ │ │ -00029140: 2020 2020 2020 2020 5365 6772 6528 412c Segre(A, │ │ │ │ -00029150: 4929 0a20 2020 2020 2020 2053 6567 7265 I). Segre │ │ │ │ -00029160: 2858 2c4a 290a 2020 2020 2020 2020 5365 (X,J). Se │ │ │ │ -00029170: 6772 6528 4368 2c58 2c4a 290a 2020 2a20 gre(Ch,X,J). * │ │ │ │ -00029180: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -00029190: 492c 2061 6e20 2a6e 6f74 6520 6964 6561 I, an *note idea │ │ │ │ -000291a0: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ -000291b0: 2949 6465 616c 2c2c 2061 206d 756c 7469 )Ideal,, a multi │ │ │ │ -000291c0: 2d68 6f6d 6f67 656e 656f 7573 2069 6465 -homogeneous ide │ │ │ │ -000291d0: 616c 2069 6e20 610a 2020 2020 2020 2020 al in a. │ │ │ │ -000291e0: 6772 6164 6564 2070 6f6c 796e 6f6d 6961 graded polynomia │ │ │ │ -000291f0: 6c20 7269 6e67 206f 7665 7220 6120 6669 l ring over a fi │ │ │ │ -00029200: 656c 6420 6465 6669 6e69 6e67 2061 2063 eld defining a c │ │ │ │ -00029210: 6c6f 7365 6420 7375 6273 6368 656d 6520 losed subscheme │ │ │ │ -00029220: 5620 6f66 0a20 2020 2020 2020 205c 5050 V of. \PP │ │ │ │ -00029230: 5e7b 6e5f 317d 782e 2e2e 785c 5050 5e7b ^{n_1}x...x\PP^{ │ │ │ │ -00029240: 6e5f 6d7d 0a20 2020 2020 202a 2041 2c20 n_m}. * A, │ │ │ │ -00029250: 6120 2a6e 6f74 6520 7175 6f74 6965 6e74 a *note quotient │ │ │ │ -00029260: 2072 696e 673a 2028 4d61 6361 756c 6179 ring: (Macaulay │ │ │ │ -00029270: 3244 6f63 2951 756f 7469 656e 7452 696e 2Doc)QuotientRin │ │ │ │ -00029280: 672c 2c0a 2020 2020 2020 2020 413d 5c5a g,,. A=\Z │ │ │ │ -00029290: 5a5b 685f 312c 2e2e 2e2c 685f 6d5d 2f28 Z[h_1,...,h_m]/( │ │ │ │ -000292a0: 685f 315e 7b6e 5f31 2b31 7d2c 2e2e 2e2c h_1^{n_1+1},..., │ │ │ │ -000292b0: 685f 6d5e 7b6e 5f6d 2b31 7d29 2071 756f h_m^{n_m+1}) quo │ │ │ │ -000292c0: 7469 656e 7420 7269 6e67 0a20 2020 2020 tient ring. │ │ │ │ -000292d0: 2020 2072 6570 7265 7365 6e74 696e 6720 representing │ │ │ │ -000292e0: 7468 6520 4368 6f77 2072 696e 6720 6f66 the Chow ring of │ │ │ │ -000292f0: 205c 5050 5e7b 6e5f 317d 782e 2e2e 785c \PP^{n_1}x...x\ │ │ │ │ -00029300: 5050 5e7b 6e5f 6d7d 2c20 7468 6973 2072 PP^{n_m}, this r │ │ │ │ -00029310: 696e 6720 7368 6f75 6c64 0a20 2020 2020 ing should. │ │ │ │ -00029320: 2020 2062 6520 6275 696c 7420 7573 696e be built usin │ │ │ │ -00029330: 6720 7468 6520 2a6e 6f74 6520 4368 6f77 g the *note Chow │ │ │ │ -00029340: 5269 6e67 3a20 4368 6f77 5269 6e67 2c20 Ring: ChowRing, │ │ │ │ -00029350: 636f 6d6d 616e 640a 2020 2020 2020 2a20 command. * │ │ │ │ -00029360: 4a2c 2061 6e20 2a6e 6f74 6520 6964 6561 J, an *note idea │ │ │ │ -00029370: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ -00029380: 2949 6465 616c 2c2c 2069 6e20 7468 6520 )Ideal,, in the │ │ │ │ -00029390: 6772 6164 6564 2070 6f6c 796e 6f6d 6961 graded polynomia │ │ │ │ -000293a0: 6c20 7269 6e67 0a20 2020 2020 2020 2077 l ring. w │ │ │ │ -000293b0: 6869 6368 2069 7320 636f 6f72 6469 6e61 hich is coordina │ │ │ │ -000293c0: 7465 2072 696e 6720 6f66 2074 6865 204e te ring of the N │ │ │ │ -000293d0: 6f72 6d61 6c20 546f 7269 6320 5661 7269 ormal Toric Vari │ │ │ │ -000293e0: 6574 7920 580a 2020 2020 2020 2a20 582c ety X. * X, │ │ │ │ -000293f0: 2061 202a 6e6f 7465 206e 6f72 6d61 6c20 a *note normal │ │ │ │ -00029400: 746f 7269 6320 7661 7269 6574 793a 0a20 toric variety:. │ │ │ │ -00029410: 2020 2020 2020 2028 4e6f 726d 616c 546f (NormalTo │ │ │ │ -00029420: 7269 6356 6172 6965 7469 6573 294e 6f72 ricVarieties)Nor │ │ │ │ -00029430: 6d61 6c54 6f72 6963 5661 7269 6574 792c malToricVariety, │ │ │ │ -00029440: 2c20 7768 6963 6820 6973 2074 6865 2061 , which is the a │ │ │ │ -00029450: 6d62 6965 6e74 2073 7061 6365 0a20 2020 mbient space. │ │ │ │ -00029460: 2020 2020 2077 6869 6368 2063 6f6e 7461 which conta │ │ │ │ -00029470: 696e 7320 5628 4a29 0a20 2020 2020 202a ins V(J). * │ │ │ │ -00029480: 2043 682c 2061 202a 6e6f 7465 2071 756f Ch, a *note quo │ │ │ │ -00029490: 7469 656e 7420 7269 6e67 3a20 284d 6163 tient ring: (Mac │ │ │ │ -000294a0: 6175 6c61 7932 446f 6329 5175 6f74 6965 aulay2Doc)Quotie │ │ │ │ -000294b0: 6e74 5269 6e67 2c2c 2074 6865 2043 686f ntRing,, the Cho │ │ │ │ -000294c0: 7720 7269 6e67 0a20 2020 2020 2020 206f w ring. o │ │ │ │ -000294d0: 6620 7468 6520 746f 7269 6320 7661 7269 f the toric vari │ │ │ │ -000294e0: 6574 7920 582c 2043 683d 2872 696e 6720 ety X, Ch=(ring │ │ │ │ -000294f0: 4a29 2f28 5352 2b4c 5229 2077 6865 7265 J)/(SR+LR) where │ │ │ │ -00029500: 2053 5220 6973 2074 6865 0a20 2020 2020 SR is the. │ │ │ │ -00029510: 2020 2053 7461 6e6c 6579 2d52 6569 736e Stanley-Reisn │ │ │ │ -00029520: 6572 2069 6465 616c 206f 6620 7468 6520 er ideal of the │ │ │ │ -00029530: 6661 6e20 6465 6669 6e69 6e67 2058 2061 fan defining X a │ │ │ │ -00029540: 6e64 204c 5220 6973 2074 6865 206c 696e nd LR is the lin │ │ │ │ -00029550: 6561 720a 2020 2020 2020 2020 7265 6c61 ear. rela │ │ │ │ -00029560: 7469 6f6e 7320 6964 6561 6c2c 2074 6869 tions ideal, thi │ │ │ │ -00029570: 7320 7269 6e67 2073 686f 756c 6420 6265 s ring should be │ │ │ │ -00029580: 2062 7569 6c74 2075 7369 6e67 2074 6865 built using the │ │ │ │ -00029590: 202a 6e6f 7465 0a20 2020 2020 2020 2054 *note. T │ │ │ │ -000295a0: 6f72 6963 4368 6f77 5269 6e67 3a20 546f oricChowRing: To │ │ │ │ -000295b0: 7269 6343 686f 7752 696e 672c 2063 6f6d ricChowRing, com │ │ │ │ -000295c0: 6d61 6e64 0a20 202a 202a 6e6f 7465 204f mand. * *note O │ │ │ │ -000295d0: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ -000295e0: 284d 6163 6175 6c61 7932 446f 6329 7573 (Macaulay2Doc)us │ │ │ │ -000295f0: 696e 6720 6675 6e63 7469 6f6e 7320 7769 ing functions wi │ │ │ │ -00029600: 7468 206f 7074 696f 6e61 6c20 696e 7075 th optional inpu │ │ │ │ -00029610: 7473 2c3a 0a20 2020 2020 202a 2043 6f6d ts,:. * Com │ │ │ │ -00029620: 704d 6574 686f 6420 286d 6973 7369 6e67 pMethod (missing │ │ │ │ -00029630: 2064 6f63 756d 656e 7461 7469 6f6e 2920 documentation) │ │ │ │ -00029640: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ -00029650: 7661 6c75 650a 2020 2020 2020 2020 5072 value. Pr │ │ │ │ -00029660: 6f6a 6563 7469 7665 4465 6772 6565 2c20 ojectiveDegree, │ │ │ │ -00029670: 5072 6f6a 6563 7469 7665 4465 6772 6565 ProjectiveDegree │ │ │ │ -00029680: 2c20 7468 6973 2061 6c67 6f72 6974 686d , this algorithm │ │ │ │ -00029690: 206d 6179 2062 6520 7573 6564 2066 6f72 may be used for │ │ │ │ -000296a0: 0a20 2020 2020 2020 2073 7562 7363 6865 . subsche │ │ │ │ -000296b0: 6d65 7320 6f66 2061 6e79 2061 7070 6c69 mes of any appli │ │ │ │ -000296c0: 6361 626c 6520 746f 7269 6320 7661 7269 cable toric vari │ │ │ │ -000296d0: 6574 7920 2874 6869 7320 6d61 7920 6265 ety (this may be │ │ │ │ -000296e0: 2063 6865 636b 6564 2075 7369 6e67 0a20 checked using. │ │ │ │ -000296f0: 2020 2020 2020 2074 6865 202a 6e6f 7465 the *note │ │ │ │ -00029700: 2043 6865 636b 546f 7269 6356 6172 6965 CheckToricVarie │ │ │ │ -00029710: 7479 5661 6c69 643a 2043 6865 636b 546f tyValid: CheckTo │ │ │ │ -00029720: 7269 6356 6172 6965 7479 5661 6c69 642c ricVarietyValid, │ │ │ │ -00029730: 2063 6f6d 6d61 6e64 290a 2020 2020 2020 command). │ │ │ │ -00029740: 2a20 436f 6d70 4d65 7468 6f64 2028 6d69 * CompMethod (mi │ │ │ │ -00029750: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ -00029760: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ -00029770: 6175 6c74 2076 616c 7565 0a20 2020 2020 ault value. │ │ │ │ -00029780: 2020 2050 726f 6a65 6374 6976 6544 6567 ProjectiveDeg │ │ │ │ -00029790: 7265 652c 2050 6e52 6573 6964 7561 6c2c ree, PnResidual, │ │ │ │ -000297a0: 2074 6869 7320 616c 676f 7269 7468 6d20 this algorithm │ │ │ │ -000297b0: 6d61 7920 6265 2075 7365 6420 666f 7220 may be used for │ │ │ │ -000297c0: 7375 6273 6368 656d 6573 0a20 2020 2020 subschemes. │ │ │ │ -000297d0: 2020 206f 6620 5c50 505e 6e20 6f6e 6c79 of \PP^n only │ │ │ │ -000297e0: 0a20 2020 2020 202a 204f 7574 7075 7420 . * Output │ │ │ │ -000297f0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ -00029800: 7661 6c75 6520 4368 6f77 5269 6e67 456c value ChowRingEl │ │ │ │ -00029810: 656d 656e 742c 2043 686f 7752 696e 6745 ement, ChowRingE │ │ │ │ -00029820: 6c65 6d65 6e74 2c20 7265 7475 726e 730a lement, returns. │ │ │ │ -00029830: 2020 2020 2020 2020 6120 5269 6e67 456c a RingEl │ │ │ │ -00029840: 656d 656e 7420 696e 2074 6865 2043 686f ement in the Cho │ │ │ │ -00029850: 7720 7269 6e67 206f 6620 7468 6520 6170 w ring of the ap │ │ │ │ -00029860: 7072 6f70 7269 6174 6520 616d 6269 656e propriate ambien │ │ │ │ -00029870: 7420 7370 6163 650a 2020 2020 2020 2a20 t space. * │ │ │ │ -00029880: 4f75 7470 7574 203d 3e20 2e2e 2e2c 2064 Output => ..., d │ │ │ │ -00029890: 6566 6175 6c74 2076 616c 7565 2043 686f efault value Cho │ │ │ │ -000298a0: 7752 696e 6745 6c65 6d65 6e74 2c20 4861 wRingElement, Ha │ │ │ │ -000298b0: 7368 466f 726d 2c20 4861 7368 466f 726d shForm, HashForm │ │ │ │ -000298c0: 0a20 2020 2020 2020 2072 6574 7572 6e73 . returns │ │ │ │ -000298d0: 2061 204d 7574 6162 6c65 4861 7368 5461 a MutableHashTa │ │ │ │ -000298e0: 626c 6520 636f 6e74 6169 6e69 6e67 2074 ble containing t │ │ │ │ -000298f0: 6865 2066 6f6c 6c6f 7769 6e67 206b 6579 he following key │ │ │ │ -00029900: 733a 2022 4722 2028 7468 650a 2020 2020 s: "G" (the. │ │ │ │ -00029910: 2020 2020 706f 6c79 6e6f 6d69 616c 2077 polynomial w │ │ │ │ -00029920: 6974 6820 636f 6566 6669 6369 656e 7473 ith coefficients │ │ │ │ -00029930: 206f 6620 7468 6520 6879 7065 7270 6c61 of the hyperpla │ │ │ │ -00029940: 6e65 2063 6c61 7373 6573 2072 6570 7265 ne classes repre │ │ │ │ -00029950: 7365 6e74 696e 6720 7468 650a 2020 2020 senting the. │ │ │ │ -00029960: 2020 2020 7072 6f6a 6563 7469 7665 2064 projective d │ │ │ │ -00029970: 6567 7265 6573 292c 2022 476c 6973 7422 egrees), "Glist" │ │ │ │ -00029980: 2028 7468 6520 6c69 7374 2066 6f72 6d20 (the list form │ │ │ │ -00029990: 6f66 2022 4722 2920 2c20 2253 6567 7265 of "G") , "Segre │ │ │ │ -000299a0: 2220 2874 6865 0a20 2020 2020 2020 2074 " (the. t │ │ │ │ -000299b0: 6f74 616c 2053 6567 7265 2063 6c61 7373 otal Segre class │ │ │ │ -000299c0: 206f 6620 7468 6520 696e 7075 7429 2c22 of the input)," │ │ │ │ -000299d0: 5365 6772 654c 6973 7422 2028 7468 6520 SegreList" (the │ │ │ │ -000299e0: 6c69 7374 2066 6f72 6d20 6f66 2022 5365 list form of "Se │ │ │ │ -000299f0: 6772 6522 290a 2020 2a20 4f75 7470 7574 gre"). * Output │ │ │ │ -00029a00: 733a 0a20 2020 2020 202a 2061 202a 6e6f s:. * a *no │ │ │ │ -00029a10: 7465 2072 696e 6720 656c 656d 656e 743a te ring element: │ │ │ │ -00029a20: 2028 4d61 6361 756c 6179 3244 6f63 2952 (Macaulay2Doc)R │ │ │ │ -00029a30: 696e 6745 6c65 6d65 6e74 2c2c 2074 6865 ingElement,, the │ │ │ │ -00029a40: 2070 7573 6866 6f72 7761 7264 206f 660a pushforward of. │ │ │ │ -00029a50: 2020 2020 2020 2020 7468 6520 746f 7461 the tota │ │ │ │ -00029a60: 6c20 5365 6772 6520 636c 6173 7320 6f66 l Segre class of │ │ │ │ -00029a70: 2074 6865 2073 6368 656d 6520 5620 6465 the scheme V de │ │ │ │ -00029a80: 6669 6e65 6420 6279 2074 6865 2069 6e70 fined by the inp │ │ │ │ -00029a90: 7574 2069 6465 616c 2074 6f20 7468 650a ut ideal to the. │ │ │ │ -00029aa0: 2020 2020 2020 2020 6170 7072 6f70 7269 appropri │ │ │ │ -00029ab0: 6174 6520 4368 6f77 2072 696e 670a 0a44 ate Chow ring..D │ │ │ │ -00029ac0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -00029ad0: 3d3d 3d3d 3d3d 0a0a 466f 7220 6120 7375 ======..For a su │ │ │ │ -00029ae0: 6273 6368 656d 6520 5620 6f66 2061 6e20 bscheme V of an │ │ │ │ -00029af0: 6170 706c 6963 6162 6c65 2074 6f72 6963 applicable toric │ │ │ │ -00029b00: 2076 6172 6965 7479 2058 2074 6869 7320 variety X this │ │ │ │ -00029b10: 636f 6d6d 616e 6420 636f 6d70 7574 6573 command computes │ │ │ │ -00029b20: 2074 6865 0a70 7573 682d 666f 7277 6172 the.push-forwar │ │ │ │ -00029b30: 6420 6f66 2074 6865 2074 6f74 616c 2053 d of the total S │ │ │ │ -00029b40: 6567 7265 2063 6c61 7373 2073 2856 2c58 egre class s(V,X │ │ │ │ -00029b50: 2920 6f66 2056 2069 6e20 5820 746f 2074 ) of V in X to t │ │ │ │ -00029b60: 6865 2043 686f 7720 7269 6e67 206f 6620 he Chow ring of │ │ │ │ -00029b70: 582e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d X...+----------- │ │ │ │ -00029b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029ba0: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e --+.|i1 : setRan │ │ │ │ -00029bb0: 646f 6d53 6565 6420 3732 3b20 2020 2020 domSeed 72; │ │ │ │ -00029bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029bd0: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67 |.| -- setting │ │ │ │ -00029be0: 2072 616e 646f 6d20 7365 6564 2074 6f20 random seed to │ │ │ │ -00029bf0: 3732 2020 2020 2020 2020 2020 2020 2020 72 │ │ │ │ -00029c00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00029c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029c30: 2d2d 2b0a 7c69 3220 3a20 5220 3d20 5a5a --+.|i2 : R = ZZ │ │ │ │ -00029c40: 2f33 3237 3439 5b77 2c79 2c7a 5d20 2020 /32749[w,y,z] │ │ │ │ -00029c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028f70: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +00028f80: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +00028f90: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +00028fa0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +00028fb0: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ +00028fc0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+ds/M2/Macaul │ │ │ │ +00028fd0: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ +00028fe0: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ +00028ff0: 7365 732e 6d32 3a32 3337 383a 302e 0a1f ses.m2:2378:0... │ │ │ │ +00029000: 0a46 696c 653a 2043 6861 7261 6374 6572 .File: Character │ │ │ │ +00029010: 6973 7469 6343 6c61 7373 6573 2e69 6e66 isticClasses.inf │ │ │ │ +00029020: 6f2c 204e 6f64 653a 2053 6567 7265 2c20 o, Node: Segre, │ │ │ │ +00029030: 4e65 7874 3a20 546f 7269 6343 686f 7752 Next: ToricChowR │ │ │ │ +00029040: 696e 672c 2050 7265 763a 2070 726f 6261 ing, Prev: proba │ │ │ │ +00029050: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ +00029060: 686d 2c20 5570 3a20 546f 700a 0a53 6567 hm, Up: Top..Seg │ │ │ │ +00029070: 7265 202d 2d20 5468 6520 5365 6772 6520 re -- The Segre │ │ │ │ +00029080: 636c 6173 7320 6f66 2061 2073 7562 7363 class of a subsc │ │ │ │ +00029090: 6865 6d65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a heme.*********** │ │ │ │ +000290a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000290b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +000290c0: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +000290d0: 2020 5365 6772 6520 490a 2020 2020 2020 Segre I. │ │ │ │ +000290e0: 2020 5365 6772 6528 412c 4929 0a20 2020 Segre(A,I). │ │ │ │ +000290f0: 2020 2020 2053 6567 7265 2858 2c4a 290a Segre(X,J). │ │ │ │ +00029100: 2020 2020 2020 2020 5365 6772 6528 4368 Segre(Ch │ │ │ │ +00029110: 2c58 2c4a 290a 2020 2a20 496e 7075 7473 ,X,J). * Inputs │ │ │ │ +00029120: 3a0a 2020 2020 2020 2a20 492c 2061 6e20 :. * I, an │ │ │ │ +00029130: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61 *note ideal: (Ma │ │ │ │ +00029140: 6361 756c 6179 3244 6f63 2949 6465 616c caulay2Doc)Ideal │ │ │ │ +00029150: 2c2c 2061 206d 756c 7469 2d68 6f6d 6f67 ,, a multi-homog │ │ │ │ +00029160: 656e 656f 7573 2069 6465 616c 2069 6e20 eneous ideal in │ │ │ │ +00029170: 610a 2020 2020 2020 2020 6772 6164 6564 a. graded │ │ │ │ +00029180: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ +00029190: 206f 7665 7220 6120 6669 656c 6420 6465 over a field de │ │ │ │ +000291a0: 6669 6e69 6e67 2061 2063 6c6f 7365 6420 fining a closed │ │ │ │ +000291b0: 7375 6273 6368 656d 6520 5620 6f66 0a20 subscheme V of. │ │ │ │ +000291c0: 2020 2020 2020 205c 5050 5e7b 6e5f 317d \PP^{n_1} │ │ │ │ +000291d0: 782e 2e2e 785c 5050 5e7b 6e5f 6d7d 0a20 x...x\PP^{n_m}. │ │ │ │ +000291e0: 2020 2020 202a 2041 2c20 6120 2a6e 6f74 * A, a *not │ │ │ │ +000291f0: 6520 7175 6f74 6965 6e74 2072 696e 673a e quotient ring: │ │ │ │ +00029200: 2028 4d61 6361 756c 6179 3244 6f63 2951 (Macaulay2Doc)Q │ │ │ │ +00029210: 756f 7469 656e 7452 696e 672c 2c0a 2020 uotientRing,,. │ │ │ │ +00029220: 2020 2020 2020 413d 5c5a 5a5b 685f 312c A=\ZZ[h_1, │ │ │ │ +00029230: 2e2e 2e2c 685f 6d5d 2f28 685f 315e 7b6e ...,h_m]/(h_1^{n │ │ │ │ +00029240: 5f31 2b31 7d2c 2e2e 2e2c 685f 6d5e 7b6e _1+1},...,h_m^{n │ │ │ │ +00029250: 5f6d 2b31 7d29 2071 756f 7469 656e 7420 _m+1}) quotient │ │ │ │ +00029260: 7269 6e67 0a20 2020 2020 2020 2072 6570 ring. rep │ │ │ │ +00029270: 7265 7365 6e74 696e 6720 7468 6520 4368 resenting the Ch │ │ │ │ +00029280: 6f77 2072 696e 6720 6f66 205c 5050 5e7b ow ring of \PP^{ │ │ │ │ +00029290: 6e5f 317d 782e 2e2e 785c 5050 5e7b 6e5f n_1}x...x\PP^{n_ │ │ │ │ +000292a0: 6d7d 2c20 7468 6973 2072 696e 6720 7368 m}, this ring sh │ │ │ │ +000292b0: 6f75 6c64 0a20 2020 2020 2020 2062 6520 ould. be │ │ │ │ +000292c0: 6275 696c 7420 7573 696e 6720 7468 6520 built using the │ │ │ │ +000292d0: 2a6e 6f74 6520 4368 6f77 5269 6e67 3a20 *note ChowRing: │ │ │ │ +000292e0: 4368 6f77 5269 6e67 2c20 636f 6d6d 616e ChowRing, comman │ │ │ │ +000292f0: 640a 2020 2020 2020 2a20 4a2c 2061 6e20 d. * J, an │ │ │ │ +00029300: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61 *note ideal: (Ma │ │ │ │ +00029310: 6361 756c 6179 3244 6f63 2949 6465 616c caulay2Doc)Ideal │ │ │ │ +00029320: 2c2c 2069 6e20 7468 6520 6772 6164 6564 ,, in the graded │ │ │ │ +00029330: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ +00029340: 0a20 2020 2020 2020 2077 6869 6368 2069 . which i │ │ │ │ +00029350: 7320 636f 6f72 6469 6e61 7465 2072 696e s coordinate rin │ │ │ │ +00029360: 6720 6f66 2074 6865 204e 6f72 6d61 6c20 g of the Normal │ │ │ │ +00029370: 546f 7269 6320 5661 7269 6574 7920 580a Toric Variety X. │ │ │ │ +00029380: 2020 2020 2020 2a20 582c 2061 202a 6e6f * X, a *no │ │ │ │ +00029390: 7465 206e 6f72 6d61 6c20 746f 7269 6320 te normal toric │ │ │ │ +000293a0: 7661 7269 6574 793a 0a20 2020 2020 2020 variety:. │ │ │ │ +000293b0: 2028 4e6f 726d 616c 546f 7269 6356 6172 (NormalToricVar │ │ │ │ +000293c0: 6965 7469 6573 294e 6f72 6d61 6c54 6f72 ieties)NormalTor │ │ │ │ +000293d0: 6963 5661 7269 6574 792c 2c20 7768 6963 icVariety,, whic │ │ │ │ +000293e0: 6820 6973 2074 6865 2061 6d62 6965 6e74 h is the ambient │ │ │ │ +000293f0: 2073 7061 6365 0a20 2020 2020 2020 2077 space. w │ │ │ │ +00029400: 6869 6368 2063 6f6e 7461 696e 7320 5628 hich contains V( │ │ │ │ +00029410: 4a29 0a20 2020 2020 202a 2043 682c 2061 J). * Ch, a │ │ │ │ +00029420: 202a 6e6f 7465 2071 756f 7469 656e 7420 *note quotient │ │ │ │ +00029430: 7269 6e67 3a20 284d 6163 6175 6c61 7932 ring: (Macaulay2 │ │ │ │ +00029440: 446f 6329 5175 6f74 6965 6e74 5269 6e67 Doc)QuotientRing │ │ │ │ +00029450: 2c2c 2074 6865 2043 686f 7720 7269 6e67 ,, the Chow ring │ │ │ │ +00029460: 0a20 2020 2020 2020 206f 6620 7468 6520 . of the │ │ │ │ +00029470: 746f 7269 6320 7661 7269 6574 7920 582c toric variety X, │ │ │ │ +00029480: 2043 683d 2872 696e 6720 4a29 2f28 5352 Ch=(ring J)/(SR │ │ │ │ +00029490: 2b4c 5229 2077 6865 7265 2053 5220 6973 +LR) where SR is │ │ │ │ +000294a0: 2074 6865 0a20 2020 2020 2020 2053 7461 the. Sta │ │ │ │ +000294b0: 6e6c 6579 2d52 6569 736e 6572 2069 6465 nley-Reisner ide │ │ │ │ +000294c0: 616c 206f 6620 7468 6520 6661 6e20 6465 al of the fan de │ │ │ │ +000294d0: 6669 6e69 6e67 2058 2061 6e64 204c 5220 fining X and LR │ │ │ │ +000294e0: 6973 2074 6865 206c 696e 6561 720a 2020 is the linear. │ │ │ │ +000294f0: 2020 2020 2020 7265 6c61 7469 6f6e 7320 relations │ │ │ │ +00029500: 6964 6561 6c2c 2074 6869 7320 7269 6e67 ideal, this ring │ │ │ │ +00029510: 2073 686f 756c 6420 6265 2062 7569 6c74 should be built │ │ │ │ +00029520: 2075 7369 6e67 2074 6865 202a 6e6f 7465 using the *note │ │ │ │ +00029530: 0a20 2020 2020 2020 2054 6f72 6963 4368 . ToricCh │ │ │ │ +00029540: 6f77 5269 6e67 3a20 546f 7269 6343 686f owRing: ToricCho │ │ │ │ +00029550: 7752 696e 672c 2063 6f6d 6d61 6e64 0a20 wRing, command. │ │ │ │ +00029560: 202a 202a 6e6f 7465 204f 7074 696f 6e61 * *note Optiona │ │ │ │ +00029570: 6c20 696e 7075 7473 3a20 284d 6163 6175 l inputs: (Macau │ │ │ │ +00029580: 6c61 7932 446f 6329 7573 696e 6720 6675 lay2Doc)using fu │ │ │ │ +00029590: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +000295a0: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20 ional inputs,:. │ │ │ │ +000295b0: 2020 2020 202a 2043 6f6d 704d 6574 686f * CompMetho │ │ │ │ +000295c0: 6420 286d 6973 7369 6e67 2064 6f63 756d d (missing docum │ │ │ │ +000295d0: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ +000295e0: 2c20 6465 6661 756c 7420 7661 6c75 650a , default value. │ │ │ │ +000295f0: 2020 2020 2020 2020 5072 6f6a 6563 7469 Projecti │ │ │ │ +00029600: 7665 4465 6772 6565 2c20 5072 6f6a 6563 veDegree, Projec │ │ │ │ +00029610: 7469 7665 4465 6772 6565 2c20 7468 6973 tiveDegree, this │ │ │ │ +00029620: 2061 6c67 6f72 6974 686d 206d 6179 2062 algorithm may b │ │ │ │ +00029630: 6520 7573 6564 2066 6f72 0a20 2020 2020 e used for. │ │ │ │ +00029640: 2020 2073 7562 7363 6865 6d65 7320 6f66 subschemes of │ │ │ │ +00029650: 2061 6e79 2061 7070 6c69 6361 626c 6520 any applicable │ │ │ │ +00029660: 746f 7269 6320 7661 7269 6574 7920 2874 toric variety (t │ │ │ │ +00029670: 6869 7320 6d61 7920 6265 2063 6865 636b his may be check │ │ │ │ +00029680: 6564 2075 7369 6e67 0a20 2020 2020 2020 ed using. │ │ │ │ +00029690: 2074 6865 202a 6e6f 7465 2043 6865 636b the *note Check │ │ │ │ +000296a0: 546f 7269 6356 6172 6965 7479 5661 6c69 ToricVarietyVali │ │ │ │ +000296b0: 643a 2043 6865 636b 546f 7269 6356 6172 d: CheckToricVar │ │ │ │ +000296c0: 6965 7479 5661 6c69 642c 2063 6f6d 6d61 ietyValid, comma │ │ │ │ +000296d0: 6e64 290a 2020 2020 2020 2a20 436f 6d70 nd). * Comp │ │ │ │ +000296e0: 4d65 7468 6f64 2028 6d69 7373 696e 6720 Method (missing │ │ │ │ +000296f0: 646f 6375 6d65 6e74 6174 696f 6e29 203d documentation) = │ │ │ │ +00029700: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ +00029710: 616c 7565 0a20 2020 2020 2020 2050 726f alue. Pro │ │ │ │ +00029720: 6a65 6374 6976 6544 6567 7265 652c 2050 jectiveDegree, P │ │ │ │ +00029730: 6e52 6573 6964 7561 6c2c 2074 6869 7320 nResidual, this │ │ │ │ +00029740: 616c 676f 7269 7468 6d20 6d61 7920 6265 algorithm may be │ │ │ │ +00029750: 2075 7365 6420 666f 7220 7375 6273 6368 used for subsch │ │ │ │ +00029760: 656d 6573 0a20 2020 2020 2020 206f 6620 emes. of │ │ │ │ +00029770: 5c50 505e 6e20 6f6e 6c79 0a20 2020 2020 \PP^n only. │ │ │ │ +00029780: 202a 204f 7574 7075 7420 3d3e 202e 2e2e * Output => ... │ │ │ │ +00029790: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +000297a0: 4368 6f77 5269 6e67 456c 656d 656e 742c ChowRingElement, │ │ │ │ +000297b0: 2043 686f 7752 696e 6745 6c65 6d65 6e74 ChowRingElement │ │ │ │ +000297c0: 2c20 7265 7475 726e 730a 2020 2020 2020 , returns. │ │ │ │ +000297d0: 2020 6120 5269 6e67 456c 656d 656e 7420 a RingElement │ │ │ │ +000297e0: 696e 2074 6865 2043 686f 7720 7269 6e67 in the Chow ring │ │ │ │ +000297f0: 206f 6620 7468 6520 6170 7072 6f70 7269 of the appropri │ │ │ │ +00029800: 6174 6520 616d 6269 656e 7420 7370 6163 ate ambient spac │ │ │ │ +00029810: 650a 2020 2020 2020 2a20 4f75 7470 7574 e. * Output │ │ │ │ +00029820: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ +00029830: 2076 616c 7565 2043 686f 7752 696e 6745 value ChowRingE │ │ │ │ +00029840: 6c65 6d65 6e74 2c20 4861 7368 466f 726d lement, HashForm │ │ │ │ +00029850: 2c20 4861 7368 466f 726d 0a20 2020 2020 , HashForm. │ │ │ │ +00029860: 2020 2072 6574 7572 6e73 2061 204d 7574 returns a Mut │ │ │ │ +00029870: 6162 6c65 4861 7368 5461 626c 6520 636f ableHashTable co │ │ │ │ +00029880: 6e74 6169 6e69 6e67 2074 6865 2066 6f6c ntaining the fol │ │ │ │ +00029890: 6c6f 7769 6e67 206b 6579 733a 2022 4722 lowing keys: "G" │ │ │ │ +000298a0: 2028 7468 650a 2020 2020 2020 2020 706f (the. po │ │ │ │ +000298b0: 6c79 6e6f 6d69 616c 2077 6974 6820 636f lynomial with co │ │ │ │ +000298c0: 6566 6669 6369 656e 7473 206f 6620 7468 efficients of th │ │ │ │ +000298d0: 6520 6879 7065 7270 6c61 6e65 2063 6c61 e hyperplane cla │ │ │ │ +000298e0: 7373 6573 2072 6570 7265 7365 6e74 696e sses representin │ │ │ │ +000298f0: 6720 7468 650a 2020 2020 2020 2020 7072 g the. pr │ │ │ │ +00029900: 6f6a 6563 7469 7665 2064 6567 7265 6573 ojective degrees │ │ │ │ +00029910: 292c 2022 476c 6973 7422 2028 7468 6520 ), "Glist" (the │ │ │ │ +00029920: 6c69 7374 2066 6f72 6d20 6f66 2022 4722 list form of "G" │ │ │ │ +00029930: 2920 2c20 2253 6567 7265 2220 2874 6865 ) , "Segre" (the │ │ │ │ +00029940: 0a20 2020 2020 2020 2074 6f74 616c 2053 . total S │ │ │ │ +00029950: 6567 7265 2063 6c61 7373 206f 6620 7468 egre class of th │ │ │ │ +00029960: 6520 696e 7075 7429 2c22 5365 6772 654c e input),"SegreL │ │ │ │ +00029970: 6973 7422 2028 7468 6520 6c69 7374 2066 ist" (the list f │ │ │ │ +00029980: 6f72 6d20 6f66 2022 5365 6772 6522 290a orm of "Segre"). │ │ │ │ +00029990: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +000299a0: 2020 202a 2061 202a 6e6f 7465 2072 696e * a *note rin │ │ │ │ +000299b0: 6720 656c 656d 656e 743a 2028 4d61 6361 g element: (Maca │ │ │ │ +000299c0: 756c 6179 3244 6f63 2952 696e 6745 6c65 ulay2Doc)RingEle │ │ │ │ +000299d0: 6d65 6e74 2c2c 2074 6865 2070 7573 6866 ment,, the pushf │ │ │ │ +000299e0: 6f72 7761 7264 206f 660a 2020 2020 2020 orward of. │ │ │ │ +000299f0: 2020 7468 6520 746f 7461 6c20 5365 6772 the total Segr │ │ │ │ +00029a00: 6520 636c 6173 7320 6f66 2074 6865 2073 e class of the s │ │ │ │ +00029a10: 6368 656d 6520 5620 6465 6669 6e65 6420 cheme V defined │ │ │ │ +00029a20: 6279 2074 6865 2069 6e70 7574 2069 6465 by the input ide │ │ │ │ +00029a30: 616c 2074 6f20 7468 650a 2020 2020 2020 al to the. │ │ │ │ +00029a40: 2020 6170 7072 6f70 7269 6174 6520 4368 appropriate Ch │ │ │ │ +00029a50: 6f77 2072 696e 670a 0a44 6573 6372 6970 ow ring..Descrip │ │ │ │ +00029a60: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +00029a70: 0a0a 466f 7220 6120 7375 6273 6368 656d ..For a subschem │ │ │ │ +00029a80: 6520 5620 6f66 2061 6e20 6170 706c 6963 e V of an applic │ │ │ │ +00029a90: 6162 6c65 2074 6f72 6963 2076 6172 6965 able toric varie │ │ │ │ +00029aa0: 7479 2058 2074 6869 7320 636f 6d6d 616e ty X this comman │ │ │ │ +00029ab0: 6420 636f 6d70 7574 6573 2074 6865 0a70 d computes the.p │ │ │ │ +00029ac0: 7573 682d 666f 7277 6172 6420 6f66 2074 ush-forward of t │ │ │ │ +00029ad0: 6865 2074 6f74 616c 2053 6567 7265 2063 he total Segre c │ │ │ │ +00029ae0: 6c61 7373 2073 2856 2c58 2920 6f66 2056 lass s(V,X) of V │ │ │ │ +00029af0: 2069 6e20 5820 746f 2074 6865 2043 686f in X to the Cho │ │ │ │ +00029b00: 7720 7269 6e67 206f 6620 582e 0a0a 2b2d w ring of X...+- │ │ │ │ +00029b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00029b40: 3120 3a20 7365 7452 616e 646f 6d53 6565 1 : setRandomSee │ │ │ │ +00029b50: 6420 3732 3b20 2020 2020 2020 2020 2020 d 72; │ │ │ │ +00029b60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00029b70: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ +00029b80: 6d20 7365 6564 2074 6f20 3732 2020 2020 m seed to 72 │ │ │ │ +00029b90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00029ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00029bd0: 3220 3a20 5220 3d20 5a5a 2f33 3237 3439 2 : R = ZZ/32749 │ │ │ │ +00029be0: 5b77 2c79 2c7a 5d20 2020 2020 2020 2020 [w,y,z] │ │ │ │ +00029bf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00029c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00029c30: 3220 3d20 5220 2020 2020 2020 2020 2020 2 = R │ │ │ │ +00029c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00029c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c90: 2020 7c0a 7c6f 3220 3d20 5220 2020 2020 |.|o2 = R │ │ │ │ -00029ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00029cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cf0: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f |.|o2 : Polyno │ │ │ │ -00029d00: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ -00029d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d20: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00029d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029d50: 2d2d 2b0a 7c69 3320 3a20 5365 6772 6528 --+.|i3 : Segre( │ │ │ │ -00029d60: 6964 6561 6c28 772a 7929 2c43 6f6d 704d ideal(w*y),CompM │ │ │ │ -00029d70: 6574 686f 643d 3e50 6e52 6573 6964 7561 ethod=>PnResidua │ │ │ │ -00029d80: 6c29 7c0a 7c20 2020 2020 2020 2020 2020 l)|.| │ │ │ │ +00029c80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00029c90: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ +00029ca0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ +00029cb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00029cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00029cf0: 3320 3a20 5365 6772 6528 6964 6561 6c28 3 : Segre(ideal( │ │ │ │ +00029d00: 772a 7929 2c43 6f6d 704d 6574 686f 643d w*y),CompMethod= │ │ │ │ +00029d10: 3e50 6e52 6573 6964 7561 6c29 7c0a 7c20 >PnResidual)|.| │ │ │ │ +00029d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029d40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00029d50: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00029d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029d70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00029d80: 3320 3d20 2d20 3448 2020 2b20 3248 2020 3 = - 4H + 2H │ │ │ │ 00029d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029db0: 2020 7c0a 7c20 2020 2020 2020 2020 3220 |.| 2 │ │ │ │ +00029da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00029db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029de0: 2020 7c0a 7c6f 3320 3d20 2d20 3448 2020 |.|o3 = - 4H │ │ │ │ -00029df0: 2b20 3248 2020 2020 2020 2020 2020 2020 + 2H │ │ │ │ -00029e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00029dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00029de0: 2020 2020 5a5a 5b48 5d20 2020 2020 2020 ZZ[H] │ │ │ │ +00029df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00029e10: 3320 3a20 2d2d 2d2d 2d20 2020 2020 2020 3 : ----- │ │ │ │ 00029e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e40: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20 |.| ZZ[H] │ │ │ │ +00029e30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00029e40: 2020 2020 2020 2033 2020 2020 2020 2020 3 │ │ │ │ 00029e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e70: 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d 2d20 |.|o3 : ----- │ │ │ │ +00029e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00029e70: 2020 2020 2020 4820 2020 2020 2020 2020 H │ │ │ │ 00029e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ea0: 2020 7c0a 7c20 2020 2020 2020 2033 2020 |.| 3 │ │ │ │ -00029eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ed0: 2020 7c0a 7c20 2020 2020 2020 4820 2020 |.| H │ │ │ │ -00029ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029f00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00029f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029f30: 2d2d 2b0a 7c69 3420 3a20 413d 4368 6f77 --+.|i4 : A=Chow │ │ │ │ -00029f40: 5269 6e67 2852 2920 2020 2020 2020 2020 Ring(R) │ │ │ │ -00029f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029f60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00029e90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00029ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00029ed0: 3420 3a20 413d 4368 6f77 5269 6e67 2852 4 : A=ChowRing(R │ │ │ │ +00029ee0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00029ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00029f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00029f30: 3420 3d20 4120 2020 2020 2020 2020 2020 4 = A │ │ │ │ +00029f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00029f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029f90: 2020 7c0a 7c6f 3420 3d20 4120 2020 2020 |.|o4 = A │ │ │ │ +00029f80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00029f90: 3420 3a20 5175 6f74 6965 6e74 5269 6e67 4 : QuotientRing │ │ │ │ 00029fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029fc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00029fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ff0: 2020 7c0a 7c6f 3420 3a20 5175 6f74 6965 |.|o4 : Quotie │ │ │ │ -0002a000: 6e74 5269 6e67 2020 2020 2020 2020 2020 ntRing │ │ │ │ -0002a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a020: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a050: 2d2d 2b0a 7c69 3520 3a20 5365 6772 6528 --+.|i5 : Segre( │ │ │ │ -0002a060: 412c 6964 6561 6c28 775e 322a 792c 772a A,ideal(w^2*y,w* │ │ │ │ -0002a070: 795e 3229 2920 2020 2020 2020 2020 2020 y^2)) │ │ │ │ -0002a080: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00029fb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00029fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00029ff0: 3520 3a20 5365 6772 6528 412c 6964 6561 5 : Segre(A,idea │ │ │ │ +0002a000: 6c28 775e 322a 792c 772a 795e 3229 2920 l(w^2*y,w*y^2)) │ │ │ │ +0002a010: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002a020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a040: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002a050: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0002a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a070: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0002a080: 3520 3d20 2d20 3368 2020 2b20 3268 2020 5 = - 3h + 2h │ │ │ │ 0002a090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0b0: 2020 7c0a 7c20 2020 2020 2020 2020 3220 |.| 2 │ │ │ │ +0002a0a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002a0b0: 2020 2020 2020 2020 3120 2020 2020 3120 1 1 │ │ │ │ 0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0e0: 2020 7c0a 7c6f 3520 3d20 2d20 3368 2020 |.|o5 = - 3h │ │ │ │ -0002a0f0: 2b20 3268 2020 2020 2020 2020 2020 2020 + 2h │ │ │ │ -0002a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a110: 2020 7c0a 7c20 2020 2020 2020 2020 3120 |.| 1 │ │ │ │ -0002a120: 2020 2020 3120 2020 2020 2020 2020 2020 1 │ │ │ │ -0002a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a140: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a170: 2020 7c0a 7c6f 3520 3a20 4120 2020 2020 |.|o5 : A │ │ │ │ -0002a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a1a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -0002a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a1d0: 2d2d 2b0a 0a4e 6f77 2063 6f6e 7369 6465 --+..Now conside │ │ │ │ -0002a1e0: 7220 616e 2065 7861 6d70 6c65 2069 6e20 r an example in │ │ │ │ -0002a1f0: 5c50 505e 3220 5c74 696d 6573 205c 5050 \PP^2 \times \PP │ │ │ │ -0002a200: 5e32 2c20 6966 2077 6520 696e 7075 7420 ^2, if we input │ │ │ │ -0002a210: 7468 6520 4368 6f77 2072 696e 6720 4120 the Chow ring A │ │ │ │ -0002a220: 7468 650a 6f75 7470 7574 2077 696c 6c20 the.output will │ │ │ │ -0002a230: 6265 2072 6574 7572 6e65 6420 696e 2074 be returned in t │ │ │ │ -0002a240: 6865 2073 616d 6520 7269 6e67 2e20 546f he same ring. To │ │ │ │ -0002a250: 2065 6e73 7572 6520 7072 6f70 6572 2066 ensure proper f │ │ │ │ -0002a260: 756e 6374 696f 6e20 6f66 2074 6865 0a6d unction of the.m │ │ │ │ -0002a270: 6574 686f 6473 2077 6520 6275 696c 6420 ethods we build │ │ │ │ -0002a280: 7468 6520 4368 6f77 2072 696e 6720 7573 the Chow ring us │ │ │ │ -0002a290: 696e 6720 7468 6520 2a6e 6f74 6520 4368 ing the *note Ch │ │ │ │ -0002a2a0: 6f77 5269 6e67 3a20 4368 6f77 5269 6e67 owRing: ChowRing │ │ │ │ -0002a2b0: 2c20 636f 6d6d 616e 642e 2057 650a 6d61 , command. We.ma │ │ │ │ -0002a2c0: 7920 616c 736f 2072 6574 7572 6e20 6120 y also return a │ │ │ │ -0002a2d0: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ -0002a2e0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ -0002a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a330: 2d2b 0a7c 6936 203a 2052 3d4d 756c 7469 -+.|i6 : R=Multi │ │ │ │ -0002a340: 5072 6f6a 436f 6f72 6452 696e 6728 7b32 ProjCoordRing({2 │ │ │ │ -0002a350: 2c32 7d29 2020 2020 2020 2020 2020 2020 ,2}) │ │ │ │ -0002a360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a380: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002a0d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a100: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0002a110: 3520 3a20 4120 2020 2020 2020 2020 2020 5 : A │ │ │ │ +0002a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a130: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002a140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a4e ------------+..N │ │ │ │ +0002a170: 6f77 2063 6f6e 7369 6465 7220 616e 2065 ow consider an e │ │ │ │ +0002a180: 7861 6d70 6c65 2069 6e20 5c50 505e 3220 xample in \PP^2 │ │ │ │ +0002a190: 5c74 696d 6573 205c 5050 5e32 2c20 6966 \times \PP^2, if │ │ │ │ +0002a1a0: 2077 6520 696e 7075 7420 7468 6520 4368 we input the Ch │ │ │ │ +0002a1b0: 6f77 2072 696e 6720 4120 7468 650a 6f75 ow ring A the.ou │ │ │ │ +0002a1c0: 7470 7574 2077 696c 6c20 6265 2072 6574 tput will be ret │ │ │ │ +0002a1d0: 7572 6e65 6420 696e 2074 6865 2073 616d urned in the sam │ │ │ │ +0002a1e0: 6520 7269 6e67 2e20 546f 2065 6e73 7572 e ring. To ensur │ │ │ │ +0002a1f0: 6520 7072 6f70 6572 2066 756e 6374 696f e proper functio │ │ │ │ +0002a200: 6e20 6f66 2074 6865 0a6d 6574 686f 6473 n of the.methods │ │ │ │ +0002a210: 2077 6520 6275 696c 6420 7468 6520 4368 we build the Ch │ │ │ │ +0002a220: 6f77 2072 696e 6720 7573 696e 6720 7468 ow ring using th │ │ │ │ +0002a230: 6520 2a6e 6f74 6520 4368 6f77 5269 6e67 e *note ChowRing │ │ │ │ +0002a240: 3a20 4368 6f77 5269 6e67 2c20 636f 6d6d : ChowRing, comm │ │ │ │ +0002a250: 616e 642e 2057 650a 6d61 7920 616c 736f and. We.may also │ │ │ │ +0002a260: 2072 6574 7572 6e20 6120 4d75 7461 626c return a Mutabl │ │ │ │ +0002a270: 6548 6173 6854 6162 6c65 2e0a 0a2b 2d2d eHashTable...+-- │ │ │ │ +0002a280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ +0002a2d0: 203a 2052 3d4d 756c 7469 5072 6f6a 436f : R=MultiProjCo │ │ │ │ +0002a2e0: 6f72 6452 696e 6728 7b32 2c32 7d29 2020 ordRing({2,2}) │ │ │ │ +0002a2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a310: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002a320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a360: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +0002a370: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +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 2020 2020 2020 │ │ │ │ +0002a3b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a3d0: 207c 0a7c 6f36 203d 2052 2020 2020 2020 |.|o6 = R │ │ │ │ +0002a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a420: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002a400: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +0002a410: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ +0002a420: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 0002a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a470: 207c 0a7c 6f36 203a 2050 6f6c 796e 6f6d |.|o6 : Polynom │ │ │ │ -0002a480: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ -0002a490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a4c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a510: 2d2b 0a7c 6937 203a 2072 3d67 656e 7320 -+.|i7 : r=gens │ │ │ │ -0002a520: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0002a450: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002a460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ +0002a4b0: 203a 2072 3d67 656e 7320 5220 2020 2020 : r=gens R │ │ │ │ +0002a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a4f0: 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2020 2020 2020 │ │ │ │ -0002a7e0: 207c 0a7c 6f38 203d 2041 2020 2020 2020 |.|o8 = A │ │ │ │ +0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a830: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002a810: 2020 2020 2020 2020 2020 207c 0a7c 6f38 |.|o8 │ │ │ │ +0002a820: 203a 2051 756f 7469 656e 7452 696e 6720 : QuotientRing │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a880: 207c 0a7c 6f38 203a 2051 756f 7469 656e |.|o8 : Quotien │ │ │ │ -0002a890: 7452 696e 6720 2020 2020 2020 2020 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2020 │ │ │ │ +0002a950: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002a960: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0002a970: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ 0002a980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a9c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002a9d0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002a9e0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002a9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aa10: 207c 0a7c 6f39 203d 2069 6465 616c 2028 |.|o9 = ideal ( │ │ │ │ -0002aa20: 7820 7820 202d 2078 2078 2078 202c 2078 x x - x x x , x │ │ │ │ -0002aa30: 2078 2029 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ -0002aa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a9a0: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ +0002a9b0: 203d 2069 6465 616c 2028 7820 7820 202d = ideal (x x - │ │ │ │ +0002a9c0: 2078 2078 2078 202c 2078 2078 2029 2020 x x x , x x ) │ │ │ │ +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 2020 |.| │ │ │ │ +0002aa00: 2020 2020 2020 2020 2020 2030 2033 2020 0 3 │ │ │ │ +0002aa10: 2020 3120 3220 3420 2020 3220 3520 2020 1 2 4 2 5 │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002aa60: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002aa70: 2030 2033 2020 2020 3120 3220 3420 2020 0 3 1 2 4 │ │ │ │ -0002aa80: 3220 3520 2020 2020 2020 2020 2020 2020 2 5 │ │ │ │ -0002aa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aab0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002aa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002aa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002aa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002aa90: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ +0002aaa0: 203a 2049 6465 616c 206f 6620 5220 2020 : Ideal of R │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ab00: 207c 0a7c 6f39 203a 2049 6465 616c 206f |.|o9 : Ideal o │ │ │ │ -0002ab10: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ -0002ab20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ab30: 2020 2020 2020 2020 2020 2020 2020 2020 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6931 -----------+.|i1 │ │ │ │ +0002ab40: 3020 3a20 5365 6772 6520 4920 2020 2020 0 : Segre I │ │ │ │ +0002ab50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ab60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ab70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ab80: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002ab90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002aba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002abb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002abc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002abd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002abe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002abf0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002ac00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002abd0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002abe0: 2020 2020 2020 2032 2032 2020 2020 2020 2 2 │ │ │ │ +0002abf0: 3220 2020 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ +0002ac00: 2032 2020 2020 2020 2020 2020 2020 3220 2 2 │ │ │ │ 0002ac10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ac20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ac30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ac40: 207c 0a7c 2020 2020 2020 2020 2032 2032 |.| 2 2 │ │ │ │ -0002ac50: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0002ac60: 2032 2020 2020 2032 2020 2020 2020 2020 2 2 │ │ │ │ -0002ac70: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0002ac80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ac90: 207c 0a7c 6f31 3020 3d20 3732 6820 6820 |.|o10 = 72h h │ │ │ │ -0002aca0: 202d 2032 3468 2068 2020 2d20 3132 6820 - 24h h - 12h │ │ │ │ -0002acb0: 6820 202b 2034 6820 202b 2034 6820 6820 h + 4h + 4h h │ │ │ │ -0002acc0: 202b 2068 2020 2020 2020 2020 2020 2020 + h │ │ │ │ +0002ac20: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0002ac30: 3020 3d20 3732 6820 6820 202d 2032 3468 0 = 72h h - 24h │ │ │ │ +0002ac40: 2068 2020 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│ │ +0002ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002acf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002ad20: 2020 2020 5a5a 5b68 202e 2e68 205d 2020 ZZ[h ..h ] │ │ │ │ +0002ad30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ad40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ad50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ad80: 207c 0a7c 2020 2020 2020 5a5a 5b68 202e |.| ZZ[h . │ │ │ │ -0002ad90: 2e68 205d 2020 2020 2020 2020 2020 2020 .h ] │ │ │ │ +0002ad60: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002ad70: 2020 2020 2020 2020 3120 2020 3220 2020 1 2 │ │ │ │ +0002ad80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ada0: 2020 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---------------- │ │ │ │ +0002b390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0002b3b0: 3320 3a20 7065 656b 2053 6567 4861 7368 3 : peek SegHash │ │ │ │ +0002b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b3f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b460: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b440: 2020 2020 2020 2020 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|.| │ │ │ │ 0002b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b500: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002b510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b520: 2020 3120 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ -0002b530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b500: 2020 2020 2047 6c69 7374 203d 3e20 7b31 Glist => {1 │ │ │ │ +0002b510: 2c20 3268 2020 2b20 6820 2c20 302c 2030 , 2h + h , 0, 0 │ │ │ │ +0002b520: 2c20 307d 2020 2020 2020 2020 2020 2020 , 0} │ │ │ │ +0002b530: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b550: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002b560: 2020 2020 2020 2020 2020 2047 6c69 7374 Glist │ │ │ │ -0002b570: 203d 3e20 7b31 2c20 3268 2020 2b20 6820 => {1, 2h + h │ │ │ │ -0002b580: 2c20 302c 2030 2c20 307d 2020 2020 2020 , 0, 0, 0} │ │ │ │ +0002b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b560: 2020 2020 3120 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│ │ │ +0002b5f0: 2020 2020 2053 6567 7265 4c69 7374 203d SegreList = │ │ │ │ +0002b600: 3e20 7b30 2c20 302c 2034 6820 202b 2034 > {0, 0, 4h + 4 │ │ │ │ +0002b610: 6820 6820 202b 2068 202c 202d 2020 2020 h h + h , - │ │ │ │ +0002b620: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002b630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b640: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002b650: 2020 2020 2020 2020 2020 2053 6567 7265 Segre │ │ │ │ -0002b660: 4c69 7374 203d 3e20 7b30 2c20 302c 2034 List => {0, 0, 4 │ │ │ │ -0002b670: 6820 202b 2034 6820 6820 202b 2068 202c h + 4h h + h , │ │ │ │ -0002b680: 202d 2020 2020 2020 2020 2020 2020 2020 - │ │ │ │ -0002b690: 207c 0a7c 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: 2031 2020 2020 2031 2032 2020 2020 3220 1 1 2 2 │ │ │ │ +0002b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b650: 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│ │ │ │ +0002b6e0: 2020 2020 2053 6567 7265 203d 3e20 3732 Segre => 72 │ │ │ │ +0002b6f0: 6820 6820 202d 2032 3468 2068 2020 2d20 h h - 24h h - │ │ │ │ +0002b700: 3132 6820 6820 202b 2034 6820 2020 2020 12h h + 4h │ │ │ │ +0002b710: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b740: 2031 2032 2020 2020 2020 3120 3220 2020 1 2 1 2 │ │ │ │ +0002b750: 2020 2031 2032 2020 2020 2031 2020 2020 1 2 1 │ │ │ │ +0002b760: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ +0002b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ +0002b7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b7d0: 2020 2020 2020 7d20 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 2020 207c 0a7c 2020 |.| │ │ │ │ +0002b810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b850: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b870: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 |.| 2 │ │ │ │ -0002b970: 2020 3220 2020 2020 3220 3220 2020 2020 2 2 2 │ │ │ │ +0002b940: 2020 2020 2020 2020 2020 207c 0a7c 3234 |.|24 │ │ │ │ +0002b950: 6820 6820 202d 2031 3268 2068 202c 2037 h h - 12h h , 7 │ │ │ │ +0002b960: 3268 2068 207d 2020 2020 2020 2020 2020 2h h } │ │ │ │ +0002b970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9b0: 207c 0a7c 3234 6820 6820 202d 2031 3268 |.|24h h - 12h │ │ │ │ -0002b9c0: 2068 202c 2037 3268 2068 207d 2020 2020 h , 72h h } │ │ │ │ +0002b990: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002b9a0: 2031 2032 2020 2020 2020 3120 3220 2020 1 2 1 2 │ │ │ │ +0002b9b0: 2020 3120 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ +0002b9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba00: 207c 0a7c 2020 2031 2032 2020 2020 2020 |.| 1 2 │ │ │ │ -0002ba10: 3120 3220 2020 2020 3120 3220 2020 2020 1 2 1 2 │ │ │ │ +0002b9e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002b9f0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ba10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ba20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba50: 207c 0a7c 2020 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ +0002ba30: 2020 2020 2020 2020 2020 207c 0a7c 2b20 |.|+ │ │ │ │ +0002ba40: 3468 2068 2020 2b20 6820 2020 2020 2020 4h h + h │ │ │ │ +0002ba50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ba70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002baa0: 207c 0a7c 2b20 3468 2068 2020 2b20 6820 |.|+ 4h h + h │ │ │ │ +0002ba80: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002ba90: 2020 3120 3220 2020 2032 2020 2020 2020 1 2 2 │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002bae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002baf0: 207c 0a7c 2020 2020 3120 3220 2020 2032 |.| 1 2 2 │ │ │ │ -0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002bb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bb90: 2d2b 0a7c 6931 3420 3a20 7331 3d3d 5365 -+.|i14 : s1==Se │ │ │ │ -0002bba0: 6748 6173 6823 2253 6567 7265 2220 2020 gHash#"Segre" │ │ │ │ +0002bad0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002baf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0002bb30: 3420 3a20 7331 3d3d 5365 6748 6173 6823 4 : s1==SegHash# │ │ │ │ +0002bb40: 2253 6567 7265 2220 2020 2020 2020 2020 "Segre" │ │ │ │ +0002bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb70: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bbe0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002bbc0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0002bbd0: 3420 3d20 7472 7565 2020 2020 2020 2020 4 = true │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc30: 207c 0a7c 6f31 3420 3d20 7472 7565 2020 |.|o14 = true │ │ │ │ -0002bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc80: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bcd0: 2d2b 0a0a 496e 2074 6865 2063 6173 6520 -+..In the case │ │ │ │ -0002bce0: 7768 6572 6520 7468 6520 616d 6269 656e where the ambien │ │ │ │ -0002bcf0: 7420 7370 6163 6520 6973 2061 2074 6f72 t space is a tor │ │ │ │ -0002bd00: 6963 2076 6172 6965 7479 2077 6869 6368 ic variety which │ │ │ │ -0002bd10: 2069 7320 6e6f 7420 6120 7072 6f64 7563 is not a produc │ │ │ │ -0002bd20: 740a 6f66 2070 726f 6a65 6374 6976 6520 t.of projective │ │ │ │ -0002bd30: 7370 6163 6573 2077 6520 6d75 7374 206c spaces we must l │ │ │ │ -0002bd40: 6f61 6420 7468 6520 4e6f 726d 616c 546f oad the NormalTo │ │ │ │ -0002bd50: 7269 6356 6172 6965 7469 6573 2070 6163 ricVarieties pac │ │ │ │ -0002bd60: 6b61 6765 2061 6e64 206d 7573 740a 616c kage and must.al │ │ │ │ -0002bd70: 736f 2069 6e70 7574 2074 6865 2074 6f72 so input the tor │ │ │ │ -0002bd80: 6963 2076 6172 6965 7479 2e20 4966 2074 ic variety. If t │ │ │ │ -0002bd90: 6865 2074 6f72 6963 2076 6172 6965 7479 he toric variety │ │ │ │ -0002bda0: 2069 7320 6120 7072 6f64 7563 7420 6f66 is a product of │ │ │ │ -0002bdb0: 2070 726f 6a65 6374 6976 650a 7370 6163 projective.spac │ │ │ │ -0002bdc0: 6520 6974 2069 7320 7265 636f 6d6d 656e e it is recommen │ │ │ │ -0002bdd0: 6465 6420 746f 2075 7365 2074 6865 2066 ded to use the f │ │ │ │ -0002bde0: 6f72 6d20 6162 6f76 6520 7261 7468 6572 orm above rather │ │ │ │ -0002bdf0: 2074 6861 6e20 696e 7075 7474 696e 6720 than inputting │ │ │ │ -0002be00: 7468 6520 746f 7269 630a 7661 7269 6574 the toric.variet │ │ │ │ -0002be10: 7920 666f 7220 6566 6669 6369 656e 6379 y for efficiency │ │ │ │ -0002be20: 2072 6561 736f 6e73 2e0a 0a2b 2d2d 2d2d reasons...+---- │ │ │ │ -0002be30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be70: 2d2d 2d2d 2b0a 7c69 3135 203a 206e 6565 ----+.|i15 : nee │ │ │ │ -0002be80: 6473 5061 636b 6167 6520 224e 6f72 6d61 dsPackage "Norma │ │ │ │ -0002be90: 6c54 6f72 6963 5661 7269 6574 6965 7322 lToricVarieties" │ │ │ │ -0002bea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002beb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bec0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002bc10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002bc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 496e -----------+..In │ │ │ │ +0002bc70: 2074 6865 2063 6173 6520 7768 6572 6520 the case where │ │ │ │ +0002bc80: 7468 6520 616d 6269 656e 7420 7370 6163 the ambient spac │ │ │ │ +0002bc90: 6520 6973 2061 2074 6f72 6963 2076 6172 e is a toric var │ │ │ │ +0002bca0: 6965 7479 2077 6869 6368 2069 7320 6e6f iety which is no │ │ │ │ +0002bcb0: 7420 6120 7072 6f64 7563 740a 6f66 2070 t a product.of p │ │ │ │ +0002bcc0: 726f 6a65 6374 6976 6520 7370 6163 6573 rojective spaces │ │ │ │ +0002bcd0: 2077 6520 6d75 7374 206c 6f61 6420 7468 we must load th │ │ │ │ +0002bce0: 6520 4e6f 726d 616c 546f 7269 6356 6172 e NormalToricVar │ │ │ │ +0002bcf0: 6965 7469 6573 2070 6163 6b61 6765 2061 ieties package a │ │ │ │ +0002bd00: 6e64 206d 7573 740a 616c 736f 2069 6e70 nd must.also inp │ │ │ │ +0002bd10: 7574 2074 6865 2074 6f72 6963 2076 6172 ut the toric var │ │ │ │ +0002bd20: 6965 7479 2e20 4966 2074 6865 2074 6f72 iety. If the tor │ │ │ │ +0002bd30: 6963 2076 6172 6965 7479 2069 7320 6120 ic variety is a │ │ │ │ +0002bd40: 7072 6f64 7563 7420 6f66 2070 726f 6a65 product of proje │ │ │ │ +0002bd50: 6374 6976 650a 7370 6163 6520 6974 2069 ctive.space it i │ │ │ │ +0002bd60: 7320 7265 636f 6d6d 656e 6465 6420 746f s recommended to │ │ │ │ +0002bd70: 2075 7365 2074 6865 2066 6f72 6d20 6162 use the form ab │ │ │ │ +0002bd80: 6f76 6520 7261 7468 6572 2074 6861 6e20 ove rather than │ │ │ │ +0002bd90: 696e 7075 7474 696e 6720 7468 6520 746f inputting the to │ │ │ │ +0002bda0: 7269 630a 7661 7269 6574 7920 666f 7220 ric.variety for │ │ │ │ +0002bdb0: 6566 6669 6369 656e 6379 2072 6561 736f efficiency reaso │ │ │ │ +0002bdc0: 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ns...+---------- │ │ │ │ +0002bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002be10: 7c69 3135 203a 206e 6565 6473 5061 636b |i15 : needsPack │ │ │ │ +0002be20: 6167 6520 224e 6f72 6d61 6c54 6f72 6963 age "NormalToric │ │ │ │ +0002be30: 5661 7269 6574 6965 7322 2020 2020 2020 Varieties" │ │ │ │ +0002be40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002be60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bea0: 2020 2020 7c0a 7c6f 3135 203d 204e 6f72 |.|o15 = Nor │ │ │ │ +0002beb0: 6d61 6c54 6f72 6963 5661 7269 6574 6965 malToricVarietie │ │ │ │ +0002bec0: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ 0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf00: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ -0002bf10: 203d 204e 6f72 6d61 6c54 6f72 6963 5661 = NormalToricVa │ │ │ │ -0002bf20: 7269 6574 6965 7320 2020 2020 2020 2020 rieties │ │ │ │ -0002bf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf50: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002bee0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002bef0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002bf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf30: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ +0002bf40: 203a 2050 6163 6b61 6765 2020 2020 2020 : Package │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfa0: 7c0a 7c6f 3135 203a 2050 6163 6b61 6765 |.|o15 : Package │ │ │ │ -0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfe0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002bff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c030: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2052 ------+.|i16 : R │ │ │ │ -0002c040: 686f 203d 207b 7b31 2c30 2c30 7d2c 7b30 ho = {{1,0,0},{0 │ │ │ │ -0002c050: 2c31 2c30 7d2c 7b30 2c30 2c31 7d2c 7b2d ,1,0},{0,0,1},{- │ │ │ │ -0002c060: 312c 2d31 2c30 7d2c 7b30 2c30 2c2d 317d 1,-1,0},{0,0,-1} │ │ │ │ -0002c070: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ -0002c080: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002c0d0: 3136 203d 207b 7b31 2c20 302c 2030 7d2c 16 = {{1, 0, 0}, │ │ │ │ -0002c0e0: 207b 302c 2031 2c20 307d 2c20 7b30 2c20 {0, 1, 0}, {0, │ │ │ │ -0002c0f0: 302c 2031 7d2c 207b 2d31 2c20 2d31 2c20 0, 1}, {-1, -1, │ │ │ │ -0002c100: 307d 2c20 7b30 2c20 302c 202d 317d 7d20 0}, {0, 0, -1}} │ │ │ │ -0002c110: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002bf80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002bf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bfd0: 2b0a 7c69 3136 203a 2052 686f 203d 207b +.|i16 : Rho = { │ │ │ │ +0002bfe0: 7b31 2c30 2c30 7d2c 7b30 2c31 2c30 7d2c {1,0,0},{0,1,0}, │ │ │ │ +0002bff0: 7b30 2c30 2c31 7d2c 7b2d 312c 2d31 2c30 {0,0,1},{-1,-1,0 │ │ │ │ +0002c000: 7d2c 7b30 2c30 2c2d 317d 7d20 2020 2020 },{0,0,-1}} │ │ │ │ +0002c010: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c060: 2020 2020 2020 7c0a 7c6f 3136 203d 207b |.|o16 = { │ │ │ │ +0002c070: 7b31 2c20 302c 2030 7d2c 207b 302c 2031 {1, 0, 0}, {0, 1 │ │ │ │ +0002c080: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c , 0}, {0, 0, 1}, │ │ │ │ +0002c090: 207b 2d31 2c20 2d31 2c20 307d 2c20 7b30 {-1, -1, 0}, {0 │ │ │ │ +0002c0a0: 2c20 302c 202d 317d 7d20 2020 2020 2020 , 0, -1}} │ │ │ │ +0002c0b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002c0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0002c100: 3136 203a 204c 6973 7420 2020 2020 2020 16 : List │ │ │ │ +0002c110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c160: 2020 7c0a 7c6f 3136 203a 204c 6973 7420 |.|o16 : List │ │ │ │ -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 207c 0a2b |.+ │ │ │ │ -0002c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a --------+.|i17 : │ │ │ │ -0002c200: 2053 6967 6d61 203d 207b 7b30 2c31 2c32 Sigma = {{0,1,2 │ │ │ │ -0002c210: 7d2c 7b31 2c32 2c33 7d2c 7b30 2c32 2c33 },{1,2,3},{0,2,3 │ │ │ │ -0002c220: 7d2c 7b30 2c31 2c34 7d2c 7b31 2c33 2c34 },{0,1,4},{1,3,4 │ │ │ │ -0002c230: 7d2c 7b30 2c33 2c34 7d7d 2020 2020 2020 },{0,3,4}} │ │ │ │ -0002c240: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002c250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c280: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c290: 7c6f 3137 203d 207b 7b30 2c20 312c 2032 |o17 = {{0, 1, 2 │ │ │ │ -0002c2a0: 7d2c 207b 312c 2032 2c20 337d 2c20 7b30 }, {1, 2, 3}, {0 │ │ │ │ -0002c2b0: 2c20 322c 2033 7d2c 207b 302c 2031 2c20 , 2, 3}, {0, 1, │ │ │ │ -0002c2c0: 347d 2c20 7b31 2c20 332c 2034 7d2c 207b 4}, {1, 3, 4}, { │ │ │ │ -0002c2d0: 302c 2033 2c20 347d 7d7c 0a7c 2020 2020 0, 3, 4}}|.| │ │ │ │ +0002c140: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0002c150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c190: 2d2d 2b0a 7c69 3137 203a 2053 6967 6d61 --+.|i17 : Sigma │ │ │ │ +0002c1a0: 203d 207b 7b30 2c31 2c32 7d2c 7b31 2c32 = {{0,1,2},{1,2 │ │ │ │ +0002c1b0: 2c33 7d2c 7b30 2c32 2c33 7d2c 7b30 2c31 ,3},{0,2,3},{0,1 │ │ │ │ +0002c1c0: 2c34 7d2c 7b31 2c33 2c34 7d2c 7b30 2c33 ,4},{1,3,4},{0,3 │ │ │ │ +0002c1d0: 2c34 7d7d 2020 2020 2020 2020 207c 0a7c ,4}} |.| │ │ │ │ +0002c1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c220: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ +0002c230: 207b 7b30 2c20 312c 2032 7d2c 207b 312c {{0, 1, 2}, {1, │ │ │ │ +0002c240: 2032 2c20 337d 2c20 7b30 2c20 322c 2033 2, 3}, {0, 2, 3 │ │ │ │ +0002c250: 7d2c 207b 302c 2031 2c20 347d 2c20 7b31 }, {0, 1, 4}, {1 │ │ │ │ +0002c260: 2c20 332c 2034 7d2c 207b 302c 2033 2c20 , 3, 4}, {0, 3, │ │ │ │ +0002c270: 347d 7d7c 0a7c 2020 2020 2020 2020 2020 4}}|.| │ │ │ │ +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 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c2c0: 7c6f 3137 203a 204c 6973 7420 2020 2020 |o17 : List │ │ │ │ +0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c320: 2020 2020 7c0a 7c6f 3137 203a 204c 6973 |.|o17 : Lis │ │ │ │ -0002c330: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -0002c340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c360: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c370: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002c380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 ----------+.|i18 │ │ │ │ -0002c3c0: 203a 2058 203d 206e 6f72 6d61 6c54 6f72 : X = normalTor │ │ │ │ -0002c3d0: 6963 5661 7269 6574 7928 5268 6f2c 5369 icVariety(Rho,Si │ │ │ │ -0002c3e0: 676d 612c 436f 6566 6669 6369 656e 7452 gma,CoefficientR │ │ │ │ -0002c3f0: 696e 6720 3d3e 5a5a 2f33 3237 3439 2920 ing =>ZZ/32749) │ │ │ │ -0002c400: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002c300: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002c310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c350: 2d2d 2d2d 2b0a 7c69 3138 203a 2058 203d ----+.|i18 : X = │ │ │ │ +0002c360: 206e 6f72 6d61 6c54 6f72 6963 5661 7269 normalToricVari │ │ │ │ +0002c370: 6574 7928 5268 6f2c 5369 676d 612c 436f ety(Rho,Sigma,Co │ │ │ │ +0002c380: 6566 6669 6369 656e 7452 696e 6720 3d3e efficientRing => │ │ │ │ +0002c390: 5a5a 2f33 3237 3439 2920 2020 2020 207c ZZ/32749) | │ │ │ │ +0002c3a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002c3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c3e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ +0002c3f0: 203d 2058 2020 2020 2020 2020 2020 2020 = X │ │ │ │ +0002c400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c430: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0002c440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c450: 7c0a 7c6f 3138 203d 2058 2020 2020 2020 |.|o18 = X │ │ │ │ +0002c450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c490: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002c480: 7c0a 7c6f 3138 203a 204e 6f72 6d61 6c54 |.|o18 : NormalT │ │ │ │ +0002c490: 6f72 6963 5661 7269 6574 7920 2020 2020 oricVariety │ │ │ │ 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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c4e0: 2020 2020 2020 7c0a 7c6f 3138 203a 204e |.|o18 : N │ │ │ │ -0002c4f0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574 ormalToricVariet │ │ │ │ -0002c500: 7920 2020 2020 2020 2020 2020 2020 2020 y │ │ │ │ -0002c510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c530: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002c580: 3139 203a 2043 6865 636b 546f 7269 6356 19 : CheckToricV │ │ │ │ -0002c590: 6172 6965 7479 5661 6c69 6428 5829 2020 arietyValid(X) │ │ │ │ -0002c5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002c4c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002c4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c510: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2043 ------+.|i19 : C │ │ │ │ +0002c520: 6865 636b 546f 7269 6356 6172 6965 7479 heckToricVariety │ │ │ │ +0002c530: 5661 6c69 6428 5829 2020 2020 2020 2020 Valid(X) │ │ │ │ +0002c540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c560: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002c570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c5a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0002c5b0: 3139 203d 2074 7275 6520 2020 2020 2020 19 = true │ │ │ │ +0002c5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c610: 2020 7c0a 7c6f 3139 203d 2074 7275 6520 |.|o19 = true │ │ │ │ -0002c620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c650: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002c660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c6a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a --------+.|i20 : │ │ │ │ -0002c6b0: 2052 3d72 696e 6728 5829 2020 2020 2020 R=ring(X) │ │ │ │ +0002c5f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0002c600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c640: 2d2d 2b0a 7c69 3230 203a 2052 3d72 696e --+.|i20 : R=rin │ │ │ │ +0002c650: 6728 5829 2020 2020 2020 2020 2020 2020 g(X) │ │ │ │ +0002c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c680: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002c690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c6f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002c6d0: 2020 2020 2020 2020 7c0a 7c6f 3230 203d |.|o20 = │ │ │ │ +0002c6e0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +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 2020 2020 2020 2020 │ │ │ │ -0002c730: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c740: 7c6f 3230 203d 2052 2020 2020 2020 2020 |o20 = R │ │ │ │ +0002c720: 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c780: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002c760: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c770: 7c6f 3230 203a 2050 6f6c 796e 6f6d 6961 |o20 : Polynomia │ │ │ │ +0002c780: 6c52 696e 6720 2020 2020 2020 2020 2020 lRing │ │ │ │ 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 2020 │ │ │ │ -0002c7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c7d0: 2020 2020 7c0a 7c6f 3230 203a 2050 6f6c |.|o20 : Pol │ │ │ │ -0002c7e0: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ -0002c7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c820: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2020 2020 4 2 2 │ │ │ │ +0002c8e0: 2020 2020 207c 0a7c 6f32 3120 3d20 6964 |.|o21 = id │ │ │ │ +0002c8f0: 6561 6c20 2878 2078 202c 202d 2078 2078 eal (x x , - x x │ │ │ │ +0002c900: 2020 2b20 7820 7820 7820 7820 2920 2020 + x x x x ) │ │ │ │ +0002c910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c940: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0002c950: 3120 3d20 6964 6561 6c20 2878 2078 202c 1 = ideal (x x , │ │ │ │ -0002c960: 202d 2078 2078 2020 2b20 7820 7820 7820 - x x + x x x │ │ │ │ -0002c970: 7820 2920 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ +0002c930: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002c940: 2030 2031 2020 2020 2030 2032 2020 2020 0 1 0 2 │ │ │ │ +0002c950: 3020 3220 3320 3420 2020 2020 2020 2020 0 2 3 4 │ │ │ │ +0002c960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c970: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002c980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c990: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002c9a0: 2020 2020 2020 2030 2031 2020 2020 2030 0 1 0 │ │ │ │ -0002c9b0: 2032 2020 2020 3020 3220 3320 3420 2020 2 0 2 3 4 │ │ │ │ -0002c9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c9e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002c990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c9c0: 2020 2020 2020 7c0a 7c6f 3231 203a 2049 |.|o21 : I │ │ │ │ +0002c9d0: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ +0002c9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ca00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ca10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ca20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002ca30: 3231 203a 2049 6465 616c 206f 6620 5220 21 : Ideal of R │ │ │ │ -0002ca40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ca50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ca60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ca70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cac0: 2d2d 2b0a 7c69 3232 203a 2053 6567 7265 --+.|i22 : Segre │ │ │ │ -0002cad0: 2858 2c49 2920 2020 2020 2020 2020 2020 (X,I) │ │ │ │ +0002ca10: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002ca20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ca30: 2d2d 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Read more │ │ │ │ -0002d430: 756e 6465 7220 2a6e 6f74 650a 7072 6f62 under *note.prob │ │ │ │ -0002d440: 6162 696c 6973 7469 6320 616c 676f 7269 abilistic algori │ │ │ │ -0002d450: 7468 6d3a 2070 726f 6261 6269 6c69 7374 thm: probabilist │ │ │ │ -0002d460: 6963 2061 6c67 6f72 6974 686d 2c2e 0a0a ic algorithm,... │ │ │ │ -0002d470: 5761 7973 2074 6f20 7573 6520 5365 6772 Ways to use Segr │ │ │ │ -0002d480: 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d e:.============= │ │ │ │ -0002d490: 3d3d 3d3d 3d0a 0a20 202a 2022 5365 6772 =====.. * "Segr │ │ │ │ -0002d4a0: 6528 4964 6561 6c29 220a 2020 2a20 2253 e(Ideal)". * "S │ │ │ │ -0002d4b0: 6567 7265 2849 6465 616c 2c53 796d 626f egre(Ideal,Symbo │ │ │ │ -0002d4c0: 6c29 220a 2020 2a20 2253 6567 7265 2851 l)". * "Segre(Q │ │ │ │ -0002d4d0: 756f 7469 656e 7452 696e 672c 4964 6561 uotientRing,Idea │ │ │ │ -0002d4e0: 6c29 220a 0a46 6f72 2074 6865 2070 726f l)"..For the pro │ │ │ │ -0002d4f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -0002d500: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -0002d510: 6f62 6a65 6374 202a 6e6f 7465 2053 6567 object *note Seg │ │ │ │ -0002d520: 7265 3a20 5365 6772 652c 2069 7320 6120 re: Segre, is a │ │ │ │ -0002d530: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e *note method fun │ │ │ │ -0002d540: 6374 696f 6e20 7769 7468 206f 7074 696f ction with optio │ │ │ │ -0002d550: 6e73 3a0a 284d 6163 6175 6c61 7932 446f ns:.(Macaulay2Do │ │ │ │ -0002d560: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -0002d570: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d WithOptions,...- │ │ │ │ -0002d580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -0002d5d0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -0002d5e0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -0002d5f0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -0002d600: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -0002d610: 6c61 7932 2d31 2e32 352e 3131 2b64 732f lay2-1.25.11+ds/ │ │ │ │ -0002d620: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -0002d630: 6b61 6765 732f 0a43 6861 7261 6374 6572 kages/.Character │ │ │ │ -0002d640: 6973 7469 6343 6c61 7373 6573 2e6d 323a isticClasses.m2: │ │ │ │ -0002d650: 3137 3633 3a30 2e0a 1f0a 4669 6c65 3a20 1763:0....File: │ │ │ │ -0002d660: 4368 6172 6163 7465 7269 7374 6963 436c CharacteristicCl │ │ │ │ -0002d670: 6173 7365 732e 696e 666f 2c20 4e6f 6465 asses.info, Node │ │ │ │ -0002d680: 3a20 546f 7269 6343 686f 7752 696e 672c : ToricChowRing, │ │ │ │ -0002d690: 2050 7265 763a 2053 6567 7265 2c20 5570 Prev: Segre, Up │ │ │ │ -0002d6a0: 3a20 546f 700a 0a54 6f72 6963 4368 6f77 : Top..ToricChow │ │ │ │ -0002d6b0: 5269 6e67 202d 2d20 436f 6d70 7574 6573 Ring -- Computes │ │ │ │ -0002d6c0: 2074 6865 2043 686f 7720 7269 6e67 206f the Chow ring o │ │ │ │ -0002d6d0: 6620 6120 6e6f 726d 616c 2074 6f72 6963 f a normal toric │ │ │ │ -0002d6e0: 2076 6172 6965 7479 0a2a 2a2a 2a2a 2a2a variety.******* │ │ │ │ -0002d6f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d720: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -0002d730: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -0002d740: 546f 7269 6343 686f 7752 696e 6720 580a ToricChowRing X. │ │ │ │ -0002d750: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -0002d760: 2020 2a20 522c 2061 202a 6e6f 7465 206e * R, a *note n │ │ │ │ -0002d770: 6f72 6d61 6c20 746f 7269 6320 7661 7269 ormal toric vari │ │ │ │ -0002d780: 6574 793a 0a20 2020 2020 2020 2028 4e6f ety:. (No │ │ │ │ -0002d790: 726d 616c 546f 7269 6356 6172 6965 7469 rmalToricVarieti │ │ │ │ -0002d7a0: 6573 294e 6f72 6d61 6c54 6f72 6963 5661 es)NormalToricVa │ │ │ │ -0002d7b0: 7269 6574 792c 2c20 4120 6e6f 726d 616c riety,, A normal │ │ │ │ -0002d7c0: 2074 6f72 6963 2076 6172 6965 7479 0a20 toric variety. │ │ │ │ -0002d7d0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -0002d7e0: 2020 2a20 6120 2a6e 6f74 6520 7175 6f74 * a *note quot │ │ │ │ -0002d7f0: 6965 6e74 2072 696e 673a 2028 4d61 6361 ient ring: (Maca │ │ │ │ -0002d800: 756c 6179 3244 6f63 2951 756f 7469 656e ulay2Doc)Quotien │ │ │ │ -0002d810: 7452 696e 672c 2c20 0a0a 4465 7363 7269 tRing,, ..Descri │ │ │ │ -0002d820: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -0002d830: 3d0a 0a4c 6574 2058 2062 6520 6120 746f =..Let X be a to │ │ │ │ -0002d840: 7269 6320 7661 7269 6574 7920 7769 7468 ric variety with │ │ │ │ -0002d850: 2074 6f74 616c 2063 6f6f 7264 696e 6174 total coordinat │ │ │ │ -0002d860: 6520 7269 6e67 2028 436f 7820 7269 6e67 e ring (Cox ring │ │ │ │ -0002d870: 2920 522e 2054 6869 7320 6d65 7468 6f64 ) R. This method │ │ │ │ -0002d880: 0a63 6f6d 7075 7465 7320 7468 6520 4368 .computes the Ch │ │ │ │ -0002d890: 6f77 2072 696e 6720 2043 686f 7720 7269 ow ring Chow ri │ │ │ │ -0002d8a0: 6e67 2043 683d 522f 2853 522b 4c52 2920 ng Ch=R/(SR+LR) │ │ │ │ -0002d8b0: 6f66 2058 3b20 6865 7265 2053 5220 6973 of X; here SR is │ │ │ │ -0002d8c0: 2074 6865 0a53 7461 6e6c 6579 2d52 6569 the.Stanley-Rei │ │ │ │ -0002d8d0: 736e 6572 2069 6465 616c 206f 6620 7468 sner ideal of th │ │ │ │ -0002d8e0: 6520 636f 7272 6573 706f 6e64 696e 6720 e corresponding │ │ │ │ -0002d8f0: 6661 6e20 616e 6420 4c52 2069 7320 7468 fan and LR is th │ │ │ │ -0002d900: 6520 6964 6561 6c20 6f66 206c 696e 6561 e ideal of linea │ │ │ │ -0002d910: 720a 7265 6c61 7469 6f6e 7320 616d 6f75 r.relations amou │ │ │ │ -0002d920: 6e74 2074 6865 2072 6179 732e 2049 7420 nt the rays. It │ │ │ │ -0002d930: 6973 206e 6565 6465 6420 666f 7220 696e is needed for in │ │ │ │ -0002d940: 7075 7420 696e 746f 2074 6865 206d 6574 put into the met │ │ │ │ -0002d950: 686f 6473 202a 6e6f 7465 2053 6567 7265 hods *note Segre │ │ │ │ -0002d960: 3a0a 5365 6772 652c 2c20 2a6e 6f74 6520 :.Segre,, *note │ │ │ │ -0002d970: 4368 6572 6e3a 2043 6865 726e 2c20 616e Chern: Chern, an │ │ │ │ -0002d980: 6420 2a6e 6f74 6520 4353 4d3a 2043 534d d *note CSM: CSM │ │ │ │ -0002d990: 2c20 696e 2074 6865 2063 6173 6573 2077 , in the cases w │ │ │ │ -0002d9a0: 6865 7265 2061 2074 6f72 6963 0a76 6172 here a toric.var │ │ │ │ -0002d9b0: 6965 7479 2069 7320 616c 736f 2069 6e70 iety is also inp │ │ │ │ -0002d9c0: 7574 2074 6f20 656e 7375 7265 2074 6861 ut to ensure tha │ │ │ │ -0002d9d0: 7420 7468 6573 6520 6d65 7468 6f64 7320 t these methods │ │ │ │ -0002d9e0: 7265 7475 726e 2072 6573 756c 7473 2069 return results i │ │ │ │ -0002d9f0: 6e20 7468 6520 7361 6d65 0a72 696e 672e n the same.ring. │ │ │ │ -0002da00: 2057 6520 6769 7665 2061 6e20 6578 616d We give an exam │ │ │ │ -0002da10: 706c 6520 6f66 2074 6865 2075 7365 206f ple of the use o │ │ │ │ -0002da20: 6620 7468 6973 206d 6574 686f 6420 746f f this method to │ │ │ │ -0002da30: 2077 6f72 6b20 7769 7468 2065 6c65 6d65 work with eleme │ │ │ │ -0002da40: 6e74 7320 6f66 2074 6865 0a43 686f 7720 nts of the.Chow │ │ │ │ -0002da50: 7269 6e67 206f 6620 6120 746f 7269 6320 ring of a toric │ │ │ │ -0002da60: 7661 7269 6574 790a 0a2b 2d2d 2d2d 2d2d variety..+------ │ │ │ │ -0002da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dab0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206e -------+.|i1 : n │ │ │ │ -0002dac0: 6565 6473 5061 636b 6167 6520 224e 6f72 eedsPackage "Nor │ │ │ │ -0002dad0: 6d61 6c54 6f72 6963 5661 7269 6574 6965 malToricVarietie │ │ │ │ -0002dae0: 7322 2020 2020 2020 2020 2020 2020 2020 s" │ │ │ │ -0002daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002d1a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002d1b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002d1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 6c6c ----------+..All │ │ │ │ +0002d200: 2074 6865 2065 7861 6d70 6c65 7320 7765 the examples we │ │ │ │ +0002d210: 7265 2064 6f6e 6520 7573 696e 6720 7379 re done using sy │ │ │ │ +0002d220: 6d62 6f6c 6963 2063 6f6d 7075 7461 7469 mbolic computati │ │ │ │ +0002d230: 6f6e 7320 7769 7468 2047 725c 226f 626e ons with Gr\"obn │ │ │ │ +0002d240: 6572 2062 6173 6573 2e0a 4368 616e 6769 er bases..Changi │ │ │ │ +0002d250: 6e67 2074 6865 206f 7074 696f 6e20 2a6e ng the option *n │ │ │ │ +0002d260: 6f74 6520 436f 6d70 4d65 7468 6f64 3a20 ote CompMethod: │ │ │ │ +0002d270: 436f 6d70 4d65 7468 6f64 2c20 746f 2062 CompMethod, to b │ │ │ │ +0002d280: 6572 7469 6e69 2077 696c 6c20 646f 2074 ertini will do t │ │ │ │ +0002d290: 6865 206d 6169 6e0a 636f 6d70 7574 6174 he main.computat │ │ │ │ +0002d2a0: 696f 6e73 206e 756d 6572 6963 616c 6c79 ions numerically │ │ │ │ +0002d2b0: 2c20 7072 6f76 6964 6564 2042 6572 7469 , provided Berti │ │ │ │ +0002d2c0: 6e69 2069 7320 202a 6e6f 7465 2069 6e73 ni is *note ins │ │ │ │ +0002d2d0: 7461 6c6c 6564 2061 6e64 2063 6f6e 6669 talled and confi │ │ │ │ +0002d2e0: 6775 7265 643a 0a63 6f6e 6669 6775 7269 gured:.configuri │ │ │ │ +0002d2f0: 6e67 2042 6572 7469 6e69 2c2e 204e 6f74 ng Bertini,. Not │ │ │ │ +0002d300: 6520 7468 6174 2074 6865 2062 6572 7469 e that the berti │ │ │ │ +0002d310: 6e69 206f 7074 696f 6e20 6973 206f 6e6c ni option is onl │ │ │ │ +0002d320: 7920 6176 6169 6c61 626c 6520 666f 720a y available for. │ │ │ │ +0002d330: 7375 6273 6368 656d 6573 206f 6620 5c50 subschemes of \P │ │ │ │ +0002d340: 505e 6e2e 0a0a 4f62 7365 7276 6520 7468 P^n...Observe th │ │ │ │ +0002d350: 6174 2074 6865 2061 6c67 6f72 6974 686d at the algorithm │ │ │ │ +0002d360: 2069 7320 6120 7072 6f62 6162 696c 6973 is a probabilis │ │ │ │ +0002d370: 7469 6320 616c 676f 7269 7468 6d20 616e tic algorithm an │ │ │ │ +0002d380: 6420 6d61 7920 6769 7665 2061 2077 726f d may give a wro │ │ │ │ +0002d390: 6e67 0a61 6e73 7765 7220 7769 7468 2061 ng.answer with a │ │ │ │ +0002d3a0: 2073 6d61 6c6c 2062 7574 206e 6f6e 7a65 small but nonze │ │ │ │ +0002d3b0: 726f 2070 726f 6261 6269 6c69 7479 2e20 ro probability. │ │ │ │ +0002d3c0: 5265 6164 206d 6f72 6520 756e 6465 7220 Read more under │ │ │ │ +0002d3d0: 2a6e 6f74 650a 7072 6f62 6162 696c 6973 *note.probabilis │ │ │ │ +0002d3e0: 7469 6320 616c 676f 7269 7468 6d3a 2070 tic algorithm: p │ │ │ │ +0002d3f0: 726f 6261 6269 6c69 7374 6963 2061 6c67 robabilistic alg │ │ │ │ +0002d400: 6f72 6974 686d 2c2e 0a0a 5761 7973 2074 orithm,...Ways t │ │ │ │ +0002d410: 6f20 7573 6520 5365 6772 653a 0a3d 3d3d o use Segre:.=== │ │ │ │ +0002d420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0002d430: 0a20 202a 2022 5365 6772 6528 4964 6561 . * "Segre(Idea │ │ │ │ +0002d440: 6c29 220a 2020 2a20 2253 6567 7265 2849 l)". * "Segre(I │ │ │ │ +0002d450: 6465 616c 2c53 796d 626f 6c29 220a 2020 deal,Symbol)". │ │ │ │ +0002d460: 2a20 2253 6567 7265 2851 756f 7469 656e * "Segre(Quotien │ │ │ │ +0002d470: 7452 696e 672c 4964 6561 6c29 220a 0a46 tRing,Ideal)"..F │ │ │ │ +0002d480: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +0002d490: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +0002d4a0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +0002d4b0: 202a 6e6f 7465 2053 6567 7265 3a20 5365 *note Segre: Se │ │ │ │ +0002d4c0: 6772 652c 2069 7320 6120 2a6e 6f74 6520 gre, is a *note │ │ │ │ +0002d4d0: 6d65 7468 6f64 2066 756e 6374 696f 6e20 method function │ │ │ │ +0002d4e0: 7769 7468 206f 7074 696f 6e73 3a0a 284d with options:.(M │ │ │ │ +0002d4f0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ +0002d500: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70 odFunctionWithOp │ │ │ │ +0002d510: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d tions,...------- │ │ │ │ +0002d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d560: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0002d570: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0002d580: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0002d590: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0002d5a0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0002d5b0: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163 .25.11+ds/M2/Mac │ │ │ │ +0002d5c0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0002d5d0: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ +0002d5e0: 6c61 7373 6573 2e6d 323a 3137 3633 3a30 lasses.m2:1763:0 │ │ │ │ +0002d5f0: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ +0002d600: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +0002d610: 696e 666f 2c20 4e6f 6465 3a20 546f 7269 info, Node: Tori │ │ │ │ +0002d620: 6343 686f 7752 696e 672c 2050 7265 763a cChowRing, Prev: │ │ │ │ +0002d630: 2053 6567 7265 2c20 5570 3a20 546f 700a Segre, Up: Top. │ │ │ │ +0002d640: 0a54 6f72 6963 4368 6f77 5269 6e67 202d .ToricChowRing - │ │ │ │ +0002d650: 2d20 436f 6d70 7574 6573 2074 6865 2043 - Computes the C │ │ │ │ +0002d660: 686f 7720 7269 6e67 206f 6620 6120 6e6f how ring of a no │ │ │ │ +0002d670: 726d 616c 2074 6f72 6963 2076 6172 6965 rmal toric varie │ │ │ │ +0002d680: 7479 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ty.************* │ │ │ │ +0002d690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002d6a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002d6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002d6c0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ +0002d6d0: 200a 2020 2020 2020 2020 546f 7269 6343 . ToricC │ │ │ │ +0002d6e0: 686f 7752 696e 6720 580a 2020 2a20 496e howRing X. * In │ │ │ │ +0002d6f0: 7075 7473 3a0a 2020 2020 2020 2a20 522c puts:. * R, │ │ │ │ +0002d700: 2061 202a 6e6f 7465 206e 6f72 6d61 6c20 a *note normal │ │ │ │ +0002d710: 746f 7269 6320 7661 7269 6574 793a 0a20 toric variety:. │ │ │ │ +0002d720: 2020 2020 2020 2028 4e6f 726d 616c 546f (NormalTo │ │ │ │ +0002d730: 7269 6356 6172 6965 7469 6573 294e 6f72 ricVarieties)Nor │ │ │ │ +0002d740: 6d61 6c54 6f72 6963 5661 7269 6574 792c malToricVariety, │ │ │ │ +0002d750: 2c20 4120 6e6f 726d 616c 2074 6f72 6963 , A normal toric │ │ │ │ +0002d760: 2076 6172 6965 7479 0a20 202a 204f 7574 variety. * Out │ │ │ │ +0002d770: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ +0002d780: 2a6e 6f74 6520 7175 6f74 6965 6e74 2072 *note quotient r │ │ │ │ +0002d790: 696e 673a 2028 4d61 6361 756c 6179 3244 ing: (Macaulay2D │ │ │ │ +0002d7a0: 6f63 2951 756f 7469 656e 7452 696e 672c oc)QuotientRing, │ │ │ │ +0002d7b0: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ +0002d7c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a4c 6574 ===========..Let │ │ │ │ +0002d7d0: 2058 2062 6520 6120 746f 7269 6320 7661 X be a toric va │ │ │ │ +0002d7e0: 7269 6574 7920 7769 7468 2074 6f74 616c riety with total │ │ │ │ +0002d7f0: 2063 6f6f 7264 696e 6174 6520 7269 6e67 coordinate ring │ │ │ │ +0002d800: 2028 436f 7820 7269 6e67 2920 522e 2054 (Cox ring) R. T │ │ │ │ +0002d810: 6869 7320 6d65 7468 6f64 0a63 6f6d 7075 his method.compu │ │ │ │ +0002d820: 7465 7320 7468 6520 4368 6f77 2072 696e tes the Chow rin │ │ │ │ +0002d830: 6720 2043 686f 7720 7269 6e67 2043 683d g Chow ring Ch= │ │ │ │ +0002d840: 522f 2853 522b 4c52 2920 6f66 2058 3b20 R/(SR+LR) of X; │ │ │ │ +0002d850: 6865 7265 2053 5220 6973 2074 6865 0a53 here SR is the.S │ │ │ │ +0002d860: 7461 6e6c 6579 2d52 6569 736e 6572 2069 tanley-Reisner i │ │ │ │ +0002d870: 6465 616c 206f 6620 7468 6520 636f 7272 deal of the corr │ │ │ │ +0002d880: 6573 706f 6e64 696e 6720 6661 6e20 616e esponding fan an │ │ │ │ +0002d890: 6420 4c52 2069 7320 7468 6520 6964 6561 d LR is the idea │ │ │ │ +0002d8a0: 6c20 6f66 206c 696e 6561 720a 7265 6c61 l of linear.rela │ │ │ │ +0002d8b0: 7469 6f6e 7320 616d 6f75 6e74 2074 6865 tions amount the │ │ │ │ +0002d8c0: 2072 6179 732e 2049 7420 6973 206e 6565 rays. It is nee │ │ │ │ +0002d8d0: 6465 6420 666f 7220 696e 7075 7420 696e ded for input in │ │ │ │ +0002d8e0: 746f 2074 6865 206d 6574 686f 6473 202a to the methods * │ │ │ │ +0002d8f0: 6e6f 7465 2053 6567 7265 3a0a 5365 6772 note Segre:.Segr │ │ │ │ +0002d900: 652c 2c20 2a6e 6f74 6520 4368 6572 6e3a e,, *note Chern: │ │ │ │ +0002d910: 2043 6865 726e 2c20 616e 6420 2a6e 6f74 Chern, and *not │ │ │ │ +0002d920: 6520 4353 4d3a 2043 534d 2c20 696e 2074 e CSM: CSM, in t │ │ │ │ +0002d930: 6865 2063 6173 6573 2077 6865 7265 2061 he cases where a │ │ │ │ +0002d940: 2074 6f72 6963 0a76 6172 6965 7479 2069 toric.variety i │ │ │ │ +0002d950: 7320 616c 736f 2069 6e70 7574 2074 6f20 s also input to │ │ │ │ +0002d960: 656e 7375 7265 2074 6861 7420 7468 6573 ensure that thes │ │ │ │ +0002d970: 6520 6d65 7468 6f64 7320 7265 7475 726e e methods return │ │ │ │ +0002d980: 2072 6573 756c 7473 2069 6e20 7468 6520 results in the │ │ │ │ +0002d990: 7361 6d65 0a72 696e 672e 2057 6520 6769 same.ring. We gi │ │ │ │ +0002d9a0: 7665 2061 6e20 6578 616d 706c 6520 6f66 ve an example of │ │ │ │ +0002d9b0: 2074 6865 2075 7365 206f 6620 7468 6973 the use of this │ │ │ │ +0002d9c0: 206d 6574 686f 6420 746f 2077 6f72 6b20 method to work │ │ │ │ +0002d9d0: 7769 7468 2065 6c65 6d65 6e74 7320 6f66 with elements of │ │ │ │ +0002d9e0: 2074 6865 0a43 686f 7720 7269 6e67 206f the.Chow ring o │ │ │ │ +0002d9f0: 6620 6120 746f 7269 6320 7661 7269 6574 f a toric variet │ │ │ │ +0002da00: 790a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d y..+------------ │ │ │ │ +0002da10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002da50: 2d2b 0a7c 6931 203a 206e 6565 6473 5061 -+.|i1 : needsPa │ │ │ │ +0002da60: 636b 6167 6520 224e 6f72 6d61 6c54 6f72 ckage "NormalTor │ │ │ │ +0002da70: 6963 5661 7269 6574 6965 7322 2020 2020 icVarieties" │ │ │ │ +0002da80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002da90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002daa0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002daf0: 207c 0a7c 6f31 203d 204e 6f72 6d61 6c54 |.|o1 = NormalT │ │ │ │ +0002db00: 6f72 6963 5661 7269 6574 6965 7320 2020 oricVarieties │ │ │ │ 0002db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db50: 2020 2020 2020 207c 0a7c 6f31 203d 204e |.|o1 = N │ │ │ │ -0002db60: 6f72 6d61 6c54 6f72 6963 5661 7269 6574 ormalToricVariet │ │ │ │ -0002db70: 6965 7320 2020 2020 2020 2020 2020 2020 ies │ │ │ │ +0002db40: 207c 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dba0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002db90: 207c 0a7c 6f31 203a 2050 6163 6b61 6765 |.|o1 : Package │ │ │ │ +0002dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbf0: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ -0002dc00: 6163 6b61 6765 2020 2020 2020 2020 2020 ackage │ │ │ │ -0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc90: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 -------+.|i2 : R │ │ │ │ -0002dca0: 686f 203d 207b 7b31 2c30 2c30 7d2c 7b30 ho = {{1,0,0},{0 │ │ │ │ -0002dcb0: 2c31 2c30 7d2c 7b30 2c30 2c31 7d2c 7b2d ,1,0},{0,0,1},{- │ │ │ │ -0002dcc0: 312c 2d31 2c30 7d2c 7b30 2c30 2c2d 317d 1,-1,0},{0,0,-1} │ │ │ │ -0002dcd0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ -0002dce0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002dcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd30: 2020 2020 2020 207c 0a7c 6f32 203d 207b |.|o2 = { │ │ │ │ -0002dd40: 7b31 2c20 302c 2030 7d2c 207b 302c 2031 {1, 0, 0}, {0, 1 │ │ │ │ -0002dd50: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c , 0}, {0, 0, 1}, │ │ │ │ -0002dd60: 207b 2d31 2c20 2d31 2c20 307d 2c20 7b30 {-1, -1, 0}, {0 │ │ │ │ -0002dd70: 2c20 302c 202d 317d 7d20 2020 2020 2020 , 0, -1}} │ │ │ │ -0002dd80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002dbe0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002dbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dc30: 2d2b 0a7c 6932 203a 2052 686f 203d 207b -+.|i2 : Rho = { │ │ │ │ +0002dc40: 7b31 2c30 2c30 7d2c 7b30 2c31 2c30 7d2c {1,0,0},{0,1,0}, │ │ │ │ +0002dc50: 7b30 2c30 2c31 7d2c 7b2d 312c 2d31 2c30 {0,0,1},{-1,-1,0 │ │ │ │ +0002dc60: 7d2c 7b30 2c30 2c2d 317d 7d20 2020 2020 },{0,0,-1}} │ │ │ │ +0002dc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc80: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dcd0: 207c 0a7c 6f32 203d 207b 7b31 2c20 302c |.|o2 = {{1, 0, │ │ │ │ +0002dce0: 2030 7d2c 207b 302c 2031 2c20 307d 2c20 0}, {0, 1, 0}, │ │ │ │ +0002dcf0: 7b30 2c20 302c 2031 7d2c 207b 2d31 2c20 {0, 0, 1}, {-1, │ │ │ │ +0002dd00: 2d31 2c20 307d 2c20 7b30 2c20 302c 202d -1, 0}, {0, 0, - │ │ │ │ +0002dd10: 317d 7d20 2020 2020 2020 2020 2020 2020 1}} │ │ │ │ +0002dd20: 207c 0a7c 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 2020 2020 2020 │ │ │ │ +0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd70: 207c 0a7c 6f32 203a 204c 6973 7420 2020 |.|o2 : List │ │ │ │ +0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ddd0: 2020 2020 2020 207c 0a7c 6f32 203a 204c |.|o2 : L │ │ │ │ -0002dde0: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ -0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de20: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de70: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2053 -------+.|i3 : S │ │ │ │ -0002de80: 6967 6d61 203d 207b 7b30 2c31 2c32 7d2c igma = {{0,1,2}, │ │ │ │ -0002de90: 7b31 2c32 2c33 7d2c 7b30 2c32 2c33 7d2c {1,2,3},{0,2,3}, │ │ │ │ -0002dea0: 7b30 2c31 2c34 7d2c 7b31 2c33 2c34 7d2c {0,1,4},{1,3,4}, │ │ │ │ -0002deb0: 7b30 2c33 2c34 7d7d 2020 2020 2020 2020 {0,3,4}} │ │ │ │ -0002dec0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002def0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df10: 2020 2020 2020 207c 0a7c 6f33 203d 207b |.|o3 = { │ │ │ │ -0002df20: 7b30 2c20 312c 2032 7d2c 207b 312c 2032 {0, 1, 2}, {1, 2 │ │ │ │ -0002df30: 2c20 337d 2c20 7b30 2c20 322c 2033 7d2c , 3}, {0, 2, 3}, │ │ │ │ -0002df40: 207b 302c 2031 2c20 347d 2c20 7b31 2c20 {0, 1, 4}, {1, │ │ │ │ -0002df50: 332c 2034 7d2c 207b 302c 2033 2c20 347d 3, 4}, {0, 3, 4} │ │ │ │ -0002df60: 7d20 2020 2020 207c 0a7c 2020 2020 2020 } |.| │ │ │ │ +0002ddc0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002ddd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ddf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002de10: 2d2b 0a7c 6933 203a 2053 6967 6d61 203d -+.|i3 : Sigma = │ │ │ │ +0002de20: 207b 7b30 2c31 2c32 7d2c 7b31 2c32 2c33 {{0,1,2},{1,2,3 │ │ │ │ +0002de30: 7d2c 7b30 2c32 2c33 7d2c 7b30 2c31 2c34 },{0,2,3},{0,1,4 │ │ │ │ +0002de40: 7d2c 7b31 2c33 2c34 7d2c 7b30 2c33 2c34 },{1,3,4},{0,3,4 │ │ │ │ +0002de50: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ +0002de60: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002de70: 2020 2020 2020 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: 207c 0a7c 6f33 203d 207b 7b30 2c20 312c |.|o3 = {{0, 1, │ │ │ │ +0002dec0: 2032 7d2c 207b 312c 2032 2c20 337d 2c20 2}, {1, 2, 3}, │ │ │ │ +0002ded0: 7b30 2c20 322c 2033 7d2c 207b 302c 2031 {0, 2, 3}, {0, 1 │ │ │ │ +0002dee0: 2c20 347d 2c20 7b31 2c20 332c 2034 7d2c , 4}, {1, 3, 4}, │ │ │ │ +0002def0: 207b 302c 2033 2c20 347d 7d20 2020 2020 {0, 3, 4}} │ │ │ │ +0002df00: 207c 0a7c 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 2020 2020 2020 │ │ │ │ +0002df40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df50: 207c 0a7c 6f33 203a 204c 6973 7420 2020 |.|o3 : List │ │ │ │ +0002df60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002df80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002df90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfb0: 2020 2020 2020 207c 0a7c 6f33 203a 204c |.|o3 : L │ │ │ │ -0002dfc0: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ -0002dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e000: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e050: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2058 -------+.|i4 : X │ │ │ │ -0002e060: 203d 206e 6f72 6d61 6c54 6f72 6963 5661 = normalToricVa │ │ │ │ -0002e070: 7269 6574 7928 5268 6f2c 5369 676d 612c riety(Rho,Sigma, │ │ │ │ -0002e080: 436f 6566 6669 6369 656e 7452 696e 6720 CoefficientRing │ │ │ │ -0002e090: 3d3e 5a5a 2f33 3237 3439 2920 2020 2020 =>ZZ/32749) │ │ │ │ -0002e0a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002dfa0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dff0: 2d2b 0a7c 6934 203a 2058 203d 206e 6f72 -+.|i4 : X = nor │ │ │ │ +0002e000: 6d61 6c54 6f72 6963 5661 7269 6574 7928 malToricVariety( │ │ │ │ +0002e010: 5268 6f2c 5369 676d 612c 436f 6566 6669 Rho,Sigma,Coeffi │ │ │ │ +0002e020: 6369 656e 7452 696e 6720 3d3e 5a5a 2f33 cientRing =>ZZ/3 │ │ │ │ +0002e030: 3237 3439 2920 2020 2020 2020 2020 2020 2749) │ │ │ │ +0002e040: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e090: 207c 0a7c 6f34 203d 2058 2020 2020 2020 |.|o4 = X │ │ │ │ +0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0f0: 2020 2020 2020 207c 0a7c 6f34 203d 2058 |.|o4 = X │ │ │ │ +0002e0e0: 207c 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e140: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002e130: 207c 0a7c 6f34 203a 204e 6f72 6d61 6c54 |.|o4 : NormalT │ │ │ │ +0002e140: 6f72 6963 5661 7269 6574 7920 2020 2020 oricVariety │ │ │ │ 0002e150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e190: 2020 2020 2020 207c 0a7c 6f34 203a 204e |.|o4 : N │ │ │ │ -0002e1a0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574 ormalToricVariet │ │ │ │ -0002e1b0: 7920 2020 2020 2020 2020 2020 2020 2020 y │ │ │ │ -0002e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e230: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2052 -------+.|i5 : R │ │ │ │ -0002e240: 3d72 696e 6720 5820 2020 2020 2020 2020 =ring X │ │ │ │ +0002e180: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e1d0: 2d2b 0a7c 6935 203a 2052 3d72 696e 6720 -+.|i5 : R=ring │ │ │ │ +0002e1e0: 5820 2020 2020 2020 2020 2020 2020 2020 X │ │ │ │ +0002e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e220: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e280: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002e270: 207c 0a7c 6f35 203d 2052 2020 2020 2020 |.|o5 = R │ │ │ │ +0002e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2d0: 2020 2020 2020 207c 0a7c 6f35 203d 2052 |.|o5 = R │ │ │ │ +0002e2c0: 207c 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e320: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002e310: 207c 0a7c 6f35 203a 2050 6f6c 796e 6f6d |.|o5 : Polynom │ │ │ │ +0002e320: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ 0002e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e370: 2020 2020 2020 207c 0a7c 6f35 203a 2050 |.|o5 : P │ │ │ │ -0002e380: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ -0002e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e410: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2043 -------+.|i6 : C │ │ │ │ -0002e420: 683d 546f 7269 6343 686f 7752 696e 6728 h=ToricChowRing( │ │ │ │ -0002e430: 5829 2020 2020 2020 2020 2020 2020 2020 X) │ │ │ │ +0002e360: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002e370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e3b0: 2d2b 0a7c 6936 203a 2043 683d 546f 7269 -+.|i6 : Ch=Tori │ │ │ │ +0002e3c0: 6343 686f 7752 696e 6728 5829 2020 2020 cChowRing(X) │ │ │ │ +0002e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e400: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e460: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002e450: 207c 0a7c 6f36 203d 2043 6820 2020 2020 |.|o6 = Ch │ │ │ │ +0002e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4b0: 2020 2020 2020 207c 0a7c 6f36 203d 2043 |.|o6 = C │ │ │ │ -0002e4c0: 6820 2020 2020 2020 2020 2020 2020 2020 h │ │ │ │ +0002e4a0: 207c 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e500: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002e4f0: 207c 0a7c 6f36 203a 2051 756f 7469 656e |.|o6 : Quotien │ │ │ │ +0002e500: 7452 696e 6720 2020 2020 2020 2020 2020 tRing │ │ │ │ 0002e510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e550: 2020 2020 2020 207c 0a7c 6f36 203a 2051 |.|o6 : Q │ │ │ │ -0002e560: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ -0002e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e5a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2064 -------+.|i7 : d │ │ │ │ -0002e600: 6573 6372 6962 6520 4368 2020 2020 2020 escribe Ch │ │ │ │ +0002e540: 207c 0a2b 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e590: 2d2b 0a7c 6937 203a 2064 6573 6372 6962 -+.|i7 : describ │ │ │ │ +0002e5a0: 6520 4368 2020 2020 2020 2020 2020 2020 e Ch │ │ │ │ +0002e5b0: 2020 2020 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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e640: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e630: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e640: 2020 2020 2020 2020 205a 5a5b 7820 2e2e ZZ[x .. │ │ │ │ +0002e650: 7820 5d20 2020 2020 2020 2020 2020 2020 x ] │ │ │ │ 0002e660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e690: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002e6a0: 2020 2020 2020 2020 2020 2020 2020 205a Z │ │ │ │ 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2020 2020 2020 │ │ │ │ +0002e880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e8c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002e8b0: 207c 0a7c 6f38 203d 207b 7820 2c20 7820 |.|o8 = {x , x │ │ │ │ +0002e8c0: 2c20 7820 2c20 7820 2c20 7820 7d20 2020 , x , x , x } │ │ │ │ 0002e8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e910: 2020 2020 2020 207c 0a7c 6f38 203d 207b |.|o8 = { │ │ │ │ -0002e920: 7820 2c20 7820 2c20 7820 2c20 7820 2c20 x , x , x , x , │ │ │ │ -0002e930: 7820 7d20 2020 2020 2020 2020 2020 2020 x } │ │ │ │ +0002e900: 207c 0a7c 2020 2020 2020 2030 2020 2031 |.| 0 1 │ │ │ │ +0002e910: 2020 2032 2020 2033 2020 2034 2020 2020 2 3 4 │ │ │ │ +0002e920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e960: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002e970: 2030 2020 2031 2020 2032 2020 2033 2020 0 1 2 3 │ │ │ │ -0002e980: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +0002e950: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e9b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002e9a0: 207c 0a7c 6f38 203a 204c 6973 7420 2020 |.|o8 : List │ │ │ │ +0002e9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea00: 2020 2020 2020 207c 0a7c 6f38 203a 204c |.|o8 : L │ │ │ │ -0002ea10: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ -0002ea20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002ea60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ea70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ea90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eaa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2049 -------+.|i9 : I │ │ │ │ -0002eab0: 3d69 6465 616c 2872 616e 646f 6d28 7b31 =ideal(random({1 │ │ │ │ -0002eac0: 2c30 7d2c 5229 2920 2020 2020 2020 2020 ,0},R)) │ │ │ │ +0002e9f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ea40: 2d2b 0a7c 6939 203a 2049 3d69 6465 616c -+.|i9 : I=ideal │ │ │ │ +0002ea50: 2872 616e 646f 6d28 7b31 2c30 7d2c 5229 (random({1,0},R) │ │ │ │ +0002ea60: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0002ea70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ea80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ea90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002eaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eaf0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002eb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eae0: 207c 0a7c 6f39 203d 2069 6465 616c 2831 |.|o9 = ideal(1 │ │ │ │ +0002eaf0: 3037 7820 202b 2034 3337 3678 2020 2d20 07x + 4376x - │ │ │ │ +0002eb00: 3633 3136 7820 2920 2020 2020 2020 2020 6316x ) │ │ │ │ 0002eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb40: 2020 2020 2020 207c 0a7c 6f39 203d 2069 |.|o9 = i │ │ │ │ -0002eb50: 6465 616c 2831 3037 7820 202b 2034 3337 deal(107x + 437 │ │ │ │ -0002eb60: 3678 2020 2d20 3633 3136 7820 2920 2020 6x - 6316x ) │ │ │ │ +0002eb30: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002eb40: 2020 2030 2020 2020 2020 2020 3120 2020 0 1 │ │ │ │ +0002eb50: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ +0002eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002eba0: 2020 2020 2020 2020 2030 2020 2020 2020 0 │ │ │ │ -0002ebb0: 2020 3120 2020 2020 2020 2033 2020 2020 1 3 │ │ │ │ +0002eb80: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002eb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ebd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ebe0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002ebd0: 207c 0a7c 6f39 203a 2049 6465 616c 206f |.|o9 : Ideal o │ │ │ │ +0002ebe0: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ 0002ebf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec30: 2020 2020 2020 207c 0a7c 6f39 203a 2049 |.|o9 : I │ │ │ │ -0002ec40: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ -0002ec50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ecd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ -0002ece0: 4b3d 6964 6561 6c28 7261 6e64 6f6d 287b K=ideal(random({ │ │ │ │ -0002ecf0: 312c 317d 2c52 2929 2020 2020 2020 2020 1,1},R)) │ │ │ │ +0002ec20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002ec30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec70: 2d2b 0a7c 6931 3020 3a20 4b3d 6964 6561 -+.|i10 : K=idea │ │ │ │ +0002ec80: 6c28 7261 6e64 6f6d 287b 312c 317d 2c52 l(random({1,1},R │ │ │ │ +0002ec90: 2929 2020 2020 2020 2020 2020 2020 2020 )) │ │ │ │ +0002eca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ecb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ecc0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002ecd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ece0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ed00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002ed30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed70: 2020 2020 2020 207c 0a7c 6f31 3020 3d20 |.|o10 = │ │ │ │ -0002ed80: 6964 6561 6c28 3331 3837 7820 7820 202d ideal(3187x x - │ │ │ │ -0002ed90: 2036 3035 3378 2078 2020 2d20 3136 3039 6053x x - 1609 │ │ │ │ -0002eda0: 3078 2078 2020 2b20 3337 3833 7820 7820 0x x + 3783x x │ │ │ │ -0002edb0: 202b 2038 3537 3078 2078 2020 2b20 3834 + 8570x x + 84 │ │ │ │ -0002edc0: 3434 7820 7820 297c 0a7c 2020 2020 2020 44x x )|.| │ │ │ │ -0002edd0: 2020 2020 2020 2020 2020 2030 2032 2020 0 2 │ │ │ │ -0002ede0: 2020 2020 2020 3120 3220 2020 2020 2020 1 2 │ │ │ │ -0002edf0: 2020 3220 3320 2020 2020 2020 2030 2034 2 3 0 4 │ │ │ │ -0002ee00: 2020 2020 2020 2020 3120 3420 2020 2020 1 4 │ │ │ │ -0002ee10: 2020 2033 2034 207c 0a7c 2020 2020 2020 3 4 |.| │ │ │ │ +0002ed10: 207c 0a7c 6f31 3020 3d20 6964 6561 6c28 |.|o10 = ideal( │ │ │ │ +0002ed20: 3331 3837 7820 7820 202d 2036 3035 3378 3187x x - 6053x │ │ │ │ +0002ed30: 2078 2020 2d20 3136 3039 3078 2078 2020 x - 16090x x │ │ │ │ +0002ed40: 2b20 3337 3833 7820 7820 202b 2038 3537 + 3783x x + 857 │ │ │ │ +0002ed50: 3078 2078 2020 2b20 3834 3434 7820 7820 0x x + 8444x x │ │ │ │ +0002ed60: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ +0002ed70: 2020 2020 2030 2032 2020 2020 2020 2020 0 2 │ │ │ │ +0002ed80: 3120 3220 2020 2020 2020 2020 3220 3320 1 2 2 3 │ │ │ │ +0002ed90: 2020 2020 2020 2030 2034 2020 2020 2020 0 4 │ │ │ │ +0002eda0: 2020 3120 3420 2020 2020 2020 2033 2034 1 4 3 4 │ │ │ │ +0002edb0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002edc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002edd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ee00: 207c 0a7c 6f31 3020 3a20 4964 6561 6c20 |.|o10 : Ideal │ │ │ │ +0002ee10: 6f66 2052 2020 2020 2020 2020 2020 2020 of R │ │ │ │ 0002ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee60: 2020 2020 2020 207c 0a7c 6f31 3020 3a20 |.|o10 : │ │ │ │ -0002ee70: 4964 6561 6c20 6f66 2052 2020 2020 2020 Ideal of R │ │ │ │ -0002ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eeb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ef00: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ -0002ef10: 633d 4368 6572 6e28 4368 2c58 2c49 2920 c=Chern(Ch,X,I) │ │ │ │ +0002ee50: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002ee60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ee90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eea0: 2d2b 0a7c 6931 3120 3a20 633d 4368 6572 -+.|i11 : c=Cher │ │ │ │ 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2020 2020 2020 │ │ │ │ -0002efa0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002efb0: 2020 3220 2020 2020 2020 3220 2020 2020 2 2 │ │ │ │ +0002ef90: 207c 0a7c 6f31 3120 3d20 3478 2078 2020 |.|o11 = 4x x │ │ │ │ +0002efa0: 2b20 3278 2020 2b20 3278 2078 2020 2b20 + 2x + 2x x + │ │ │ │ +0002efb0: 7820 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ 0002efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002efd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eff0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20 |.|o11 = │ │ │ │ -0002f000: 3478 2078 2020 2b20 3278 2020 2b20 3278 4x x + 2x + 2x │ │ │ │ -0002f010: 2078 2020 2b20 7820 2020 2020 2020 2020 x + x │ │ │ │ +0002efe0: 207c 0a7c 2020 2020 2020 2020 3320 3420 |.| 3 4 │ │ │ │ +0002eff0: 2020 2020 3320 2020 2020 3320 3420 2020 3 3 4 │ │ │ │ +0002f000: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0002f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f020: 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2d20 7820 202d 2032 7820 3x x - x - 2x │ │ │ │ -0002f290: 7820 202b 2078 2020 2b20 7820 2020 2020 x + x + x │ │ │ │ +0002f260: 207c 0a7c 2020 2020 2020 2020 3320 3420 |.| 3 4 │ │ │ │ +0002f270: 2020 2033 2020 2020 2033 2034 2020 2020 3 3 4 │ │ │ │ +0002f280: 3320 2020 2034 2020 2020 2020 2020 2020 3 4 │ │ │ │ +0002f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f2c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002f2d0: 2020 3320 3420 2020 2033 2020 2020 2033 3 4 3 3 │ │ │ │ -0002f2e0: 2034 2020 2020 3320 2020 2034 2020 2020 4 3 4 │ │ │ │ +0002f2b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f300: 2020 2020 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7820 202b - 3x - 4x x + │ │ │ │ +0002f4b0: 2078 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ 0002f4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f4f0: 2020 2020 2020 207c 0a7c 6f31 3320 3d20 |.|o13 = │ │ │ │ -0002f500: 2d20 7820 7820 202d 2033 7820 202d 2034 - x x - 3x - 4 │ │ │ │ -0002f510: 7820 7820 202b 2078 2020 2020 2020 2020 x x + x │ │ │ │ +0002f4e0: 207c 0a7c 2020 2020 2020 2020 2033 2034 |.| 3 4 │ │ │ │ +0002f4f0: 2020 2020 2033 2020 2020 2033 2034 2020 3 3 4 │ │ │ │ +0002f500: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +0002f510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f540: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002f550: 2020 2033 2034 2020 2020 2033 2020 2020 3 4 3 │ │ │ │ -0002f560: 2033 2034 2020 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2020 │ │ │ │ -0002f610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f630: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002f640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f680: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 -------+.|i14 : │ │ │ │ -0002f690: 732a 6320 2020 2020 2020 2020 2020 2020 s*c │ │ │ │ +0002f5d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002f5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f620: 2d2b 0a7c 6931 3420 3a20 732a 6320 2020 -+.|i14 : s*c │ │ │ │ +0002f630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f670: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f6d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002f6c0: 207c 0a7c 2020 2020 2020 2020 3220 2020 |.| 2 │ │ │ │ +0002f6d0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0002f6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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===============. │ │ │ │ -0002f930: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0002f940: 6520 546f 7269 6343 686f 7752 696e 673a e ToricChowRing: │ │ │ │ -0002f950: 2054 6f72 6963 4368 6f77 5269 6e67 2c20 ToricChowRing, │ │ │ │ -0002f960: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -0002f970: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -0002f980: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -0002f990: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ -0002f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0002f9f0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0002fa00: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0002fa10: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0002fa20: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0002fa30: 322d 312e 3235 2e31 312b 6473 2f4d 322f 2-1.25.11+ds/M2/ │ │ │ │ -0002fa40: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0002fa50: 6573 2f0a 4368 6172 6163 7465 7269 7374 es/.Characterist │ │ │ │ -0002fa60: 6963 436c 6173 7365 732e 6d32 3a31 3935 icClasses.m2:195 │ │ │ │ -0002fa70: 313a 302e 0a1f 0a54 6167 2054 6162 6c65 1:0....Tag Table │ │ │ │ -0002fa80: 3a0a 4e6f 6465 3a20 546f 707f 3239 310a :.Node: Top.291. │ │ │ │ -0002fa90: 4e6f 6465 3a20 6265 7274 696e 6943 6865 Node: bertiniChe │ │ │ │ -0002faa0: 636b 7f31 3637 3035 0a4e 6f64 653a 2043 ck.16705.Node: C │ │ │ │ -0002fab0: 6865 636b 536d 6f6f 7468 7f31 3739 3232 heckSmooth.17922 │ │ │ │ -0002fac0: 0a4e 6f64 653a 2043 6865 636b 546f 7269 .Node: CheckTori │ │ │ │ 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320a 4e6f 6465 3a20 4575 6c65 7241 6666 2.Node: EulerAff │ │ │ │ -0002fbb0: 696e 657f 3131 3936 3739 0a4e 6f64 653a ine.119679.Node: │ │ │ │ -0002fbc0: 2049 6e64 734f 6653 6d6f 6f74 687f 3132 IndsOfSmooth.12 │ │ │ │ -0002fbd0: 3231 3931 0a4e 6f64 653a 2049 6e70 7574 2191.Node: Input │ │ │ │ -0002fbe0: 4973 536d 6f6f 7468 7f31 3236 3134 360a IsSmooth.126146. │ │ │ │ -0002fbf0: 4e6f 6465 3a20 6973 4d75 6c74 6948 6f6d Node: isMultiHom │ │ │ │ -0002fc00: 6f67 656e 656f 7573 7f31 3330 3132 390a ogeneous.130129. │ │ │ │ -0002fc10: 4e6f 6465 3a20 4d65 7468 6f64 7f31 3334 Node: Method.134 │ │ │ │ -0002fc20: 3230 370a 4e6f 6465 3a20 4d75 6c74 6950 207.Node: MultiP │ │ │ │ -0002fc30: 726f 6a43 6f6f 7264 5269 6e67 7f31 3338 rojCoordRing.138 │ │ │ │ -0002fc40: 3135 360a 4e6f 6465 3a20 4f75 7470 7574 156.Node: Output │ │ │ │ -0002fc50: 7f31 3434 3832 340a 4e6f 6465 3a20 7072 .144824.Node: pr │ │ │ │ -0002fc60: 6f62 6162 696c 6973 7469 6320 616c 676f obabilistic algo │ │ │ │ -0002fc70: 7269 7468 6d7f 3136 3333 3432 0a4e 6f64 rithm.163342.Nod │ │ │ │ -0002fc80: 653a 2053 6567 7265 7f31 3638 3033 370a e: Segre.168037. │ │ │ │ -0002fc90: 4e6f 6465 3a20 546f 7269 6343 686f 7752 Node: ToricChowR │ │ │ │ -0002fca0: 696e 677f 3138 3539 3434 0a1f 0a45 6e64 ing.185944...End │ │ │ │ -0002fcb0: 2054 6167 2054 6162 6c65 0a Tag Table. │ │ │ │ +0002f850: 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f8a0: 2d2b 0a0a 466f 7220 7468 6520 7072 6f67 -+..For the prog │ │ │ │ +0002f8b0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +0002f8c0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +0002f8d0: 626a 6563 7420 2a6e 6f74 6520 546f 7269 bject *note Tori │ │ │ │ +0002f8e0: 6343 686f 7752 696e 673a 2054 6f72 6963 cChowRing: Toric │ │ │ │ +0002f8f0: 4368 6f77 5269 6e67 2c20 6973 2061 202a ChowRing, is a * │ │ │ │ +0002f900: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ +0002f910: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ +0002f920: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ +0002f930: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0002f940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f980: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0002f990: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0002f9a0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0002f9b0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0002f9c0: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ +0002f9d0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+ds/M2/Macaul │ │ │ │ +0002f9e0: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ +0002f9f0: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ +0002fa00: 7365 732e 6d32 3a31 3935 313a 302e 0a1f ses.m2:1951:0... │ │ │ │ +0002fa10: 0a54 6167 2054 6162 6c65 3a0a 4e6f 6465 .Tag Table:.Node │ │ │ │ +0002fa20: 3a20 546f 707f 3239 310a 4e6f 6465 3a20 : Top.291.Node: │ │ │ │ +0002fa30: 6265 7274 696e 6943 6865 636b 7f31 3637 bertiniCheck.167 │ │ │ │ +0002fa40: 3035 0a4e 6f64 653a 2043 6865 636b 536d 05.Node: CheckSm │ │ │ │ +0002fa50: 6f6f 7468 7f31 3739 3232 0a4e 6f64 653a ooth.17922.Node: │ │ │ │ +0002fa60: 2043 6865 636b 546f 7269 6356 6172 6965 CheckToricVarie │ │ │ │ +0002fa70: 7479 5661 6c69 647f 3232 3731 380a 4e6f tyValid.22718.No │ │ │ │ +0002fa80: 6465 3a20 4368 6572 6e7f 3333 3730 390a de: Chern.33709. │ │ │ │ +0002fa90: 4e6f 6465 3a20 4368 6f77 5269 6e67 7f35 Node: ChowRing.5 │ │ │ │ +0002faa0: 3132 3137 0a4e 6f64 653a 2043 6c61 7373 1217.Node: Class │ │ │ │ +0002fab0: 496e 4368 6f77 5269 6e67 7f35 3839 3438 InChowRing.58948 │ │ │ │ +0002fac0: 0a4e 6f64 653a 2043 6c61 7373 496e 546f .Node: ClassInTo │ │ │ │ +0002fad0: 7269 6343 686f 7752 696e 677f 3631 3330 ricChowRing.6130 │ │ │ │ +0002fae0: 380a 4e6f 6465 3a20 436f 6d70 4d65 7468 8.Node: CompMeth │ │ │ │ +0002faf0: 6f64 7f36 3635 3531 0a4e 6f64 653a 2063 od.66551.Node: c │ │ │ │ +0002fb00: 6f6e 6669 6775 7269 6e67 2042 6572 7469 onfiguring Berti │ │ │ │ +0002fb10: 6e69 7f37 3634 3738 0a4e 6f64 653a 2043 ni.76478.Node: C │ │ │ │ +0002fb20: 534d 7f37 3831 3133 0a4e 6f64 653a 2045 SM.78113.Node: E │ │ │ │ +0002fb30: 756c 6572 7f31 3031 3730 300a 4e6f 6465 uler.101700.Node │ │ │ │ +0002fb40: 3a20 4575 6c65 7241 6666 696e 657f 3131 : EulerAffine.11 │ │ │ │ +0002fb50: 3935 3737 0a4e 6f64 653a 2049 6e64 734f 9577.Node: IndsO │ │ │ │ +0002fb60: 6653 6d6f 6f74 687f 3132 3230 3839 0a4e fSmooth.122089.N │ │ │ │ +0002fb70: 6f64 653a 2049 6e70 7574 4973 536d 6f6f ode: InputIsSmoo │ │ │ │ +0002fb80: 7468 7f31 3236 3034 340a 4e6f 6465 3a20 th.126044.Node: │ │ │ │ +0002fb90: 6973 4d75 6c74 6948 6f6d 6f67 656e 656f isMultiHomogeneo │ │ │ │ +0002fba0: 7573 7f31 3330 3032 370a 4e6f 6465 3a20 us.130027.Node: │ │ │ │ +0002fbb0: 4d65 7468 6f64 7f31 3334 3130 350a 4e6f Method.134105.No │ │ │ │ +0002fbc0: 6465 3a20 4d75 6c74 6950 726f 6a43 6f6f de: MultiProjCoo │ │ │ │ +0002fbd0: 7264 5269 6e67 7f31 3338 3035 340a 4e6f rdRing.138054.No │ │ │ │ +0002fbe0: 6465 3a20 4f75 7470 7574 7f31 3434 3732 de: Output.14472 │ │ │ │ +0002fbf0: 320a 4e6f 6465 3a20 7072 6f62 6162 696c 2.Node: probabil │ │ │ │ +0002fc00: 6973 7469 6320 616c 676f 7269 7468 6d7f istic algorithm. │ │ │ │ +0002fc10: 3136 3332 3430 0a4e 6f64 653a 2053 6567 163240.Node: Seg │ │ │ │ +0002fc20: 7265 7f31 3637 3933 350a 4e6f 6465 3a20 re.167935.Node: │ │ │ │ +0002fc30: 546f 7269 6343 686f 7752 696e 677f 3138 ToricChowRing.18 │ │ │ │ +0002fc40: 3538 3432 0a1f 0a45 6e64 2054 6167 2054 5842...End Tag T │ │ │ │ +0002fc50: 6162 6c65 0a able. │ │ ├── ./usr/share/info/CohomCalg.info.gz │ │ │ ├── CohomCalg.info │ │ │ │ @@ -1042,15 +1042,15 @@ │ │ │ │ 00004110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00004130: 7c69 3230 203a 2065 6c61 7073 6564 5469 |i20 : elapsedTi │ │ │ │ 00004140: 6d65 2068 7665 6373 203d 2063 6f68 6f6d me hvecs = cohom │ │ │ │ 00004150: 4361 6c67 2858 2c20 4432 2920 2020 2020 Calg(X, D2) │ │ │ │ 00004160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004170: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00004180: 7c20 2d2d 2033 2e34 3337 3134 7320 656c | -- 3.43714s el │ │ │ │ +00004180: 7c20 2d2d 2033 2e32 3735 3638 7320 656c | -- 3.27568s el │ │ │ │ 00004190: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 000041a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000041b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000041c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000041d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000041e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000041f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1677,15 +1677,15 @@ │ │ │ │ 000068c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000068d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000068e0: 7c69 3233 203a 2065 6c61 7073 6564 5469 |i23 : elapsedTi │ │ │ │ 000068f0: 6d65 2063 6f68 6f6d 7665 6331 203d 2063 me cohomvec1 = c │ │ │ │ 00006900: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058 ohomCalg(X_3 + X │ │ │ │ 00006910: 5f37 202b 2058 5f38 2920 2020 2020 2020 _7 + X_8) │ │ │ │ 00006920: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00006930: 7c20 2d2d 202e 3333 3339 3134 7320 656c | -- .333914s el │ │ │ │ +00006930: 7c20 2d2d 202e 3533 3736 3331 7320 656c | -- .537631s el │ │ │ │ 00006940: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00006950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006970: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00006980: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00006990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000069a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1712,15 +1712,15 @@ │ │ │ │ 00006af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00006b10: 7c69 3234 203a 2065 6c61 7073 6564 5469 |i24 : elapsedTi │ │ │ │ 00006b20: 6d65 2063 6f68 6f6d 7665 6332 203d 2066 me cohomvec2 = f │ │ │ │ 00006b30: 6f72 206a 2066 726f 6d20 3020 746f 2064 or j from 0 to d │ │ │ │ 00006b40: 696d 2058 206c 6973 7420 7261 6e6b 2048 im X list rank H │ │ │ │ 00006b50: 485e 6a28 582c 2020 2020 2020 2020 7c0a H^j(X, |. │ │ │ │ -00006b60: 7c20 2d2d 2031 312e 3932 3533 7320 656c | -- 11.9253s el │ │ │ │ +00006b60: 7c20 2d2d 2031 302e 3433 3332 7320 656c | -- 10.4332s 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 3337 3336 3032 7320 656c | -- .373602s el │ │ │ │ +000070b0: 7c20 2d2d 202e 3532 3135 3631 7320 656c | -- .521561s 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 3539 3230 3431 7320 656c | -- .592041s el │ │ │ │ +000072e0: 7c20 2d2d 202e 3530 3737 3032 7320 656c | -- .507702s 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 3539 3230 3734 7320 656c | -- .592074s el │ │ │ │ +00007330: 7c20 2d2d 202e 3530 3737 3334 7320 656c | -- .507734s el │ │ │ │ 00007340: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00007350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007370: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00007380: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00007390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000073a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/CompleteIntersectionResolutions.info.gz │ │ │ ├── CompleteIntersectionResolutions.info │ │ │ │ @@ -4343,17 +4343,17 @@ │ │ │ │ 00010f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f80: 2d2d 2d2b 0a7c 6937 203a 2074 696d 6520 ---+.|i7 : time │ │ │ │ 00010f90: 4720 3d20 4569 7365 6e62 7564 5368 616d G = EisenbudSham │ │ │ │ 00010fa0: 6173 6828 6666 2c46 2c6c 656e 2920 2020 ash(ff,F,len) │ │ │ │ 00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2036 |.| -- used 6 │ │ │ │ -00010fe0: 2e37 3936 3631 7320 2863 7075 293b 2035 .79661s (cpu); 5 │ │ │ │ -00010ff0: 2e32 3336 3634 7320 2874 6872 6561 6429 .23664s (thread) │ │ │ │ +00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2039 |.| -- used 9 │ │ │ │ +00010fe0: 2e34 3932 3631 7320 2863 7075 293b 2036 .49261s (cpu); 6 │ │ │ │ +00010ff0: 2e35 3635 3731 7320 2874 6872 6561 6429 .56571s (thread) │ │ │ │ 00011000: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00011010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011020: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4884,17 +4884,17 @@ │ │ │ │ 00013130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013150: 2d2b 0a7c 6932 3020 3a20 4646 203d 2074 -+.|i20 : FF = t │ │ │ │ 00013160: 696d 6520 5368 616d 6173 6828 5231 2c46 ime Shamash(R1,F │ │ │ │ 00013170: 2c34 2920 2020 2020 2020 2020 2020 2020 ,4) │ │ │ │ 00013180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013190: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -000131a0: 2e30 3731 3937 3737 7320 2863 7075 293b .0719777s (cpu); │ │ │ │ -000131b0: 2030 2e30 3730 3932 3238 7320 2874 6872 0.0709228s (thr │ │ │ │ -000131c0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +000131a0: 2e32 3636 3237 3773 2028 6370 7529 3b20 .266277s (cpu); │ │ │ │ +000131b0: 302e 3135 3437 3736 7320 2874 6872 6561 0.154776s (threa │ │ │ │ +000131c0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 000131d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000131e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000131f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013210: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00013220: 2020 3120 2020 2020 2020 3620 2020 2020 1 6 │ │ │ │ 00013230: 2020 3138 2020 2020 2020 2033 3820 2020 18 38 │ │ │ │ @@ -4925,17 +4925,17 @@ │ │ │ │ 000133c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000133d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000133e0: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 4747 -----+.|i21 : GG │ │ │ │ 000133f0: 203d 2074 696d 6520 4569 7365 6e62 7564 = time Eisenbud │ │ │ │ 00013400: 5368 616d 6173 6828 6666 2c46 2c34 2920 Shamash(ff,F,4) │ │ │ │ 00013410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013420: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00013430: 6564 2030 2e39 3638 3434 3273 2028 6370 ed 0.968442s (cp │ │ │ │ -00013440: 7529 3b20 302e 3734 3034 3032 7320 2874 u); 0.740402s (t │ │ │ │ -00013450: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00013430: 6564 2031 2e34 3038 3433 7320 2863 7075 ed 1.40843s (cpu │ │ │ │ +00013440: 293b 2031 2e30 3830 3673 2028 7468 7265 ); 1.0806s (thre │ │ │ │ +00013450: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00013460: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00013470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000134a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000134b0: 2020 2020 2f20 525c 3120 2020 2020 2f20 / R\1 / │ │ │ │ 000134c0: 525c 3620 2020 2020 2f20 525c 3138 2020 R\6 / R\18 │ │ │ │ @@ -4977,24033 +4977,24031 @@ │ │ │ │ 00013700: 5468 6520 6675 6e63 7469 6f6e 2061 6c73 The function als │ │ │ │ 00013710: 6f20 6465 616c 7320 636f 7272 6563 746c o deals correctl │ │ │ │ 00013720: 7920 7769 7468 2063 6f6d 706c 6578 6573 y with complexes │ │ │ │ 00013730: 2046 2077 6865 7265 206d 696e 2046 2069 F where min F i │ │ │ │ 00013740: 7320 6e6f 7420 303a 0a0a 2b2d 2d2d 2d2d s not 0:..+----- │ │ │ │ 00013750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013780: 2b0a 7c69 3232 203a 2047 4720 3d20 7469 +.|i22 : GG = ti │ │ │ │ -00013790: 6d65 2045 6973 656e 6275 6453 6861 6d61 me EisenbudShama │ │ │ │ -000137a0: 7368 2852 312c 465b 325d 2c34 2920 2020 sh(R1,F[2],4) │ │ │ │ -000137b0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000137c0: 7365 6420 302e 3932 3037 3631 7320 2863 sed 0.920761s (c │ │ │ │ -000137d0: 7075 293b 2030 2e37 3239 3736 3673 2028 pu); 0.729766s ( │ │ │ │ -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 3330 3931 3673 2028 6370 7529 d 1.30916s (cpu) │ │ │ │ +000137d0: 3b20 302e 3939 3931 3335 7320 2874 6872 ; 0.999135s (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 352e /macaulay2-1.25. │ │ │ │ -00013c70: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61 11+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 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+ds/M2/Ma │ │ │ │ +00013c70: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00013c80: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ +00013c90: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +00013ca0: 732e 6d32 3a34 3834 323a 302e 0a1f 0a46 s.m2:4842:0....F │ │ │ │ +00013cb0: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ +00013cc0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00013cd0: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ +00013ce0: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ +00013cf0: 546f 7461 6c2c 204e 6578 743a 2065 7665 Total, Next: eve │ │ │ │ +00013d00: 6e45 7874 4d6f 6475 6c65 2c20 5072 6576 nExtModule, Prev │ │ │ │ +00013d10: 3a20 4569 7365 6e62 7564 5368 616d 6173 : EisenbudShamas │ │ │ │ +00013d20: 682c 2055 703a 2054 6f70 0a0a 4569 7365 h, Up: Top..Eise │ │ │ │ +00013d30: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00013d40: 202d 2d20 5072 6563 7572 736f 7220 636f -- Precursor co │ │ │ │ +00013d50: 6d70 6c65 7820 6f66 2074 6f74 616c 2045 mplex of total E │ │ │ │ +00013d60: 7874 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a xt.************* │ │ │ │ 00013d70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00013d80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00013d90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00013da0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -00013db0: 200a 2020 2020 2020 2020 2864 302c 6431 . (d0,d1 │ │ │ │ -00013dc0: 2920 3d20 2045 6973 656e 6275 6453 6861 ) = EisenbudSha │ │ │ │ -00013dd0: 6d61 7368 546f 7461 6c20 4d0a 2020 2a20 mashTotal M. * │ │ │ │ -00013de0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -00013df0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ -00013e00: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -00013e10: 294d 6f64 756c 652c 2c20 6f76 6572 2061 )Module,, over a │ │ │ │ -00013e20: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -00013e30: 6563 7469 6f6e 0a20 202a 202a 6e6f 7465 ection. * *note │ │ │ │ -00013e40: 204f 7074 696f 6e61 6c20 696e 7075 7473 Optional inputs │ │ │ │ -00013e50: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00013e60: 7573 696e 6720 6675 6e63 7469 6f6e 7320 using functions │ │ │ │ -00013e70: 7769 7468 206f 7074 696f 6e61 6c20 696e with optional in │ │ │ │ -00013e80: 7075 7473 2c3a 0a20 2020 2020 202a 2043 puts,:. * C │ │ │ │ -00013e90: 6865 636b 203d 3e20 2e2e 2e2c 2064 6566 heck => ..., def │ │ │ │ -00013ea0: 6175 6c74 2076 616c 7565 2066 616c 7365 ault value false │ │ │ │ -00013eb0: 0a20 2020 2020 202a 2047 7261 6469 6e67 . * Grading │ │ │ │ -00013ec0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ -00013ed0: 2076 616c 7565 2032 0a20 2020 2020 202a value 2. * │ │ │ │ -00013ee0: 2056 6172 6961 626c 6573 203d 3e20 2e2e Variables => .. │ │ │ │ -00013ef0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -00013f00: 2073 0a20 202a 204f 7574 7075 7473 3a0a s. * Outputs:. │ │ │ │ -00013f10: 2020 2020 2020 2a20 6430 2c20 6120 2a6e * d0, a *n │ │ │ │ -00013f20: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ -00013f30: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ -00013f40: 2c2c 206d 6170 206f 6620 6672 6565 206d ,, map of free m │ │ │ │ -00013f50: 6f64 756c 6573 206f 7665 7220 616e 0a20 odules over an. │ │ │ │ -00013f60: 2020 2020 2020 2065 6e6c 6172 6765 6420 enlarged │ │ │ │ -00013f70: 7269 6e67 0a20 2020 2020 202a 2064 312c ring. * d1, │ │ │ │ -00013f80: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ -00013f90: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -00013fa0: 6174 7269 782c 2c20 6d61 7020 6f66 2066 atrix,, map of f │ │ │ │ -00013fb0: 7265 6520 6d6f 6475 6c65 7320 6f76 6572 ree modules over │ │ │ │ -00013fc0: 2061 6e0a 2020 2020 2020 2020 656e 6c61 an. enla │ │ │ │ -00013fd0: 7267 6564 2072 696e 670a 0a44 6573 6372 rged ring..Descr │ │ │ │ -00013fe0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00013ff0: 3d3d 0a0a 4173 7375 6d65 2074 6861 7420 ==..Assume that │ │ │ │ -00014000: 4d20 6973 2064 6566 696e 6564 206f 7665 M is defined ove │ │ │ │ -00014010: 7220 6120 7269 6e67 206f 6620 7468 6520 r a ring of the │ │ │ │ -00014020: 666f 726d 2052 6261 7220 3d20 522f 2866 form Rbar = R/(f │ │ │ │ -00014030: 5f30 2e2e 665f 7b63 2d31 7d29 2c20 610a _0..f_{c-1}), a. │ │ │ │ -00014040: 636f 6d70 6c65 7465 2069 6e74 6572 7365 complete interse │ │ │ │ -00014050: 6374 696f 6e2c 2061 6e64 2074 6861 7420 ction, and that │ │ │ │ -00014060: 4d20 6861 7320 6120 6669 6e69 7465 2066 M has a finite f │ │ │ │ -00014070: 7265 6520 7265 736f 6c75 7469 6f6e 2047 ree resolution G │ │ │ │ -00014080: 206f 7665 7220 522e 2049 6e0a 7468 6973 over R. In.this │ │ │ │ -00014090: 2063 6173 6520 4d20 6861 7320 6120 6672 case M has a fr │ │ │ │ -000140a0: 6565 2072 6573 6f6c 7574 696f 6e20 4620 ee resolution F │ │ │ │ -000140b0: 6f76 6572 2052 6261 7220 7768 6f73 6520 over Rbar whose │ │ │ │ -000140c0: 6475 616c 2c20 465e 2a20 6973 2061 2066 dual, F^* is a f │ │ │ │ -000140d0: 696e 6974 656c 790a 6765 6e65 7261 7465 initely.generate │ │ │ │ -000140e0: 642c 205a 2d67 7261 6465 6420 6672 6565 d, Z-graded free │ │ │ │ -000140f0: 206d 6f64 756c 6520 6f76 6572 2061 2072 module over a r │ │ │ │ -00014100: 696e 6720 5362 6172 5c63 6f6e 6720 6b6b ing Sbar\cong kk │ │ │ │ -00014110: 5b73 5f30 2e2e 735f 7b63 2d31 7d2c 6765 [s_0..s_{c-1},ge │ │ │ │ -00014120: 6e73 0a52 6261 725d 2c20 7768 6572 6520 ns.Rbar], where │ │ │ │ -00014130: 7468 6520 6465 6772 6565 7320 6f66 2074 the degrees of t │ │ │ │ -00014140: 6865 2073 5f69 2061 7265 207b 2d32 2c20 he s_i are {-2, │ │ │ │ -00014150: 2d64 6567 7265 6520 665f 697d 2e20 5468 -degree f_i}. Th │ │ │ │ -00014160: 6973 2072 6573 6f6c 7574 696f 6e20 6973 is resolution is │ │ │ │ -00014170: 0a69 7320 636f 6e73 7472 7563 7465 6420 .is constructed │ │ │ │ -00014180: 6672 6f6d 2074 6865 2064 7561 6c20 6f66 from the dual of │ │ │ │ -00014190: 2047 2c20 746f 6765 7468 6572 2077 6974 G, together wit │ │ │ │ -000141a0: 6820 7468 6520 6475 616c 7320 6f66 2074 h the duals of t │ │ │ │ -000141b0: 6865 2068 6967 6865 720a 686f 6d6f 746f he higher.homoto │ │ │ │ -000141c0: 7069 6573 206f 6e20 4720 6465 6669 6e65 pies on G define │ │ │ │ -000141d0: 6420 6279 2045 6973 656e 6275 642e 0a0a d by Eisenbud... │ │ │ │ -000141e0: 5468 6520 6675 6e63 7469 6f6e 2072 6574 The function ret │ │ │ │ -000141f0: 7572 6e73 2074 6865 2064 6966 6665 7265 urns the differe │ │ │ │ -00014200: 6e74 6961 6c73 2064 303a 465e 2a5f 7b65 ntials d0:F^*_{e │ │ │ │ -00014210: 7665 6e7d 205c 746f 2046 5e2a 5f7b 6f64 ven} \to F^*_{od │ │ │ │ -00014220: 647d 2061 6e64 0a64 313a 465e 2a5f 7b6f d} and.d1:F^*_{o │ │ │ │ -00014230: 6464 7d5c 746f 2046 5e2a 5f7b 6576 656e dd}\to F^*_{even │ │ │ │ -00014240: 7d2e 0a0a 5468 6520 6d61 7073 2064 302c }...The maps d0, │ │ │ │ -00014250: 6431 2066 6f72 6d20 6120 6d61 7472 6978 d1 form a matrix │ │ │ │ -00014260: 2066 6163 746f 7269 7a61 7469 6f6e 206f factorization o │ │ │ │ -00014270: 6620 7375 6d28 632c 2069 2d3e 735f 692a f sum(c, i->s_i* │ │ │ │ -00014280: 665f 6929 2e20 5468 6520 6861 7665 2074 f_i). The have t │ │ │ │ -00014290: 6865 0a70 726f 7065 7274 7920 7468 6174 he.property that │ │ │ │ -000142a0: 2066 6f72 2061 6e79 2052 6261 7220 6d6f for any Rbar mo │ │ │ │ -000142b0: 6475 6c65 204e 2c0a 0a48 485f 3120 636f dule N,..HH_1 co │ │ │ │ -000142c0: 6d70 6c65 7820 5c7b 6430 2a2a 4e2c 2064 mplex \{d0**N, d │ │ │ │ -000142d0: 312a 2a4e 5c7d 203d 2045 7874 5e7b 6576 1**N\} = Ext^{ev │ │ │ │ -000142e0: 656e 7d5f 7b52 6261 727d 284d 2c4e 290a en}_{Rbar}(M,N). │ │ │ │ -000142f0: 0a53 5e7b 7b31 2c30 7d7d 2a2a 4848 5f31 .S^{{1,0}}**HH_1 │ │ │ │ -00014300: 2063 6f6d 706c 6578 205c 7b53 5e7b 7b2d complex \{S^{{- │ │ │ │ -00014310: 322c 307d 7d2a 2a64 312a 2a4e 2c20 6430 2,0}}**d1**N, d0 │ │ │ │ -00014320: 2a2a 4e5c 7d20 3d20 4578 745e 7b6f 6464 **N\} = Ext^{odd │ │ │ │ -00014330: 7d5f 7b52 6261 727d 284d 2c4e 290a 0a54 }_{Rbar}(M,N)..T │ │ │ │ -00014340: 6869 7320 6973 2065 6e63 6f64 6564 2069 his is encoded i │ │ │ │ -00014350: 6e20 7468 6520 7363 7269 7074 206e 6577 n the script new │ │ │ │ -00014360: 4578 740a 0a4f 7074 696f 6e20 6465 6661 Ext..Option defa │ │ │ │ -00014370: 756c 7473 3a20 4368 6563 6b3d 3e66 616c ults: Check=>fal │ │ │ │ -00014380: 7365 2056 6172 6961 626c 6573 3d3e 6765 se Variables=>ge │ │ │ │ -00014390: 7453 796d 626f 6c20 2273 222c 2047 7261 tSymbol "s", Gra │ │ │ │ -000143a0: 6469 6e67 203d 3e32 7d0a 0a49 6620 4772 ding =>2}..If Gr │ │ │ │ -000143b0: 6164 696e 6720 3d3e 312c 2074 6865 6e20 ading =>1, then │ │ │ │ -000143c0: 6120 7369 6e67 6c79 2067 7261 6465 6420 a singly graded │ │ │ │ -000143d0: 7265 7375 6c74 2069 7320 7265 7475 726e result is return │ │ │ │ -000143e0: 6564 2028 6a75 7374 2066 6f72 6765 7474 ed (just forgett │ │ │ │ -000143f0: 696e 6720 7468 650a 686f 6d6f 6c6f 6769 ing the.homologi │ │ │ │ -00014400: 6361 6c20 6772 6164 696e 672e 290a 0a0a cal grading.)... │ │ │ │ -00014410: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00013d90: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00013da0: 7361 6765 3a20 0a20 2020 2020 2020 2028 sage: . ( │ │ │ │ +00013db0: 6430 2c64 3129 203d 2020 4569 7365 6e62 d0,d1) = Eisenb │ │ │ │ +00013dc0: 7564 5368 616d 6173 6854 6f74 616c 204d udShamashTotal M │ │ │ │ +00013dd0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +00013de0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ +00013df0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +00013e00: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f y2Doc)Module,, o │ │ │ │ +00013e10: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +00013e20: 6e74 6572 7365 6374 696f 6e0a 2020 2a20 ntersection. * │ │ │ │ +00013e30: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069 *note Optional i │ │ │ │ +00013e40: 6e70 7574 733a 2028 4d61 6361 756c 6179 nputs: (Macaulay │ │ │ │ +00013e50: 3244 6f63 2975 7369 6e67 2066 756e 6374 2Doc)using funct │ │ │ │ +00013e60: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ +00013e70: 616c 2069 6e70 7574 732c 3a0a 2020 2020 al inputs,:. │ │ │ │ +00013e80: 2020 2a20 4368 6563 6b20 3d3e 202e 2e2e * Check => ... │ │ │ │ +00013e90: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +00013ea0: 6661 6c73 650a 2020 2020 2020 2a20 4772 false. * Gr │ │ │ │ +00013eb0: 6164 696e 6720 3d3e 202e 2e2e 2c20 6465 ading => ..., de │ │ │ │ +00013ec0: 6661 756c 7420 7661 6c75 6520 320a 2020 fault value 2. │ │ │ │ +00013ed0: 2020 2020 2a20 5661 7269 6162 6c65 7320 * Variables │ │ │ │ +00013ee0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ +00013ef0: 7661 6c75 6520 730a 2020 2a20 4f75 7470 value s. * Outp │ │ │ │ +00013f00: 7574 733a 0a20 2020 2020 202a 2064 302c uts:. * d0, │ │ │ │ +00013f10: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ +00013f20: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +00013f30: 6174 7269 782c 2c20 6d61 7020 6f66 2066 atrix,, map of f │ │ │ │ +00013f40: 7265 6520 6d6f 6475 6c65 7320 6f76 6572 ree modules over │ │ │ │ +00013f50: 2061 6e0a 2020 2020 2020 2020 656e 6c61 an. enla │ │ │ │ +00013f60: 7267 6564 2072 696e 670a 2020 2020 2020 rged ring. │ │ │ │ +00013f70: 2a20 6431 2c20 6120 2a6e 6f74 6520 6d61 * d1, a *note ma │ │ │ │ +00013f80: 7472 6978 3a20 284d 6163 6175 6c61 7932 trix: (Macaulay2 │ │ │ │ +00013f90: 446f 6329 4d61 7472 6978 2c2c 206d 6170 Doc)Matrix,, map │ │ │ │ +00013fa0: 206f 6620 6672 6565 206d 6f64 756c 6573 of free modules │ │ │ │ +00013fb0: 206f 7665 7220 616e 0a20 2020 2020 2020 over an. │ │ │ │ +00013fc0: 2065 6e6c 6172 6765 6420 7269 6e67 0a0a enlarged ring.. │ │ │ │ +00013fd0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00013fe0: 3d3d 3d3d 3d3d 3d0a 0a41 7373 756d 6520 =======..Assume │ │ │ │ +00013ff0: 7468 6174 204d 2069 7320 6465 6669 6e65 that M is define │ │ │ │ +00014000: 6420 6f76 6572 2061 2072 696e 6720 6f66 d over a ring of │ │ │ │ +00014010: 2074 6865 2066 6f72 6d20 5262 6172 203d the form Rbar = │ │ │ │ +00014020: 2052 2f28 665f 302e 2e66 5f7b 632d 317d R/(f_0..f_{c-1} │ │ │ │ +00014030: 292c 2061 0a63 6f6d 706c 6574 6520 696e ), a.complete in │ │ │ │ +00014040: 7465 7273 6563 7469 6f6e 2c20 616e 6420 tersection, and │ │ │ │ +00014050: 7468 6174 204d 2068 6173 2061 2066 696e that M has a fin │ │ │ │ +00014060: 6974 6520 6672 6565 2072 6573 6f6c 7574 ite free resolut │ │ │ │ +00014070: 696f 6e20 4720 6f76 6572 2052 2e20 496e ion G over R. In │ │ │ │ +00014080: 0a74 6869 7320 6361 7365 204d 2068 6173 .this case M has │ │ │ │ +00014090: 2061 2066 7265 6520 7265 736f 6c75 7469 a free resoluti │ │ │ │ +000140a0: 6f6e 2046 206f 7665 7220 5262 6172 2077 on F over Rbar w │ │ │ │ +000140b0: 686f 7365 2064 7561 6c2c 2046 5e2a 2069 hose dual, F^* i │ │ │ │ +000140c0: 7320 6120 6669 6e69 7465 6c79 0a67 656e s a finitely.gen │ │ │ │ +000140d0: 6572 6174 6564 2c20 5a2d 6772 6164 6564 erated, Z-graded │ │ │ │ +000140e0: 2066 7265 6520 6d6f 6475 6c65 206f 7665 free module ove │ │ │ │ +000140f0: 7220 6120 7269 6e67 2053 6261 725c 636f r a ring Sbar\co │ │ │ │ +00014100: 6e67 206b 6b5b 735f 302e 2e73 5f7b 632d ng kk[s_0..s_{c- │ │ │ │ +00014110: 317d 2c67 656e 730a 5262 6172 5d2c 2077 1},gens.Rbar], w │ │ │ │ +00014120: 6865 7265 2074 6865 2064 6567 7265 6573 here the degrees │ │ │ │ +00014130: 206f 6620 7468 6520 735f 6920 6172 6520 of the s_i are │ │ │ │ +00014140: 7b2d 322c 202d 6465 6772 6565 2066 5f69 {-2, -degree f_i │ │ │ │ +00014150: 7d2e 2054 6869 7320 7265 736f 6c75 7469 }. This resoluti │ │ │ │ +00014160: 6f6e 2069 730a 6973 2063 6f6e 7374 7275 on is.is constru │ │ │ │ +00014170: 6374 6564 2066 726f 6d20 7468 6520 6475 cted from the du │ │ │ │ +00014180: 616c 206f 6620 472c 2074 6f67 6574 6865 al of G, togethe │ │ │ │ +00014190: 7220 7769 7468 2074 6865 2064 7561 6c73 r with the duals │ │ │ │ +000141a0: 206f 6620 7468 6520 6869 6768 6572 0a68 of the higher.h │ │ │ │ +000141b0: 6f6d 6f74 6f70 6965 7320 6f6e 2047 2064 omotopies on G d │ │ │ │ +000141c0: 6566 696e 6564 2062 7920 4569 7365 6e62 efined by Eisenb │ │ │ │ +000141d0: 7564 2e0a 0a54 6865 2066 756e 6374 696f ud...The functio │ │ │ │ +000141e0: 6e20 7265 7475 726e 7320 7468 6520 6469 n returns the di │ │ │ │ +000141f0: 6666 6572 656e 7469 616c 7320 6430 3a46 fferentials d0:F │ │ │ │ +00014200: 5e2a 5f7b 6576 656e 7d20 5c74 6f20 465e ^*_{even} \to F^ │ │ │ │ +00014210: 2a5f 7b6f 6464 7d20 616e 640a 6431 3a46 *_{odd} and.d1:F │ │ │ │ +00014220: 5e2a 5f7b 6f64 647d 5c74 6f20 465e 2a5f ^*_{odd}\to F^*_ │ │ │ │ +00014230: 7b65 7665 6e7d 2e0a 0a54 6865 206d 6170 {even}...The map │ │ │ │ +00014240: 7320 6430 2c64 3120 666f 726d 2061 206d s d0,d1 form a m │ │ │ │ +00014250: 6174 7269 7820 6661 6374 6f72 697a 6174 atrix factorizat │ │ │ │ +00014260: 696f 6e20 6f66 2073 756d 2863 2c20 692d ion of sum(c, i- │ │ │ │ +00014270: 3e73 5f69 2a66 5f69 292e 2054 6865 2068 >s_i*f_i). The h │ │ │ │ +00014280: 6176 6520 7468 650a 7072 6f70 6572 7479 ave the.property │ │ │ │ +00014290: 2074 6861 7420 666f 7220 616e 7920 5262 that for any Rb │ │ │ │ +000142a0: 6172 206d 6f64 756c 6520 4e2c 0a0a 4848 ar module N,..HH │ │ │ │ +000142b0: 5f31 2063 6f6d 706c 6578 205c 7b64 302a _1 complex \{d0* │ │ │ │ +000142c0: 2a4e 2c20 6431 2a2a 4e5c 7d20 3d20 4578 *N, d1**N\} = Ex │ │ │ │ +000142d0: 745e 7b65 7665 6e7d 5f7b 5262 6172 7d28 t^{even}_{Rbar}( │ │ │ │ +000142e0: 4d2c 4e29 0a0a 535e 7b7b 312c 307d 7d2a M,N)..S^{{1,0}}* │ │ │ │ +000142f0: 2a48 485f 3120 636f 6d70 6c65 7820 5c7b *HH_1 complex \{ │ │ │ │ +00014300: 535e 7b7b 2d32 2c30 7d7d 2a2a 6431 2a2a S^{{-2,0}}**d1** │ │ │ │ +00014310: 4e2c 2064 302a 2a4e 5c7d 203d 2045 7874 N, d0**N\} = Ext │ │ │ │ +00014320: 5e7b 6f64 647d 5f7b 5262 6172 7d28 4d2c ^{odd}_{Rbar}(M, │ │ │ │ +00014330: 4e29 0a0a 5468 6973 2069 7320 656e 636f N)..This is enco │ │ │ │ +00014340: 6465 6420 696e 2074 6865 2073 6372 6970 ded in the scrip │ │ │ │ +00014350: 7420 6e65 7745 7874 0a0a 4f70 7469 6f6e t newExt..Option │ │ │ │ +00014360: 2064 6566 6175 6c74 733a 2043 6865 636b defaults: Check │ │ │ │ +00014370: 3d3e 6661 6c73 6520 5661 7269 6162 6c65 =>false Variable │ │ │ │ +00014380: 733d 3e67 6574 5379 6d62 6f6c 2022 7322 s=>getSymbol "s" │ │ │ │ +00014390: 2c20 4772 6164 696e 6720 3d3e 327d 0a0a , Grading =>2}.. │ │ │ │ +000143a0: 4966 2047 7261 6469 6e67 203d 3e31 2c20 If Grading =>1, │ │ │ │ +000143b0: 7468 656e 2061 2073 696e 676c 7920 6772 then a singly gr │ │ │ │ +000143c0: 6164 6564 2072 6573 756c 7420 6973 2072 aded result is r │ │ │ │ +000143d0: 6574 7572 6e65 6420 286a 7573 7420 666f eturned (just fo │ │ │ │ +000143e0: 7267 6574 7469 6e67 2074 6865 0a68 6f6d rgetting the.hom │ │ │ │ +000143f0: 6f6c 6f67 6963 616c 2067 7261 6469 6e67 ological grading │ │ │ │ +00014400: 2e29 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d .)....+--------- │ │ │ │ +00014410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014460: 0a7c 6931 203a 206e 203d 2033 2020 2020 .|i1 : n = 3 │ │ │ │ +00014450: 2d2d 2d2d 2b0a 7c69 3120 3a20 6e20 3d20 ----+.|i1 : n = │ │ │ │ +00014460: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00014470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000144a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000144b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000144a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000144b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000144f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014500: 0a7c 6f31 203d 2033 2020 2020 2020 2020 .|o1 = 3 │ │ │ │ +000144f0: 2020 2020 7c0a 7c6f 3120 3d20 3320 2020 |.|o1 = 3 │ │ │ │ +00014500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014540: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014550: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014540: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000145a0: 0a7c 6932 203a 2063 203d 2032 2020 2020 .|i2 : c = 2 │ │ │ │ +00014590: 2d2d 2d2d 2b0a 7c69 3220 3a20 6320 3d20 ----+.|i2 : c = │ │ │ │ +000145a0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 000145b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000145c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000145d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000145e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000145f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000145e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000145f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014630: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014640: 0a7c 6f32 203d 2032 2020 2020 2020 2020 .|o2 = 2 │ │ │ │ +00014630: 2020 2020 7c0a 7c6f 3220 3d20 3220 2020 |.|o2 = 2 │ │ │ │ +00014640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014680: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014690: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014680: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000146a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000146b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000146c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000146d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000146e0: 0a7c 6933 203a 206b 6b20 3d20 5a5a 2f31 .|i3 : kk = ZZ/1 │ │ │ │ -000146f0: 3031 2020 2020 2020 2020 2020 2020 2020 01 │ │ │ │ +000146d0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6b6b 203d ----+.|i3 : kk = │ │ │ │ +000146e0: 205a 5a2f 3130 3120 2020 2020 2020 2020 ZZ/101 │ │ │ │ +000146f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014720: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014730: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014720: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014770: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014780: 0a7c 6f33 203d 206b 6b20 2020 2020 2020 .|o3 = kk │ │ │ │ +00014770: 2020 2020 7c0a 7c6f 3320 3d20 6b6b 2020 |.|o3 = kk │ │ │ │ +00014780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000147c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000147d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000147c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000147d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014820: 0a7c 6f33 203a 2051 756f 7469 656e 7452 .|o3 : QuotientR │ │ │ │ -00014830: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00014810: 2020 2020 7c0a 7c6f 3320 3a20 5175 6f74 |.|o3 : Quot │ │ │ │ +00014820: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +00014830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014860: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014870: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014860: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000148a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000148b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000148c0: 0a7c 6934 203a 2052 203d 206b 6b5b 785f .|i4 : R = kk[x_ │ │ │ │ -000148d0: 302e 2e78 5f28 6e2d 3129 5d20 2020 2020 0..x_(n-1)] │ │ │ │ +000148b0: 2d2d 2d2d 2b0a 7c69 3420 3a20 5220 3d20 ----+.|i4 : R = │ │ │ │ +000148c0: 6b6b 5b78 5f30 2e2e 785f 286e 2d31 295d kk[x_0..x_(n-1)] │ │ │ │ +000148d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000148e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000148f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014900: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014910: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014900: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014950: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014960: 0a7c 6f34 203d 2052 2020 2020 2020 2020 .|o4 = R │ │ │ │ +00014950: 2020 2020 7c0a 7c6f 3420 3d20 5220 2020 |.|o4 = R │ │ │ │ +00014960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000149a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000149b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000149a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000149b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000149f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014a00: 0a7c 6f34 203a 2050 6f6c 796e 6f6d 6961 .|o4 : Polynomia │ │ │ │ -00014a10: 6c52 696e 6720 2020 2020 2020 2020 2020 lRing │ │ │ │ +000149f0: 2020 2020 7c0a 7c6f 3420 3a20 506f 6c79 |.|o4 : Poly │ │ │ │ +00014a00: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ +00014a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014a40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014a50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014a40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014aa0: 0a7c 6935 203a 2049 203d 2069 6465 616c .|i5 : I = ideal │ │ │ │ -00014ab0: 2878 5f30 5e32 2c20 785f 325e 3329 2020 (x_0^2, x_2^3) │ │ │ │ +00014a90: 2d2d 2d2d 2b0a 7c69 3520 3a20 4920 3d20 ----+.|i5 : I = │ │ │ │ +00014aa0: 6964 6561 6c28 785f 305e 322c 2078 5f32 ideal(x_0^2, x_2 │ │ │ │ +00014ab0: 5e33 2920 2020 2020 2020 2020 2020 2020 ^3) │ │ │ │ 00014ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ae0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014af0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014b40: 0a7c 2020 2020 2020 2020 2020 2020 2032 .| 2 │ │ │ │ -00014b50: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00014b30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014b40: 2020 2020 3220 2020 3320 2020 2020 2020 2 3 │ │ │ │ +00014b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014b90: 0a7c 6f35 203d 2069 6465 616c 2028 7820 .|o5 = ideal (x │ │ │ │ -00014ba0: 2c20 7820 2920 2020 2020 2020 2020 2020 , x ) │ │ │ │ +00014b80: 2020 2020 7c0a 7c6f 3520 3d20 6964 6561 |.|o5 = idea │ │ │ │ +00014b90: 6c20 2878 202c 2078 2029 2020 2020 2020 l (x , x ) │ │ │ │ +00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014bd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014be0: 0a7c 2020 2020 2020 2020 2020 2020 2030 .| 0 │ │ │ │ -00014bf0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00014bd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014be0: 2020 2020 3020 2020 3220 2020 2020 2020 0 2 │ │ │ │ +00014bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014c20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014c30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014c20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014c80: 0a7c 6f35 203a 2049 6465 616c 206f 6620 .|o5 : Ideal of │ │ │ │ -00014c90: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00014c70: 2020 2020 7c0a 7c6f 3520 3a20 4964 6561 |.|o5 : Idea │ │ │ │ +00014c80: 6c20 6f66 2052 2020 2020 2020 2020 2020 l of R │ │ │ │ +00014c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014cc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014cd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014cc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014d20: 0a7c 6936 203a 2066 6620 3d20 6765 6e73 .|i6 : ff = gens │ │ │ │ -00014d30: 2049 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ +00014d10: 2d2d 2d2d 2b0a 7c69 3620 3a20 6666 203d ----+.|i6 : ff = │ │ │ │ +00014d20: 2067 656e 7320 4920 2020 2020 2020 2020 gens I │ │ │ │ +00014d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014d60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014d70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014db0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014dc0: 0a7c 6f36 203d 207c 2078 5f30 5e32 2078 .|o6 = | x_0^2 x │ │ │ │ -00014dd0: 5f32 5e33 207c 2020 2020 2020 2020 2020 _2^3 | │ │ │ │ +00014db0: 2020 2020 7c0a 7c6f 3620 3d20 7c20 785f |.|o6 = | x_ │ │ │ │ +00014dc0: 305e 3220 785f 325e 3320 7c20 2020 2020 0^2 x_2^3 | │ │ │ │ +00014dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014e00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014e10: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014e00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014e50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014e60: 0a7c 2020 2020 2020 2020 2020 2020 2031 .| 1 │ │ │ │ -00014e70: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00014e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014e60: 2020 2020 3120 2020 2020 2032 2020 2020 1 2 │ │ │ │ +00014e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ea0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014eb0: 0a7c 6f36 203a 204d 6174 7269 7820 5220 .|o6 : Matrix R │ │ │ │ -00014ec0: 203c 2d2d 2052 2020 2020 2020 2020 2020 <-- R │ │ │ │ +00014ea0: 2020 2020 7c0a 7c6f 3620 3a20 4d61 7472 |.|o6 : Matr │ │ │ │ +00014eb0: 6978 2052 2020 3c2d 2d20 5220 2020 2020 ix R <-- R │ │ │ │ +00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ef0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014f00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014ef0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014f50: 0a7c 6937 203a 2052 6261 7220 3d20 522f .|i7 : Rbar = R/ │ │ │ │ -00014f60: 4920 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ +00014f40: 2d2d 2d2d 2b0a 7c69 3720 3a20 5262 6172 ----+.|i7 : Rbar │ │ │ │ +00014f50: 203d 2052 2f49 2020 2020 2020 2020 2020 = R/I │ │ │ │ +00014f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014f90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014f90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014fe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014ff0: 0a7c 6f37 203d 2052 6261 7220 2020 2020 .|o7 = Rbar │ │ │ │ +00014fe0: 2020 2020 7c0a 7c6f 3720 3d20 5262 6172 |.|o7 = Rbar │ │ │ │ +00014ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015030: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015040: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015030: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015080: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015090: 0a7c 6f37 203a 2051 756f 7469 656e 7452 .|o7 : QuotientR │ │ │ │ -000150a0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00015080: 2020 2020 7c0a 7c6f 3720 3a20 5175 6f74 |.|o7 : Quot │ │ │ │ +00015090: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +000150a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000150b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000150c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000150d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000150e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000150d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000150e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000150f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00015130: 0a7c 6938 203a 2062 6172 203d 206d 6170 .|i8 : bar = map │ │ │ │ -00015140: 2852 6261 722c 2052 2920 2020 2020 2020 (Rbar, R) │ │ │ │ +00015120: 2d2d 2d2d 2b0a 7c69 3820 3a20 6261 7220 ----+.|i8 : bar │ │ │ │ +00015130: 3d20 6d61 7028 5262 6172 2c20 5229 2020 = map(Rbar, R) │ │ │ │ +00015140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015170: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015180: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015170: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000151a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000151b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000151c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000151d0: 0a7c 6f38 203d 206d 6170 2028 5262 6172 .|o8 = map (Rbar │ │ │ │ -000151e0: 2c20 522c 207b 7820 2c20 7820 2c20 7820 , R, {x , x , x │ │ │ │ -000151f0: 7d29 2020 2020 2020 2020 2020 2020 2020 }) │ │ │ │ +000151c0: 2020 2020 7c0a 7c6f 3820 3d20 6d61 7020 |.|o8 = map │ │ │ │ +000151d0: 2852 6261 722c 2052 2c20 7b78 202c 2078 (Rbar, R, {x , x │ │ │ │ +000151e0: 202c 2078 207d 2920 2020 2020 2020 2020 , x }) │ │ │ │ +000151f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015210: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00015230: 2020 2020 2020 2030 2020 2031 2020 2032 0 1 2 │ │ │ │ +00015210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015220: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +00015230: 3120 2020 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ 00015240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015260: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015270: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015260: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000152b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000152c0: 0a7c 6f38 203a 2052 696e 674d 6170 2052 .|o8 : RingMap R │ │ │ │ -000152d0: 6261 7220 3c2d 2d20 5220 2020 2020 2020 bar <-- R │ │ │ │ +000152b0: 2020 2020 7c0a 7c6f 3820 3a20 5269 6e67 |.|o8 : Ring │ │ │ │ +000152c0: 4d61 7020 5262 6172 203c 2d2d 2052 2020 Map Rbar <-- R │ │ │ │ +000152d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015310: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00015300: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00015310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00015360: 0a7c 6939 203a 204d 6261 7220 3d20 7072 .|i9 : Mbar = pr │ │ │ │ -00015370: 756e 6520 636f 6b65 7220 7261 6e64 6f6d une coker random │ │ │ │ -00015380: 2852 6261 725e 312c 2052 6261 725e 7b2d (Rbar^1, Rbar^{- │ │ │ │ -00015390: 327d 2920 2020 2020 2020 2020 2020 2020 2}) │ │ │ │ -000153a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000153b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015350: 2d2d 2d2d 2b0a 7c69 3920 3a20 4d62 6172 ----+.|i9 : Mbar │ │ │ │ +00015360: 203d 2070 7275 6e65 2063 6f6b 6572 2072 = prune coker r │ │ │ │ +00015370: 616e 646f 6d28 5262 6172 5e31 2c20 5262 andom(Rbar^1, Rb │ │ │ │ +00015380: 6172 5e7b 2d32 7d29 2020 2020 2020 2020 ar^{-2}) │ │ │ │ +00015390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000153a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000153b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000153f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015400: 0a7c 6f39 203d 2063 6f6b 6572 6e65 6c20 .|o9 = cokernel │ │ │ │ -00015410: 7c20 785f 3078 5f31 2b32 3478 5f31 5e32 | x_0x_1+24x_1^2 │ │ │ │ -00015420: 2b34 3978 5f30 785f 322b 3378 5f31 785f +49x_0x_2+3x_1x_ │ │ │ │ -00015430: 322b 3578 5f32 5e32 207c 2020 2020 2020 2+5x_2^2 | │ │ │ │ -00015440: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015450: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000153f0: 2020 2020 7c0a 7c6f 3920 3d20 636f 6b65 |.|o9 = coke │ │ │ │ +00015400: 726e 656c 207c 2078 5f30 785f 312b 3234 rnel | x_0x_1+24 │ │ │ │ +00015410: 785f 315e 322b 3439 785f 3078 5f32 2b33 x_1^2+49x_0x_2+3 │ │ │ │ +00015420: 785f 3178 5f32 2b35 785f 325e 3220 7c20 x_1x_2+5x_2^2 | │ │ │ │ +00015430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015440: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000154a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000154b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000154c0: 2020 2020 3120 2020 2020 2020 2020 2020 1 │ │ │ │ +00015490: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000154a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000154b0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ +000154c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000154d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000154e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000154f0: 0a7c 6f39 203a 2052 6261 722d 6d6f 6475 .|o9 : Rbar-modu │ │ │ │ -00015500: 6c65 2c20 7175 6f74 6965 6e74 206f 6620 le, quotient of │ │ │ │ -00015510: 5262 6172 2020 2020 2020 2020 2020 2020 Rbar │ │ │ │ +000154e0: 2020 2020 7c0a 7c6f 3920 3a20 5262 6172 |.|o9 : Rbar │ │ │ │ +000154f0: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ +00015500: 7420 6f66 2052 6261 7220 2020 2020 2020 t of Rbar │ │ │ │ +00015510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015530: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015540: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00015530: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00015540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00015590: 0a7c 6931 3020 3a20 2864 302c 6431 2920 .|i10 : (d0,d1) │ │ │ │ -000155a0: 3d20 4569 7365 6e62 7564 5368 616d 6173 = EisenbudShamas │ │ │ │ -000155b0: 6854 6f74 616c 284d 6261 722c 4772 6164 hTotal(Mbar,Grad │ │ │ │ -000155c0: 696e 6720 3d3e 3129 2020 2020 2020 2020 ing =>1) │ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000155e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015580: 2d2d 2d2d 2b0a 7c69 3130 203a 2028 6430 ----+.|i10 : (d0 │ │ │ │ +00015590: 2c64 3129 203d 2045 6973 656e 6275 6453 ,d1) = EisenbudS │ │ │ │ +000155a0: 6861 6d61 7368 546f 7461 6c28 4d62 6172 hamashTotal(Mbar │ │ │ │ +000155b0: 2c47 7261 6469 6e67 203d 3e31 2920 2020 ,Grading =>1) │ │ │ │ +000155c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000155d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000155e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000155f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015620: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015630: 0a7c 6f31 3020 3d20 287b 2d32 7d20 7c20 .|o10 = ({-2} | │ │ │ │ -00015640: 785f 305e 3220 2020 2020 2020 2020 2020 x_0^2 │ │ │ │ -00015650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015660: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ -00015670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015680: 0a7c 2020 2020 2020 207b 2d32 7d20 7c20 .| {-2} | │ │ │ │ -00015690: 785f 3078 5f31 2b32 3478 5f31 5e32 2b34 x_0x_1+24x_1^2+4 │ │ │ │ -000156a0: 3978 5f30 785f 322b 3378 5f31 785f 322b 9x_0x_2+3x_1x_2+ │ │ │ │ -000156b0: 3578 5f32 5e32 2033 3073 5f30 2020 2020 5x_2^2 30s_0 │ │ │ │ -000156c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000156d0: 0a7c 2020 2020 2020 207b 2d33 7d20 7c20 .| {-3} | │ │ │ │ -000156e0: 785f 325e 3320 2020 2020 2020 2020 2020 x_2^3 │ │ │ │ -000156f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015700: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ -00015710: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015720: 0a7c 2020 2020 2020 207b 2d37 7d20 7c20 .| {-7} | │ │ │ │ -00015730: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -00015740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015750: 2020 2020 2020 2078 5f32 5e33 2020 2020 x_2^3 │ │ │ │ -00015760: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015770: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ +00015620: 2020 2020 7c0a 7c6f 3130 203d 2028 7b2d |.|o10 = ({- │ │ │ │ +00015630: 327d 207c 2078 5f30 5e32 2020 2020 2020 2} | x_0^2 │ │ │ │ +00015640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +00015660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015670: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d |.| {- │ │ │ │ +00015680: 327d 207c 2078 5f30 785f 312b 3234 785f 2} | x_0x_1+24x_ │ │ │ │ +00015690: 315e 322b 3439 785f 3078 5f32 2b33 785f 1^2+49x_0x_2+3x_ │ │ │ │ +000156a0: 3178 5f32 2b35 785f 325e 3220 3330 735f 1x_2+5x_2^2 30s_ │ │ │ │ +000156b0: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +000156c0: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d |.| {- │ │ │ │ +000156d0: 337d 207c 2078 5f32 5e33 2020 2020 2020 3} | x_2^3 │ │ │ │ +000156e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000156f0: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +00015700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015710: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d |.| {- │ │ │ │ +00015720: 377d 207c 2030 2020 2020 2020 2020 2020 7} | 0 │ │ │ │ +00015730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015740: 2020 2020 2020 2020 2020 2020 785f 325e x_2^ │ │ │ │ +00015750: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00015760: 2020 2020 7c0a 7c20 2020 2020 202d 2d2d |.| --- │ │ │ │ +00015770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000157a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000157b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -000157c0: 0a7c 2020 2020 2020 2d73 5f31 2020 2020 .| -s_1 │ │ │ │ +000157b0: 2d2d 2d2d 7c0a 7c20 2020 2020 202d 735f ----|.| -s_ │ │ │ │ +000157c0: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 000157d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000157e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000157f0: 2020 3020 2020 2020 2020 207c 2c20 7b30 0 |, {0 │ │ │ │ -00015800: 7d20 207c 2020 2020 2020 2020 2020 207c } | | │ │ │ │ -00015810: 0a7c 2020 2020 2020 3020 2020 2020 2020 .| 0 │ │ │ │ +000157e0: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ +000157f0: 7c2c 207b 307d 2020 7c20 2020 2020 2020 |, {0} | │ │ │ │ +00015800: 2020 2020 7c0a 7c20 2020 2020 2030 2020 |.| 0 │ │ │ │ +00015810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015840: 2020 2d73 5f31 2020 2020 207c 2020 7b2d -s_1 | {- │ │ │ │ -00015850: 347d 207c 2020 2020 2020 2020 2020 207c 4} | | │ │ │ │ -00015860: 0a7c 2020 2020 2020 735f 3020 2020 2020 .| s_0 │ │ │ │ +00015830: 2020 2020 2020 202d 735f 3120 2020 2020 -s_1 │ │ │ │ +00015840: 7c20 207b 2d34 7d20 7c20 2020 2020 2020 | {-4} | │ │ │ │ +00015850: 2020 2020 7c0a 7c20 2020 2020 2073 5f30 |.| s_0 │ │ │ │ +00015860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015890: 2020 3020 2020 2020 2020 207c 2020 7b2d 0 | {- │ │ │ │ -000158a0: 357d 207c 2020 2020 2020 2020 2020 207c 5} | | │ │ │ │ -000158b0: 0a7c 2020 2020 2020 3337 785f 3078 5f31 .| 37x_0x_1 │ │ │ │ -000158c0: 2d32 3178 5f31 5e32 2d35 785f 3078 5f32 -21x_1^2-5x_0x_2 │ │ │ │ -000158d0: 2b31 3078 5f31 785f 322d 3137 785f 325e +10x_1x_2-17x_2^ │ │ │ │ -000158e0: 3220 2d33 3778 5f30 5e32 207c 2020 7b2d 2 -37x_0^2 | {- │ │ │ │ -000158f0: 357d 207c 2020 2020 2020 2020 2020 207c 5} | | │ │ │ │ -00015900: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ +00015880: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ +00015890: 7c20 207b 2d35 7d20 7c20 2020 2020 2020 | {-5} | │ │ │ │ +000158a0: 2020 2020 7c0a 7c20 2020 2020 2033 3778 |.| 37x │ │ │ │ +000158b0: 5f30 785f 312d 3231 785f 315e 322d 3578 _0x_1-21x_1^2-5x │ │ │ │ +000158c0: 5f30 785f 322b 3130 785f 3178 5f32 2d31 _0x_2+10x_1x_2-1 │ │ │ │ +000158d0: 3778 5f32 5e32 202d 3337 785f 305e 3220 7x_2^2 -37x_0^2 │ │ │ │ +000158e0: 7c20 207b 2d35 7d20 7c20 2020 2020 2020 | 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| │ │ │ │ -00016580: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016570: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000165a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000165b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000165c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000165d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000165e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000165f0: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -00016600: 2020 2020 2020 2020 2020 2020 2034 2020 4 │ │ │ │ -00016610: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016620: 0a7c 6f31 3220 3a20 4d61 7472 6978 2028 .|o12 : Matrix ( │ │ │ │ -00016630: 6b6b 5b73 202e 2e73 202c 2078 202e 2e78 kk[s ..s , x ..x │ │ │ │ -00016640: 205d 2920 203c 2d2d 2028 6b6b 5b73 202e ]) <-- (kk[s . │ │ │ │ -00016650: 2e73 202c 2078 202e 2e78 205d 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2020 2020 2020 2020 │ │ │ │ 00016c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016c50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016c60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00016c70: 3020 2020 3220 2020 2020 2020 2020 2020 0 2 │ │ │ │ +00016c50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016c60: 2020 2020 2030 2020 2032 2020 2020 2020 0 2 │ │ │ │ +00016c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ca0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016cb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016ca0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016cf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016d00: 0a7c 6f31 3520 3a20 4964 6561 6c20 6f66 .|o15 : Ideal of │ │ │ │ -00016d10: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00016cf0: 2020 2020 7c0a 7c6f 3135 203a 2049 6465 |.|o15 : Ide │ │ │ │ +00016d00: 616c 206f 6620 5320 2020 2020 2020 2020 al of S │ │ │ │ +00016d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016d40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016d50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00016d40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00016d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00016da0: 0a7c 6931 3620 3a20 5362 6172 203d 2053 .|i16 : Sbar = S │ │ │ │ -00016db0: 2f49 5320 2020 2020 2020 2020 2020 2020 /IS │ │ │ │ +00016d90: 2d2d 2d2d 2b0a 7c69 3136 203a 2053 6261 ----+.|i16 : Sba │ │ │ │ +00016da0: 7220 3d20 532f 4953 2020 2020 2020 2020 r = S/IS │ │ │ │ +00016db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016de0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016df0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016de0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016e40: 0a7c 6f31 3620 3d20 5362 6172 2020 2020 .|o16 = Sbar │ │ │ │ +00016e30: 2020 2020 7c0a 7c6f 3136 203d 2053 6261 |.|o16 = Sba │ │ │ │ +00016e40: 7220 2020 2020 2020 2020 2020 2020 2020 r │ │ │ │ 00016e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016e90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016e80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ed0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016ee0: 0a7c 6f31 3620 3a20 5175 6f74 6965 6e74 .|o16 : Quotient │ │ │ │ -00016ef0: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +00016ed0: 2020 2020 7c0a 7c6f 3136 203a 2051 756f |.|o16 : Quo │ │ │ │ +00016ee0: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ +00016ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016f20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016f30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00016f20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00016f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00016f80: 0a7c 6931 3720 3a20 534d 6261 7220 3d20 .|i17 : SMbar = │ │ │ │ -00016f90: 5362 6172 2a2a 4d62 6172 2020 2020 2020 Sbar**Mbar │ │ │ │ +00016f70: 2d2d 2d2d 2b0a 7c69 3137 203a 2053 4d62 ----+.|i17 : SMb │ │ │ │ +00016f80: 6172 203d 2053 6261 722a 2a4d 6261 7220 ar = Sbar**Mbar │ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016fc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016fc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017010: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017020: 0a7c 6f31 3720 3d20 636f 6b65 726e 656c .|o17 = cokernel │ │ │ │ -00017030: 207c 2078 5f30 785f 312b 3234 785f 315e | x_0x_1+24x_1^ │ │ │ │ -00017040: 322b 3439 785f 3078 5f32 2b33 785f 3178 2+49x_0x_2+3x_1x │ │ │ │ -00017050: 5f32 2b35 785f 325e 3220 7c20 2020 2020 _2+5x_2^2 | │ │ │ │ -00017060: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017070: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00017010: 2020 2020 7c0a 7c6f 3137 203d 2063 6f6b |.|o17 = cok │ │ │ │ +00017020: 6572 6e65 6c20 7c20 785f 3078 5f31 2b32 ernel | x_0x_1+2 │ │ │ │ +00017030: 3478 5f31 5e32 2b34 3978 5f30 785f 322b 4x_1^2+49x_0x_2+ │ │ │ │ +00017040: 3378 5f31 785f 322b 3578 5f32 5e32 207c 3x_1x_2+5x_2^2 | │ │ │ │ +00017050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017060: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00017070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000170a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000170b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000170c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000170d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000170e0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +000170b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000170c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000170d0: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +000170e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000170f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017100: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017110: 0a7c 6f31 3720 3a20 5362 6172 2d6d 6f64 .|o17 : Sbar-mod │ │ │ │ -00017120: 756c 652c 2071 756f 7469 656e 7420 6f66 ule, quotient of │ │ │ │ -00017130: 2053 6261 7220 2020 2020 2020 2020 2020 Sbar │ │ │ │ +00017100: 2020 2020 7c0a 7c6f 3137 203a 2053 6261 |.|o17 : Sba │ │ │ │ +00017110: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965 r-module, quotie │ │ │ │ +00017120: 6e74 206f 6620 5362 6172 2020 2020 2020 nt of Sbar │ │ │ │ +00017130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017150: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017160: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00017150: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00017160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000171a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000171b0: 0a0a 486f 6d28 6430 2c53 6261 7229 2061 ..Hom(d0,Sbar) a │ │ │ │ -000171c0: 6e64 2048 6f6d 2864 312c 5362 6172 2920 nd Hom(d1,Sbar) │ │ │ │ -000171d0: 746f 6765 7468 6572 2066 6f72 6d20 7468 together form th │ │ │ │ -000171e0: 6520 7265 736f 6c75 7469 6f6e 206f 6620 e resolution of │ │ │ │ -000171f0: 4d62 6172 3b20 7468 7573 2074 6865 0a68 Mbar; thus the.h │ │ │ │ -00017200: 6f6d 6f6c 6f67 7920 6f66 206f 6e65 2063 omology of one c │ │ │ │ -00017210: 6f6d 706f 7369 7469 6f6e 2069 7320 302c omposition is 0, │ │ │ │ -00017220: 2077 6869 6c65 2074 6865 206f 7468 6572 while the other │ │ │ │ -00017230: 2069 7320 4d62 6172 0a0a 2b2d 2d2d 2d2d is Mbar..+----- │ │ │ │ +000171a0: 2d2d 2d2d 2b0a 0a48 6f6d 2864 302c 5362 ----+..Hom(d0,Sb │ │ │ │ +000171b0: 6172 2920 616e 6420 486f 6d28 6431 2c53 ar) and Hom(d1,S │ │ │ │ +000171c0: 6261 7229 2074 6f67 6574 6865 7220 666f bar) together fo │ │ │ │ +000171d0: 726d 2074 6865 2072 6573 6f6c 7574 696f rm the resolutio │ │ │ │ +000171e0: 6e20 6f66 204d 6261 723b 2074 6875 7320 n of Mbar; thus │ │ │ │ +000171f0: 7468 650a 686f 6d6f 6c6f 6779 206f 6620 the.homology of │ │ │ │ +00017200: 6f6e 6520 636f 6d70 6f73 6974 696f 6e20 one composition │ │ │ │ +00017210: 6973 2030 2c20 7768 696c 6520 7468 6520 is 0, while the │ │ │ │ +00017220: 6f74 6865 7220 6973 204d 6261 720a 0a2b other is Mbar..+ │ │ │ │ +00017230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017280: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a --------+.|i18 : │ │ │ │ -00017290: 2070 7275 6e65 2048 485f 3120 636f 6d70 prune HH_1 comp │ │ │ │ -000172a0: 6c65 787b 6475 616c 2028 5362 6172 2a2a lex{dual (Sbar** │ │ │ │ -000172b0: 6430 292c 2064 7561 6c28 5362 6172 2a2a d0), dual(Sbar** │ │ │ │ -000172c0: 6431 297d 203d 3d20 3020 2020 2020 2020 d1)} == 0 │ │ │ │ -000172d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00017280: 6931 3820 3a20 7072 756e 6520 4848 5f31 i18 : prune HH_1 │ │ │ │ +00017290: 2063 6f6d 706c 6578 7b64 7561 6c20 2853 complex{dual (S │ │ │ │ +000172a0: 6261 722a 2a64 3029 2c20 6475 616c 2853 bar**d0), dual(S │ │ │ │ +000172b0: 6261 722a 2a64 3129 7d20 3d3d 2030 2020 bar**d1)} == 0 │ │ │ │ +000172c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000172d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000172e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000172f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017320: 2020 2020 2020 2020 7c0a 7c6f 3138 203d |.|o18 = │ │ │ │ -00017330: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ +00017310: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017320: 6f31 3820 3d20 7472 7565 2020 2020 2020 o18 = true │ │ │ │ +00017330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017370: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017360: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00017370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000173a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000173b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000173c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a --------+.|i19 : │ │ │ │ -000173d0: 204d 6261 7227 203d 2053 6261 725e 312f Mbar' = Sbar^1/ │ │ │ │ -000173e0: 2853 6261 725f 302c 2053 6261 725f 3129 (Sbar_0, Sbar_1) │ │ │ │ -000173f0: 2a2a 534d 6261 7220 2020 2020 2020 2020 **SMbar │ │ │ │ -00017400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017410: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000173b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000173c0: 6931 3920 3a20 4d62 6172 2720 3d20 5362 i19 : Mbar' = Sb │ │ │ │ +000173d0: 6172 5e31 2f28 5362 6172 5f30 2c20 5362 ar^1/(Sbar_0, Sb │ │ │ │ +000173e0: 6172 5f31 292a 2a53 4d62 6172 2020 2020 ar_1)**SMbar │ │ │ │ +000173f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017400: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017460: 2020 2020 2020 2020 7c0a 7c6f 3139 203d |.|o19 = │ │ │ │ -00017470: 2063 6f6b 6572 6e65 6c20 7c20 785f 3078 cokernel | x_0x │ │ │ │ -00017480: 5f31 2b32 3478 5f31 5e32 2b34 3978 5f30 _1+24x_1^2+49x_0 │ │ │ │ -00017490: 785f 322b 3378 5f31 785f 322b 3578 5f32 x_2+3x_1x_2+5x_2 │ │ │ │ -000174a0: 5e32 2073 5f30 2073 5f31 207c 2020 2020 ^2 s_0 s_1 | │ │ │ │ -000174b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017450: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017460: 6f31 3920 3d20 636f 6b65 726e 656c 207c o19 = cokernel | │ │ │ │ +00017470: 2078 5f30 785f 312b 3234 785f 315e 322b x_0x_1+24x_1^2+ │ │ │ │ +00017480: 3439 785f 3078 5f32 2b33 785f 3178 5f32 49x_0x_2+3x_1x_2 │ │ │ │ +00017490: 2b35 785f 325e 3220 735f 3020 735f 3120 +5x_2^2 s_0 s_1 │ │ │ │ +000174a0: 7c20 2020 2020 2020 2020 2020 207c 0a7c | |.| │ │ │ │ +000174b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000174c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000174d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000174e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000174f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017500: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000174f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017520: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ +00017520: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 00017530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017550: 2020 2020 2020 2020 7c0a 7c6f 3139 203a |.|o19 : │ │ │ │ -00017560: 2053 6261 722d 6d6f 6475 6c65 2c20 7175 Sbar-module, qu │ │ │ │ -00017570: 6f74 6965 6e74 206f 6620 5362 6172 2020 otient of Sbar │ │ │ │ +00017540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017550: 6f31 3920 3a20 5362 6172 2d6d 6f64 756c o19 : Sbar-modul │ │ │ │ +00017560: 652c 2071 756f 7469 656e 7420 6f66 2053 e, quotient of S │ │ │ │ +00017570: 6261 7220 2020 2020 2020 2020 2020 2020 bar │ │ │ │ 00017580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000175a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017590: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000175a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000175f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a --------+.|i20 : │ │ │ │ -00017600: 2069 6465 616c 2070 7265 7365 6e74 6174 ideal presentat │ │ │ │ -00017610: 696f 6e20 7072 756e 6520 4848 5f31 2063 ion prune HH_1 c │ │ │ │ -00017620: 6f6d 706c 6578 7b64 7561 6c20 2853 6261 omplex{dual (Sba │ │ │ │ -00017630: 722a 2a64 3129 2c20 6475 616c 2853 6261 r**d1), dual(Sba │ │ │ │ -00017640: 722a 2a64 3029 7d20 7c0a 7c20 2020 2020 r**d0)} |.| │ │ │ │ +000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000175f0: 6932 3020 3a20 6964 6561 6c20 7072 6573 i20 : ideal pres │ │ │ │ +00017600: 656e 7461 7469 6f6e 2070 7275 6e65 2048 entation prune H │ │ │ │ +00017610: 485f 3120 636f 6d70 6c65 787b 6475 616c H_1 complex{dual │ │ │ │ +00017620: 2028 5362 6172 2a2a 6431 292c 2064 7561 (Sbar**d1), dua │ │ │ │ +00017630: 6c28 5362 6172 2a2a 6430 297d 207c 0a7c l(Sbar**d0)} |.| │ │ │ │ +00017640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017690: 2020 2020 2020 2020 7c0a 7c6f 3230 203d |.|o20 = │ │ │ │ -000176a0: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ +00017680: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017690: 6f32 3020 3d20 7472 7565 2020 2020 2020 o20 = true │ │ │ │ +000176a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000176b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000176c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000176d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000176e0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d |.|----- │ │ │ │ +000176d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000176e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000176f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017730: 2d2d 2d2d 2d2d 2d2d 7c0a 7c3d 3d20 6964 --------|.|== id │ │ │ │ -00017740: 6561 6c20 7072 6573 656e 7461 7469 6f6e eal presentation │ │ │ │ -00017750: 204d 6261 7227 2020 2020 2020 2020 2020 Mbar' │ │ │ │ +00017720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +00017730: 3d3d 2069 6465 616c 2070 7265 7365 6e74 == ideal present │ │ │ │ +00017740: 6174 696f 6e20 4d62 6172 2720 2020 2020 ation Mbar' │ │ │ │ +00017750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017770: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00017780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000177a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000177b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000177c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000177d0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -000177e0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -000177f0: 2a20 2a6e 6f74 6520 4578 743a 2028 4d61 * *note Ext: (Ma │ │ │ │ -00017800: 6361 756c 6179 3244 6f63 2945 7874 2c20 caulay2Doc)Ext, │ │ │ │ -00017810: 2d2d 2063 6f6d 7075 7465 2061 6e20 4578 -- compute an Ex │ │ │ │ -00017820: 7420 6d6f 6475 6c65 0a20 202a 202a 6e6f t module. * *no │ │ │ │ -00017830: 7465 206e 6577 4578 743a 206e 6577 4578 te newExt: newEx │ │ │ │ -00017840: 742c 202d 2d20 476c 6f62 616c 2045 7874 t, -- Global Ext │ │ │ │ -00017850: 2066 6f72 206d 6f64 756c 6573 206f 7665 for modules ove │ │ │ │ -00017860: 7220 6120 636f 6d70 6c65 7465 0a20 2020 r a complete. │ │ │ │ -00017870: 2049 6e74 6572 7365 6374 696f 6e0a 2020 Intersection. │ │ │ │ -00017880: 2a20 2a6e 6f74 6520 6d61 6b65 486f 6d6f * *note makeHomo │ │ │ │ -00017890: 746f 7069 6573 3a20 6d61 6b65 486f 6d6f topies: makeHomo │ │ │ │ -000178a0: 746f 7069 6573 2c20 2d2d 2072 6574 7572 topies, -- retur │ │ │ │ -000178b0: 6e73 2061 2073 7973 7465 6d20 6f66 2068 ns a system of h │ │ │ │ -000178c0: 6967 6865 720a 2020 2020 686f 6d6f 746f igher. homoto │ │ │ │ -000178d0: 7069 6573 0a0a 5761 7973 2074 6f20 7573 pies..Ways to us │ │ │ │ -000178e0: 6520 4569 7365 6e62 7564 5368 616d 6173 e EisenbudShamas │ │ │ │ -000178f0: 6854 6f74 616c 3a0a 3d3d 3d3d 3d3d 3d3d hTotal:.======== │ │ │ │ -00017900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00017910: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ -00017920: 4569 7365 6e62 7564 5368 616d 6173 6854 EisenbudShamashT │ │ │ │ -00017930: 6f74 616c 284d 6f64 756c 6529 220a 0a46 otal(Module)"..F │ │ │ │ -00017940: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00017950: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00017960: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00017970: 202a 6e6f 7465 2045 6973 656e 6275 6453 *note EisenbudS │ │ │ │ -00017980: 6861 6d61 7368 546f 7461 6c3a 2045 6973 hamashTotal: Eis │ │ │ │ -00017990: 656e 6275 6453 6861 6d61 7368 546f 7461 enbudShamashTota │ │ │ │ -000179a0: 6c2c 2069 7320 6120 2a6e 6f74 6520 6d65 l, is a *note me │ │ │ │ -000179b0: 7468 6f64 0a66 756e 6374 696f 6e20 7769 thod.function wi │ │ │ │ -000179c0: 7468 206f 7074 696f 6e73 3a20 284d 6163 th options: (Mac │ │ │ │ -000179d0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -000179e0: 4675 6e63 7469 6f6e 5769 7468 4f70 7469 FunctionWithOpti │ │ │ │ -000179f0: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ons,...--------- │ │ │ │ +000177c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +000177d0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +000177e0: 3d0a 0a20 202a 202a 6e6f 7465 2045 7874 =.. * *note Ext │ │ │ │ +000177f0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00017800: 4578 742c 202d 2d20 636f 6d70 7574 6520 Ext, -- compute │ │ │ │ +00017810: 616e 2045 7874 206d 6f64 756c 650a 2020 an Ext module. │ │ │ │ +00017820: 2a20 2a6e 6f74 6520 6e65 7745 7874 3a20 * *note newExt: │ │ │ │ +00017830: 6e65 7745 7874 2c20 2d2d 2047 6c6f 6261 newExt, -- Globa │ │ │ │ +00017840: 6c20 4578 7420 666f 7220 6d6f 6475 6c65 l Ext for module │ │ │ │ +00017850: 7320 6f76 6572 2061 2063 6f6d 706c 6574 s over a complet │ │ │ │ +00017860: 650a 2020 2020 496e 7465 7273 6563 7469 e. Intersecti │ │ │ │ +00017870: 6f6e 0a20 202a 202a 6e6f 7465 206d 616b on. * *note mak │ │ │ │ +00017880: 6548 6f6d 6f74 6f70 6965 733a 206d 616b eHomotopies: mak │ │ │ │ +00017890: 6548 6f6d 6f74 6f70 6965 732c 202d 2d20 eHomotopies, -- │ │ │ │ +000178a0: 7265 7475 726e 7320 6120 7379 7374 656d returns a system │ │ │ │ +000178b0: 206f 6620 6869 6768 6572 0a20 2020 2068 of higher. h │ │ │ │ +000178c0: 6f6d 6f74 6f70 6965 730a 0a57 6179 7320 omotopies..Ways │ │ │ │ +000178d0: 746f 2075 7365 2045 6973 656e 6275 6453 to use EisenbudS │ │ │ │ +000178e0: 6861 6d61 7368 546f 7461 6c3a 0a3d 3d3d hamashTotal:.=== │ │ │ │ +000178f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00017900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00017910: 2020 2a20 2245 6973 656e 6275 6453 6861 * "EisenbudSha │ │ │ │ +00017920: 6d61 7368 546f 7461 6c28 4d6f 6475 6c65 mashTotal(Module │ │ │ │ +00017930: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +00017940: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00017950: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00017960: 626a 6563 7420 2a6e 6f74 6520 4569 7365 bject *note Eise │ │ │ │ +00017970: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00017980: 3a20 4569 7365 6e62 7564 5368 616d 6173 : EisenbudShamas │ │ │ │ +00017990: 6854 6f74 616c 2c20 6973 2061 202a 6e6f hTotal, is a *no │ │ │ │ +000179a0: 7465 206d 6574 686f 640a 6675 6e63 7469 te method.functi │ │ │ │ +000179b0: 6f6e 2077 6974 6820 6f70 7469 6f6e 733a on with options: │ │ │ │ +000179c0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +000179d0: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ +000179e0: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ +000179f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017a40: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00017a50: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00017a60: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00017a70: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00017a80: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00017a90: 352e 3131 2b64 732f 4d32 2f4d 6163 6175 5.11+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 3235 2e31 312b 6473 2f4d 322f 2-1.25.11+ds/M2/ │ │ │ │ +00017a90: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00017aa0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +00017ab0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00017ac0: 6f6e 732e 6d32 3a35 3136 393a 302e 0a1f ons.m2:5169:0... │ │ │ │ +00017ad0: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +00017ae0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +00017af0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +00017b00: 653a 2065 7665 6e45 7874 4d6f 6475 6c65 e: evenExtModule │ │ │ │ +00017b10: 2c20 4e65 7874 3a20 6578 706f 2c20 5072 , Next: expo, Pr │ │ │ │ +00017b20: 6576 3a20 4569 7365 6e62 7564 5368 616d ev: EisenbudSham │ │ │ │ +00017b30: 6173 6854 6f74 616c 2c20 5570 3a20 546f ashTotal, Up: To │ │ │ │ +00017b40: 700a 0a65 7665 6e45 7874 4d6f 6475 6c65 p..evenExtModule │ │ │ │ +00017b50: 202d 2d20 6576 656e 2070 6172 7420 6f66 -- even part of │ │ │ │ +00017b60: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572 Ext^*(M,k) over │ │ │ │ +00017b70: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ +00017b80: 7273 6563 7469 6f6e 2061 7320 6d6f 6475 rsection as modu │ │ │ │ +00017b90: 6c65 206f 7665 7220 4349 206f 7065 7261 le over CI opera │ │ │ │ +00017ba0: 746f 7220 7269 6e67 0a2a 2a2a 2a2a 2a2a tor ring.******* │ │ │ │ +00017bb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017be0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00017c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00017c10: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00017c20: 7361 6765 3a20 0a20 2020 2020 2020 2045 sage: . E │ │ │ │ -00017c30: 203d 2065 7665 6e45 7874 4d6f 6475 6c65 = evenExtModule │ │ │ │ -00017c40: 204d 0a20 202a 2049 6e70 7574 733a 0a20 M. * Inputs:. │ │ │ │ -00017c50: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ -00017c60: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ -00017c70: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ -00017c80: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ -00017c90: 2069 6e74 6572 7365 6374 696f 6e0a 2020 intersection. │ │ │ │ -00017ca0: 2020 2020 2020 7269 6e67 0a20 202a 202a ring. * * │ │ │ │ -00017cb0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e note Optional in │ │ │ │ -00017cc0: 7075 7473 3a20 284d 6163 6175 6c61 7932 puts: (Macaulay2 │ │ │ │ -00017cd0: 446f 6329 7573 696e 6720 6675 6e63 7469 Doc)using functi │ │ │ │ -00017ce0: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ -00017cf0: 6c20 696e 7075 7473 2c3a 0a20 2020 2020 l inputs,:. │ │ │ │ -00017d00: 202a 204f 7574 5269 6e67 203d 3e20 2e2e * OutRing => .. │ │ │ │ -00017d10: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -00017d20: 2030 0a20 202a 204f 7574 7075 7473 3a0a 0. * Outputs:. │ │ │ │ -00017d30: 2020 2020 2020 2a20 452c 2061 202a 6e6f * E, a *no │ │ │ │ -00017d40: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -00017d50: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -00017d60: 2c20 6f76 6572 2061 2070 6f6c 796e 6f6d , over a polynom │ │ │ │ -00017d70: 6961 6c20 7269 6e67 2077 6974 680a 2020 ial ring with. │ │ │ │ -00017d80: 2020 2020 2020 6765 6e73 2069 6e20 6465 gens in de │ │ │ │ -00017d90: 6772 6565 2031 0a0a 4465 7363 7269 7074 gree 1..Descript │ │ │ │ -00017da0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00017db0: 0a45 7874 7261 6374 7320 7468 6520 6576 .Extracts the ev │ │ │ │ -00017dc0: 656e 2064 6567 7265 6520 7061 7274 2066 en degree part f │ │ │ │ -00017dd0: 726f 6d20 4578 744d 6f64 756c 6520 4d20 rom ExtModule M │ │ │ │ -00017de0: 4966 2074 6865 206f 7074 696f 6e61 6c20 If the optional │ │ │ │ -00017df0: 6172 6775 6d65 6e74 204f 7574 5269 6e67 argument OutRing │ │ │ │ -00017e00: 0a3d 3e20 5420 6973 2067 6976 656e 2c20 .=> T is given, │ │ │ │ -00017e10: 616e 6420 636c 6173 7320 5420 3d3d 3d20 and class T === │ │ │ │ -00017e20: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20 PolynomialRing, │ │ │ │ -00017e30: 7468 656e 2074 6865 206f 7574 7075 7420 then the output │ │ │ │ -00017e40: 7769 6c6c 2062 6520 6120 6d6f 6475 6c65 will be a module │ │ │ │ -00017e50: 0a6f 7665 7220 542e 0a0a 2b2d 2d2d 2d2d .over T...+----- │ │ │ │ +00017c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00017c10: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00017c20: 2020 2020 4520 3d20 6576 656e 4578 744d E = evenExtM │ │ │ │ +00017c30: 6f64 756c 6520 4d0a 2020 2a20 496e 7075 odule M. * Inpu │ │ │ │ +00017c40: 7473 3a0a 2020 2020 2020 2a20 4d2c 2061 ts:. * M, a │ │ │ │ +00017c50: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ +00017c60: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ +00017c70: 756c 652c 2c20 6f76 6572 2061 2063 6f6d ule,, over a com │ │ │ │ +00017c80: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ +00017c90: 6f6e 0a20 2020 2020 2020 2072 696e 670a on. ring. │ │ │ │ +00017ca0: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e * *note Option │ │ │ │ +00017cb0: 616c 2069 6e70 7574 733a 2028 4d61 6361 al inputs: (Maca │ │ │ │ +00017cc0: 756c 6179 3244 6f63 2975 7369 6e67 2066 ulay2Doc)using f │ │ │ │ +00017cd0: 756e 6374 696f 6e73 2077 6974 6820 6f70 unctions with op │ │ │ │ +00017ce0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a tional inputs,:. │ │ │ │ +00017cf0: 2020 2020 2020 2a20 4f75 7452 696e 6720 * OutRing │ │ │ │ +00017d00: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ +00017d10: 7661 6c75 6520 300a 2020 2a20 4f75 7470 value 0. * Outp │ │ │ │ +00017d20: 7574 733a 0a20 2020 2020 202a 2045 2c20 uts:. * E, │ │ │ │ +00017d30: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ +00017d40: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ +00017d50: 6475 6c65 2c2c 206f 7665 7220 6120 706f dule,, over a po │ │ │ │ +00017d60: 6c79 6e6f 6d69 616c 2072 696e 6720 7769 lynomial ring wi │ │ │ │ +00017d70: 7468 0a20 2020 2020 2020 2067 656e 7320 th. gens │ │ │ │ +00017d80: 696e 2064 6567 7265 6520 310a 0a44 6573 in degree 1..Des │ │ │ │ +00017d90: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +00017da0: 3d3d 3d3d 0a0a 4578 7472 6163 7473 2074 ====..Extracts t │ │ │ │ +00017db0: 6865 2065 7665 6e20 6465 6772 6565 2070 he even degree p │ │ │ │ +00017dc0: 6172 7420 6672 6f6d 2045 7874 4d6f 6475 art from ExtModu │ │ │ │ +00017dd0: 6c65 204d 2049 6620 7468 6520 6f70 7469 le M If the opti │ │ │ │ +00017de0: 6f6e 616c 2061 7267 756d 656e 7420 4f75 onal argument Ou │ │ │ │ +00017df0: 7452 696e 670a 3d3e 2054 2069 7320 6769 tRing.=> T is gi │ │ │ │ +00017e00: 7665 6e2c 2061 6e64 2063 6c61 7373 2054 ven, and class T │ │ │ │ +00017e10: 203d 3d3d 2050 6f6c 796e 6f6d 6961 6c52 === PolynomialR │ │ │ │ +00017e20: 696e 672c 2074 6865 6e20 7468 6520 6f75 ing, then the ou │ │ │ │ +00017e30: 7470 7574 2077 696c 6c20 6265 2061 206d tput will be a m │ │ │ │ +00017e40: 6f64 756c 650a 6f76 6572 2054 2e0a 0a2b odule.over T...+ │ │ │ │ +00017e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017e90: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 3d20 ----+.|i1 : kk= │ │ │ │ -00017ea0: 5a5a 2f31 3031 2020 2020 2020 2020 2020 ZZ/101 │ │ │ │ +00017e80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ +00017e90: 206b 6b3d 205a 5a2f 3130 3120 2020 2020 kk= ZZ/101 │ │ │ │ +00017ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017ed0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00017ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00017ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00017f10: 3120 3d20 6b6b 2020 2020 2020 2020 2020 1 = kk │ │ │ │ +00017f00: 207c 0a7c 6f31 203d 206b 6b20 2020 2020 |.|o1 = kk │ │ │ │ +00017f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017f30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f80: 2020 2020 7c0a 7c6f 3120 3a20 5175 6f74 |.|o1 : Quot │ │ │ │ -00017f90: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +00017f70: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ +00017f80: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +00017f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017fc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00017fb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00017fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00018000: 3220 3a20 5320 3d20 6b6b 5b78 2c79 2c7a 2 : S = kk[x,y,z │ │ │ │ -00018010: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -00018020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018030: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017ff0: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b -+.|i2 : S = kk[ │ │ │ │ +00018000: 782c 792c 7a5d 2020 2020 2020 2020 2020 x,y,z] │ │ │ │ +00018010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018020: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018070: 2020 2020 7c0a 7c6f 3220 3d20 5320 2020 |.|o2 = S │ │ │ │ +00018060: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ +00018070: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00018080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000180a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000180b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000180c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000180d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000180f0: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ -00018100: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -00018110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018120: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000180e0: 207c 0a7c 6f32 203a 2050 6f6c 796e 6f6d |.|o2 : Polynom │ │ │ │ +000180f0: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +00018100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018110: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00018120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018160: 2d2d 2d2d 2b0a 7c69 3320 3a20 4932 203d ----+.|i3 : I2 = │ │ │ │ -00018170: 2069 6465 616c 2278 332c 797a 2220 2020 ideal"x3,yz" │ │ │ │ +00018150: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +00018160: 2049 3220 3d20 6964 6561 6c22 7833 2c79 I2 = ideal"x3,y │ │ │ │ +00018170: 7a22 2020 2020 2020 2020 2020 2020 2020 z" │ │ │ │ 00018180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000181a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018190: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000181a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000181b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000181c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000181d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000181e0: 2020 2020 2020 2020 2020 2020 3320 2020 3 │ │ │ │ +000181d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000181e0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 000181f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018210: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00018220: 6964 6561 6c20 2878 202c 2079 2a7a 2920 ideal (x , y*z) │ │ │ │ +00018200: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018210: 6f33 203d 2069 6465 616c 2028 7820 2c20 o3 = ideal (x , │ │ │ │ +00018220: 792a 7a29 2020 2020 2020 2020 2020 2020 y*z) │ │ │ │ 00018230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018250: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00018240: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018290: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66 |.|o3 : Ideal of │ │ │ │ -000182a0: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00018280: 2020 2020 207c 0a7c 6f33 203a 2049 6465 |.|o3 : Ide │ │ │ │ +00018290: 616c 206f 6620 5320 2020 2020 2020 2020 al of S │ │ │ │ +000182a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000182b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000182c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000182c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 000182d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000182e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018300: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -00018310: 5232 203d 2053 2f49 3220 2020 2020 2020 R2 = S/I2 │ │ │ │ +000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00018300: 6934 203a 2052 3220 3d20 532f 4932 2020 i4 : R2 = S/I2 │ │ │ │ +00018310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018340: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00018330: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018380: 7c0a 7c6f 3420 3d20 5232 2020 2020 2020 |.|o4 = R2 │ │ │ │ +00018370: 2020 2020 207c 0a7c 6f34 203d 2052 3220 |.|o4 = R2 │ │ │ │ +00018380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000183a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000183b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000183b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000183c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000183d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000183e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000183f0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ -00018400: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +000183e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000183f0: 6f34 203a 2051 756f 7469 656e 7452 696e o4 : QuotientRin │ │ │ │ +00018400: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 00018410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018430: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00018420: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00018430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018470: 2b0a 7c69 3520 3a20 4d32 203d 2052 325e +.|i5 : M2 = R2^ │ │ │ │ -00018480: 312f 6964 6561 6c22 7832 2c79 2c7a 2220 1/ideal"x2,y,z" │ │ │ │ +00018460: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 3220 -----+.|i5 : M2 │ │ │ │ +00018470: 3d20 5232 5e31 2f69 6465 616c 2278 322c = R2^1/ideal"x2, │ │ │ │ +00018480: 792c 7a22 2020 2020 2020 2020 2020 2020 y,z" │ │ │ │ 00018490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000184a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000184a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000184b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000184c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000184d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000184e0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -000184f0: 636f 6b65 726e 656c 207c 2078 3220 7920 cokernel | x2 y │ │ │ │ -00018500: 7a20 7c20 2020 2020 2020 2020 2020 2020 z | │ │ │ │ -00018510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018520: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000184d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000184e0: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20 o5 = cokernel | │ │ │ │ +000184f0: 7832 2079 207a 207c 2020 2020 2020 2020 x2 y z | │ │ │ │ +00018500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018510: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018560: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00018570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018580: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00018590: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000185a0: 3520 3a20 5232 2d6d 6f64 756c 652c 2071 5 : R2-module, q │ │ │ │ -000185b0: 756f 7469 656e 7420 6f66 2052 3220 2020 uotient of R2 │ │ │ │ -000185c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000185d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00018550: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018570: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +00018580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018590: 207c 0a7c 6f35 203a 2052 322d 6d6f 6475 |.|o5 : R2-modu │ │ │ │ +000185a0: 6c65 2c20 7175 6f74 6965 6e74 206f 6620 le, quotient of │ │ │ │ +000185b0: 5232 2020 2020 2020 2020 2020 2020 2020 R2 │ │ │ │ +000185c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000185d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000185e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000185f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018610: 2d2d 2d2d 2b0a 7c69 3620 3a20 6265 7474 ----+.|i6 : bett │ │ │ │ -00018620: 6920 6672 6565 5265 736f 6c75 7469 6f6e i freeResolution │ │ │ │ -00018630: 2028 4d32 2c20 4c65 6e67 7468 4c69 6d69 (M2, LengthLimi │ │ │ │ -00018640: 7420 3d3e 3130 2920 2020 2020 2020 2020 t =>10) │ │ │ │ -00018650: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018600: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ +00018610: 2062 6574 7469 2066 7265 6552 6573 6f6c betti freeResol │ │ │ │ +00018620: 7574 696f 6e20 284d 322c 204c 656e 6774 ution (M2, Lengt │ │ │ │ +00018630: 684c 696d 6974 203d 3e31 3029 2020 2020 hLimit =>10) │ │ │ │ +00018640: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018680: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00018690: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ -000186a0: 2033 2034 2020 3520 2036 2020 3720 2038 3 4 5 6 7 8 │ │ │ │ -000186b0: 2020 3920 3130 2020 2020 2020 2020 2020 9 10 │ │ │ │ -000186c0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -000186d0: 746f 7461 6c3a 2031 2033 2035 2037 2039 total: 1 3 5 7 9 │ │ │ │ -000186e0: 2031 3120 3133 2031 3520 3137 2031 3920 11 13 15 17 19 │ │ │ │ -000186f0: 3231 2020 2020 2020 2020 2020 2020 2020 21 │ │ │ │ -00018700: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00018710: 303a 2031 2032 2032 2032 2032 2020 3220 0: 1 2 2 2 2 2 │ │ │ │ -00018720: 2032 2020 3220 2032 2020 3220 2032 2020 2 2 2 2 2 │ │ │ │ -00018730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018740: 7c0a 7c20 2020 2020 2020 2020 313a 202e |.| 1: . │ │ │ │ -00018750: 2031 2033 2034 2034 2020 3420 2034 2020 1 3 4 4 4 4 │ │ │ │ -00018760: 3420 2034 2020 3420 2034 2020 2020 2020 4 4 4 4 │ │ │ │ -00018770: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00018780: 2020 2020 2020 2020 323a 202e 202e 202e 2: . . . │ │ │ │ -00018790: 2031 2033 2020 3420 2034 2020 3420 2034 1 3 4 4 4 4 │ │ │ │ -000187a0: 2020 3420 2034 2020 2020 2020 2020 2020 4 4 │ │ │ │ -000187b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000187c0: 2020 2020 333a 202e 202e 202e 202e 202e 3: . . . . . │ │ │ │ -000187d0: 2020 3120 2033 2020 3420 2034 2020 3420 1 3 4 4 4 │ │ │ │ -000187e0: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -000187f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00018800: 343a 202e 202e 202e 202e 202e 2020 2e20 4: . . . . . . │ │ │ │ -00018810: 202e 2020 3120 2033 2020 3420 2034 2020 . 1 3 4 4 │ │ │ │ -00018820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018830: 7c0a 7c20 2020 2020 2020 2020 353a 202e |.| 5: . │ │ │ │ -00018840: 202e 202e 202e 202e 2020 2e20 202e 2020 . . . . . . │ │ │ │ -00018850: 2e20 202e 2020 3120 2033 2020 2020 2020 . . 1 3 │ │ │ │ -00018860: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00018680: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018690: 3020 3120 3220 3320 3420 2035 2020 3620 0 1 2 3 4 5 6 │ │ │ │ +000186a0: 2037 2020 3820 2039 2031 3020 2020 2020 7 8 9 10 │ │ │ │ +000186b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000186c0: 6f36 203d 2074 6f74 616c 3a20 3120 3320 o6 = total: 1 3 │ │ │ │ +000186d0: 3520 3720 3920 3131 2031 3320 3135 2031 5 7 9 11 13 15 1 │ │ │ │ +000186e0: 3720 3139 2032 3120 2020 2020 2020 2020 7 19 21 │ │ │ │ +000186f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018700: 2020 2020 2030 3a20 3120 3220 3220 3220 0: 1 2 2 2 │ │ │ │ +00018710: 3220 2032 2020 3220 2032 2020 3220 2032 2 2 2 2 2 2 │ │ │ │ +00018720: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00018730: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018740: 2031 3a20 2e20 3120 3320 3420 3420 2034 1: . 1 3 4 4 4 │ │ │ │ +00018750: 2020 3420 2034 2020 3420 2034 2020 3420 4 4 4 4 4 │ │ │ │ +00018760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018770: 207c 0a7c 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ +00018780: 2e20 2e20 2e20 3120 3320 2034 2020 3420 . . . 1 3 4 4 │ │ │ │ +00018790: 2034 2020 3420 2034 2020 3420 2020 2020 4 4 4 4 │ │ │ │ +000187a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000187b0: 2020 2020 2020 2020 2033 3a20 2e20 2e20 3: . . │ │ │ │ +000187c0: 2e20 2e20 2e20 2031 2020 3320 2034 2020 . . . 1 3 4 │ │ │ │ +000187d0: 3420 2034 2020 3420 2020 2020 2020 2020 4 4 4 │ │ │ │ +000187e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000187f0: 2020 2020 2034 3a20 2e20 2e20 2e20 2e20 4: . . . . │ │ │ │ +00018800: 2e20 202e 2020 2e20 2031 2020 3320 2034 . . . 1 3 4 │ │ │ │ +00018810: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +00018820: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018830: 2035 3a20 2e20 2e20 2e20 2e20 2e20 202e 5: . . . . . . │ │ │ │ +00018840: 2020 2e20 202e 2020 2e20 2031 2020 3320 . . . 1 3 │ │ │ │ +00018850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018860: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000188a0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ -000188b0: 4265 7474 6954 616c 6c79 2020 2020 2020 BettiTally │ │ │ │ +00018890: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000188a0: 6f36 203a 2042 6574 7469 5461 6c6c 7920 o6 : BettiTally │ │ │ │ +000188b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000188c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000188d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000188e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000188d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000188e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000188f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018920: 2b0a 7c69 3720 3a20 4520 3d20 4578 744d +.|i7 : E = ExtM │ │ │ │ -00018930: 6f64 756c 6520 4d32 2020 2020 2020 2020 odule M2 │ │ │ │ +00018910: 2d2d 2d2d 2d2b 0a7c 6937 203a 2045 203d -----+.|i7 : E = │ │ │ │ +00018920: 2045 7874 4d6f 6475 6c65 204d 3220 2020 ExtModule M2 │ │ │ │ +00018930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018950: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00018950: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018990: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000189a0: 2020 2020 2020 2020 2020 2020 3820 2020 8 │ │ │ │ +00018980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000189a0: 2038 2020 2020 2020 2020 2020 2020 2020 8 │ │ │ │ 000189b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000189c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000189d0: 2020 2020 7c0a 7c6f 3720 3d20 286b 6b5b |.|o7 = (kk[ │ │ │ │ -000189e0: 5820 2e2e 5820 5d29 2020 2020 2020 2020 X ..X ]) │ │ │ │ +000189c0: 2020 2020 2020 2020 207c 0a7c 6f37 203d |.|o7 = │ │ │ │ +000189d0: 2028 6b6b 5b58 202e 2e58 205d 2920 2020 (kk[X ..X ]) │ │ │ │ +000189e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000189f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a10: 7c0a 7c20 2020 2020 2020 2020 2030 2020 |.| 0 │ │ │ │ -00018a20: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00018a00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018a10: 2020 3020 2020 3120 2020 2020 2020 2020 0 1 │ │ │ │ +00018a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00018a40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a80: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20 |.|o7 : │ │ │ │ -00018a90: 6b6b 5b58 202e 2e58 205d 2d6d 6f64 756c kk[X ..X ]-modul │ │ │ │ -00018aa0: 652c 2066 7265 652c 2064 6567 7265 6573 e, free, degrees │ │ │ │ -00018ab0: 207b 302e 2e31 2c20 323a 312c 2033 3a32 {0..1, 2:1, 3:2 │ │ │ │ -00018ac0: 2c20 337d 7c0a 7c20 2020 2020 2020 2020 , 3}|.| │ │ │ │ -00018ad0: 3020 2020 3120 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00018a70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018a80: 6f37 203a 206b 6b5b 5820 2e2e 5820 5d2d o7 : kk[X ..X ]- │ │ │ │ +00018a90: 6d6f 6475 6c65 2c20 6672 6565 2c20 6465 module, free, de │ │ │ │ +00018aa0: 6772 6565 7320 7b30 2e2e 312c 2032 3a31 grees {0..1, 2:1 │ │ │ │ +00018ab0: 2c20 333a 322c 2033 7d7c 0a7c 2020 2020 , 3:2, 3}|.| │ │ │ │ +00018ac0: 2020 2020 2030 2020 2031 2020 2020 2020 0 1 │ │ │ │ +00018ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018b00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00018af0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00018b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00018b40: 3820 3a20 6170 706c 7928 746f 4c69 7374 8 : apply(toList │ │ │ │ -00018b50: 2830 2e2e 3130 292c 2069 2d3e 6869 6c62 (0..10), i->hilb │ │ │ │ -00018b60: 6572 7446 756e 6374 696f 6e28 692c 2045 ertFunction(i, E │ │ │ │ -00018b70: 2929 2020 2020 2020 7c0a 7c20 2020 2020 )) |.| │ │ │ │ +00018b30: 2d2b 0a7c 6938 203a 2061 7070 6c79 2874 -+.|i8 : apply(t │ │ │ │ +00018b40: 6f4c 6973 7428 302e 2e31 3029 2c20 692d oList(0..10), i- │ │ │ │ +00018b50: 3e68 696c 6265 7274 4675 6e63 7469 6f6e >hilbertFunction │ │ │ │ +00018b60: 2869 2c20 4529 2920 2020 2020 207c 0a7c (i, E)) |.| │ │ │ │ +00018b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018bb0: 2020 2020 7c0a 7c6f 3820 3d20 7b31 2c20 |.|o8 = {1, │ │ │ │ -00018bc0: 332c 2035 2c20 372c 2039 2c20 3131 2c20 3, 5, 7, 9, 11, │ │ │ │ -00018bd0: 3133 2c20 3135 2c20 3137 2c20 3139 2c20 13, 15, 17, 19, │ │ │ │ -00018be0: 3231 7d20 2020 2020 2020 2020 2020 2020 21} │ │ │ │ -00018bf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018ba0: 2020 2020 2020 2020 207c 0a7c 6f38 203d |.|o8 = │ │ │ │ +00018bb0: 207b 312c 2033 2c20 352c 2037 2c20 392c {1, 3, 5, 7, 9, │ │ │ │ +00018bc0: 2031 312c 2031 332c 2031 352c 2031 372c 11, 13, 15, 17, │ │ │ │ +00018bd0: 2031 392c 2032 317d 2020 2020 2020 2020 19, 21} │ │ │ │ +00018be0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018c20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00018c30: 3820 3a20 4c69 7374 2020 2020 2020 2020 8 : List │ │ │ │ +00018c20: 207c 0a7c 6f38 203a 204c 6973 7420 2020 |.|o8 : List │ │ │ │ +00018c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018c60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00018c50: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00018c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018ca0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4565 7665 ----+.|i9 : Eeve │ │ │ │ -00018cb0: 6e20 3d20 6576 656e 4578 744d 6f64 756c n = evenExtModul │ │ │ │ -00018cc0: 6520 4d32 2020 2020 2020 2020 2020 2020 e M2 │ │ │ │ -00018cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018ce0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018c90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a ---------+.|i9 : │ │ │ │ +00018ca0: 2045 6576 656e 203d 2065 7665 6e45 7874 Eeven = evenExt │ │ │ │ +00018cb0: 4d6f 6475 6c65 204d 3220 2020 2020 2020 Module M2 │ │ │ │ +00018cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00018d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d30: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -00018d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d50: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20 |.|o9 = │ │ │ │ -00018d60: 286b 6b5b 5820 2e2e 5820 5d29 2020 2020 (kk[X ..X ]) │ │ │ │ +00018d10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018d20: 2020 2020 2034 2020 2020 2020 2020 2020 4 │ │ │ │ +00018d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018d40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018d50: 6f39 203d 2028 6b6b 5b58 202e 2e58 205d o9 = (kk[X ..X ] │ │ │ │ +00018d60: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00018d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00018da0: 2030 2020 2031 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00018d80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018d90: 2020 2020 2020 3020 2020 3120 2020 2020 0 1 │ │ │ │ +00018da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018e00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00018e10: 3920 3a20 6b6b 5b58 202e 2e58 205d 2d6d 9 : kk[X ..X ]-m │ │ │ │ -00018e20: 6f64 756c 652c 2066 7265 652c 2064 6567 odule, free, deg │ │ │ │ -00018e30: 7265 6573 207b 302e 2e31 2c20 323a 317d rees {0..1, 2:1} │ │ │ │ -00018e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00018e50: 2020 2020 3020 2020 3120 2020 2020 2020 0 1 │ │ │ │ +00018e00: 207c 0a7c 6f39 203a 206b 6b5b 5820 2e2e |.|o9 : kk[X .. │ │ │ │ +00018e10: 5820 5d2d 6d6f 6475 6c65 2c20 6672 6565 X ]-module, free │ │ │ │ +00018e20: 2c20 6465 6772 6565 7320 7b30 2e2e 312c , degrees {0..1, │ │ │ │ +00018e30: 2032 3a31 7d20 2020 2020 2020 207c 0a7c 2:1} |.| │ │ │ │ +00018e40: 2020 2020 2020 2020 2030 2020 2031 2020 0 1 │ │ │ │ +00018e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018e80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00018e70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00018e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018ec0: 2b0a 7c69 3130 203a 2061 7070 6c79 2874 +.|i10 : apply(t │ │ │ │ -00018ed0: 6f4c 6973 7428 302e 2e35 292c 2069 2d3e oList(0..5), i-> │ │ │ │ -00018ee0: 6869 6c62 6572 7446 756e 6374 696f 6e28 hilbertFunction( │ │ │ │ -00018ef0: 692c 2045 6576 656e 2929 2020 7c0a 7c20 i, Eeven)) |.| │ │ │ │ +00018eb0: 2d2d 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+00018f40: 3133 2c20 3137 2c20 3231 7d20 2020 2020 13, 17, 21} │ │ │ │ +00018f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018f60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018fb0: 7c0a 7c6f 3130 203a 204c 6973 7420 2020 |.|o10 : List │ │ │ │ +00018fa0: 2020 2020 207c 0a7c 6f31 3020 3a20 4c69 |.|o10 : Li │ │ │ │ +00018fb0: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ 00018fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018fe0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00018fe0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00018ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019020: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -00019030: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00019040: 2a20 2a6e 6f74 6520 4578 744d 6f64 756c * *note ExtModul │ │ │ │ -00019050: 653a 2045 7874 4d6f 6475 6c65 2c20 2d2d e: ExtModule, -- │ │ │ │ -00019060: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572 Ext^*(M,k) over │ │ │ │ -00019070: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ -00019080: 7273 6563 7469 6f6e 2061 730a 2020 2020 rsection as. │ │ │ │ -00019090: 6d6f 6475 6c65 206f 7665 7220 4349 206f module over CI o │ │ │ │ -000190a0: 7065 7261 746f 7220 7269 6e67 0a20 202a perator ring. * │ │ │ │ -000190b0: 202a 6e6f 7465 206f 6464 4578 744d 6f64 *note oddExtMod │ │ │ │ -000190c0: 756c 653a 206f 6464 4578 744d 6f64 756c ule: oddExtModul │ │ │ │ -000190d0: 652c 202d 2d20 6f64 6420 7061 7274 206f e, -- odd part o │ │ │ │ -000190e0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665 f Ext^*(M,k) ove │ │ │ │ -000190f0: 7220 6120 636f 6d70 6c65 7465 0a20 2020 r a complete. │ │ │ │ -00019100: 2069 6e74 6572 7365 6374 696f 6e20 6173 intersection as │ │ │ │ -00019110: 206d 6f64 756c 6520 6f76 6572 2043 4920 module over CI │ │ │ │ -00019120: 6f70 6572 6174 6f72 2072 696e 670a 2020 operator ring. │ │ │ │ -00019130: 2a20 2a6e 6f74 6520 4f75 7452 696e 673a * *note OutRing: │ │ │ │ -00019140: 204f 7574 5269 6e67 2c20 2d2d 204f 7074 OutRing, -- Opt │ │ │ │ -00019150: 696f 6e20 616c 6c6f 7769 6e67 2073 7065 ion allowing spe │ │ │ │ -00019160: 6369 6669 6361 7469 6f6e 206f 6620 7468 cification of th │ │ │ │ -00019170: 6520 7269 6e67 206f 7665 720a 2020 2020 e ring over. │ │ │ │ -00019180: 7768 6963 6820 7468 6520 6f75 7470 7574 which the output │ │ │ │ -00019190: 2069 7320 6465 6669 6e65 640a 0a57 6179 is defined..Way │ │ │ │ -000191a0: 7320 746f 2075 7365 2065 7665 6e45 7874 s to use evenExt │ │ │ │ -000191b0: 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d Module:.======== │ │ │ │ -000191c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000191d0: 3d3d 0a0a 2020 2a20 2265 7665 6e45 7874 ==.. * "evenExt │ │ │ │ -000191e0: 4d6f 6475 6c65 284d 6f64 756c 6529 220a Module(Module)". │ │ │ │ -000191f0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -00019200: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -00019210: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -00019220: 6374 202a 6e6f 7465 2065 7665 6e45 7874 ct *note evenExt │ │ │ │ -00019230: 4d6f 6475 6c65 3a20 6576 656e 4578 744d Module: evenExtM │ │ │ │ -00019240: 6f64 756c 652c 2069 7320 6120 2a6e 6f74 odule, is a *not │ │ │ │ -00019250: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ -00019260: 6e20 7769 7468 0a6f 7074 696f 6e73 3a20 n with.options: │ │ │ │ -00019270: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00019280: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ -00019290: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +00019010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00019020: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +00019030: 3d0a 0a20 202a 202a 6e6f 7465 2045 7874 =.. * *note Ext │ │ │ │ +00019040: 4d6f 6475 6c65 3a20 4578 744d 6f64 756c Module: ExtModul │ │ │ │ +00019050: 652c 202d 2d20 4578 745e 2a28 4d2c 6b29 e, -- Ext^*(M,k) │ │ │ │ +00019060: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +00019070: 2069 6e74 6572 7365 6374 696f 6e20 6173 intersection as │ │ │ │ +00019080: 0a20 2020 206d 6f64 756c 6520 6f76 6572 . module over │ │ │ │ +00019090: 2043 4920 6f70 6572 6174 6f72 2072 696e CI operator rin │ │ │ │ +000190a0: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445 g. * *note oddE │ │ │ │ +000190b0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874 xtModule: oddExt │ │ │ │ +000190c0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070 Module, -- odd p │ │ │ │ +000190d0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ +000190e0: 2920 6f76 6572 2061 2063 6f6d 706c 6574 ) over a complet │ │ │ │ +000190f0: 650a 2020 2020 696e 7465 7273 6563 7469 e. intersecti │ │ │ │ +00019100: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ +00019110: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ +00019120: 6e67 0a20 202a 202a 6e6f 7465 204f 7574 ng. * *note Out │ │ │ │ +00019130: 5269 6e67 3a20 4f75 7452 696e 672c 202d Ring: OutRing, - │ │ │ │ +00019140: 2d20 4f70 7469 6f6e 2061 6c6c 6f77 696e - Option allowin │ │ │ │ +00019150: 6720 7370 6563 6966 6963 6174 696f 6e20 g specification │ │ │ │ +00019160: 6f66 2074 6865 2072 696e 6720 6f76 6572 of the ring over │ │ │ │ +00019170: 0a20 2020 2077 6869 6368 2074 6865 206f . which the o │ │ │ │ +00019180: 7574 7075 7420 6973 2064 6566 696e 6564 utput is defined │ │ │ │ +00019190: 0a0a 5761 7973 2074 6f20 7573 6520 6576 ..Ways to use ev │ │ │ │ +000191a0: 656e 4578 744d 6f64 756c 653a 0a3d 3d3d enExtModule:.=== │ │ │ │ +000191b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000191c0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6576 =======.. * "ev │ │ │ │ +000191d0: 656e 4578 744d 6f64 756c 6528 4d6f 6475 enExtModule(Modu │ │ │ │ +000191e0: 6c65 2922 0a0a 466f 7220 7468 6520 7072 le)"..For the pr │ │ │ │ +000191f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +00019200: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +00019210: 206f 626a 6563 7420 2a6e 6f74 6520 6576 object *note ev │ │ │ │ +00019220: 656e 4578 744d 6f64 756c 653a 2065 7665 enExtModule: eve │ │ │ │ +00019230: 6e45 7874 4d6f 6475 6c65 2c20 6973 2061 nExtModule, is a │ │ │ │ +00019240: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ +00019250: 6e63 7469 6f6e 2077 6974 680a 6f70 7469 nction with.opti │ │ │ │ +00019260: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ +00019270: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +00019280: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +00019290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000192a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000192b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000192c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000192d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000192e0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -000192f0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00019300: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00019310: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00019320: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00019330: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d -1.25.11+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 3235 2e31 312b 6473 ulay2-1.25.11+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 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+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 352e 3131 2b64 732f lay2-1.25.11+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 3235 2e31 312b caulay2-1.25.11+ │ │ │ │ -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: 352e 3131 2b64 732f 4d32 2f4d 6163 6175 5.11+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 3235 2e31 macaulay2-1.25.1 │ │ │ │ -0001d930: 312b 6473 2f4d 322f 4d61 6361 756c 6179 1+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 352e 3131 2b64 732f 4d32 2f4d 6163 .25.11+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: 3235 2e31 312b 6473 2f4d 322f 4d61 6361 25.11+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 352e 3131 2b64 732f 4d32 y2-1.25.11+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 3235 2e31 macaulay2-1.25.1 │ │ │ │ -000211f0: 312b 6473 2f4d 322f 4d61 6361 756c 6179 1+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 352e 3131 2b64 732f 4d32 2f4d 6163 .25.11+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 352e 3131 2b64 732f lay2-1.25.11+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 3235 2e31 macaulay2-1.25.1 │ │ │ │ +000240e0: 312b 6473 2f4d 322f 4d61 6361 756c 6179 1+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: 352e 3131 2b64 732f 4d32 2f4d 6163 6175 5.11+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 3235 2e31 312b 6473 2f4d 322f 2-1.25.11+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 352e 3131 2b64 732f 4d32 2f4d 6163 .25.11+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 3235 2e31 312b 6473 2f4d ay2-1.25.11+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 352e 3131 acaulay2-1.25.11 │ │ │ │ -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: 3235 2e31 312b 6473 2f4d 322f 4d61 6361 25.11+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 352e 3131 acaulay2-1.25.11 │ │ │ │ -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: 3235 2e31 312b 6473 2f4d 322f 4d61 6361 25.11+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 352e /macaulay2-1.25. │ │ │ │ -00029b80: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61 11+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 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+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 3235 2e31 312b 6473 2f4d ay2-1.25.11+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 352e 3131 acaulay2-1.25.11 │ │ │ │ +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 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+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 352e 3131 2b64 732f lay2-1.25.11+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 3235 2e31 macaulay2-1.25.1 │ │ │ │ -0002ce60: 312b 6473 2f4d 322f 4d61 6361 756c 6179 1+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 352e 3131 2b64 732f 4d32 2f4d 6163 .25.11+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: 352e 3131 2b64 732f 4d32 2f4d 6163 6175 5.11+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 3235 2e31 312b 6473 2f4d 322f 2-1.25.11+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 3235 2e31 312b 6473 2f4d 322f 2-1.25.11+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 352e 3131 2b64 aulay2-1.25.11+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 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+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 352e 3131 2b64 732f lay2-1.25.11+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 352e 3131 2b64 732f lay2-1.25.11+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 3235 2e31 macaulay2-1.25.1 │ │ │ │ +0002fbb0: 312b 6473 2f4d 322f 4d61 6361 756c 6179 1+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 3235 2e31 312b 6473 2f4d 322f 2-1.25.11+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 352e 3131 2b64 aulay2-1.25.11+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 3235 2e31 macaulay2-1.25.1 │ │ │ │ +00031ee0: 312b 6473 2f4d 322f 4d61 6361 756c 6179 1+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: 352e 3131 2b64 732f 4d32 2f4d 6163 6175 5.11+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 3235 2e31 312b 6473 2f4d 322f 2-1.25.11+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 352e 3131 2b64 732f 4d32 2f4d -1.25.11+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 3235 2e31 312b 6473 ulay2-1.25.11+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 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+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 352e 3131 2b64 732f lay2-1.25.11+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 7c0a 7c6f |.|o │ │ │ │ -0003af30: 3134 203d 207c 2061 3320 6233 2063 3320 14 = | a3 b3 c3 │ │ │ │ -0003af40: 6433 207c 2020 2020 2020 2020 2020 2020 d3 | │ │ │ │ +0003af20: 207c 0a7c 6f31 3420 3d20 7c20 6133 2062 |.|o14 = | a3 b │ │ │ │ +0003af30: 3320 6333 2064 3320 7c20 2020 2020 2020 3 c3 d3 | │ │ │ │ +0003af40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003af70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003af70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003afc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0003afd0: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -0003afe0: 2020 2020 3420 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 7c20 |.| │ │ │ │ +0003b740: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b760: 2020 2020 2020 2020 2020 2020 3320 2020 3 │ │ │ │ +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 7c6f |.|o │ │ │ │ -0003b7a0: 3137 203a 2052 2d6d 6f64 756c 652c 2071 17 : R-module, q │ │ │ │ -0003b7b0: 756f 7469 656e 7420 6f66 2052 2020 2020 uotient of R │ │ │ │ +0003b790: 207c 0a7c 6f31 3720 3a20 522d 6d6f 6475 |.|o17 : R-modu │ │ │ │ +0003b7a0: 6c65 2c20 7175 6f74 6965 6e74 206f 6620 le, quotient of │ │ │ │ +0003b7b0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 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 352e 3131 2b64 732f 4d32 2f4d -1.25.11+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 3235 2e31 312b 6473 ulay2-1.25.11+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 3235 2e31 312b 6473 2f4d ay2-1.25.11+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 352e 3131 acaulay2-1.25.11 │ │ │ │ +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 352e 3131 2b64 732f 4d32 2f4d -1.25.11+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 3235 2e31 312b 6473 ulay2-1.25.11+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 │ │ │ │ +00058070: 6173 2061 206d 6f64 756c 6520 6f76 6572 as a module over │ │ │ │ +00058080: 2061 6e0a 2020 2020 6578 7465 7269 6f72 an. exterior │ │ │ │ +00058090: 2061 6c67 6562 7261 206f 7220 6269 6772 algebra or bigr │ │ │ │ +000580a0: 6164 6564 2061 6c67 6562 7261 0a20 202a aded algebra. * │ │ │ │ +000580b0: 202a 6e6f 7465 2065 7874 6572 696f 7245 *note exteriorE │ │ │ │ +000580c0: 7874 4d6f 6475 6c65 3a20 6578 7465 7269 xtModule: exteri │ │ │ │ +000580d0: 6f72 4578 744d 6f64 756c 652c 202d 2d20 orExtModule, -- │ │ │ │ +000580e0: 4578 7428 4d2c 6b29 206f 7220 4578 7428 Ext(M,k) or Ext( │ │ │ │ +000580f0: 4d2c 4e29 2061 7320 610a 2020 2020 6d6f M,N) as a. mo │ │ │ │ +00058100: 6475 6c65 206f 7665 7220 616e 2065 7874 dule over an ext │ │ │ │ +00058110: 6572 696f 7220 616c 6765 6272 610a 0a57 erior algebra..W │ │ │ │ +00058120: 6179 7320 746f 2075 7365 206d 616b 654d ays to use makeM │ │ │ │ +00058130: 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d 3d3d odule:.========= │ │ │ │ +00058140: 3d3d 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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 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+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 352e 3131 2b64 aulay2-1.25.11+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 3235 2e31 312b 6473 ulay2-1.25.11+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 352e /macaulay2-1.25. │ │ │ │ +0005b5a0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61 11+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 3235 2e31 macaulay2-1.25.1 │ │ │ │ -0005bd10: 312b 6473 2f4d 322f 4d61 6361 756c 6179 1+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 352e 3131 2b64 732f 4d32 2f4d 6163 .25.11+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 3235 2e31 312b 6473 2f4d 322f 2-1.25.11+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 352e 3131 2b64 aulay2-1.25.11+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 3235 h/macaulay2-1.25 │ │ │ │ -00064520: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+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 352e 3131 2b64 732f 4d32 2f4d -1.25.11+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: 3235 2e31 312b 6473 2f4d 322f 4d61 6361 25.11+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 352e 3131 2b64 732f 4d32 y2-1.25.11+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 3235 2e31 312b 6473 2f4d ay2-1.25.11+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 352e 3131 acaulay2-1.25.11 │ │ │ │ +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 352e 3131 2b64 aulay2-1.25.11+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 3235 h/macaulay2-1.25 │ │ │ │ +000668e0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+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 352e /macaulay2-1.25. │ │ │ │ -00066ef0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61 11+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 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+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: 3235 2e31 312b 6473 2f4d 322f 4d61 6361 25.11+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 352e 3131 2b64 732f 4d32 y2-1.25.11+ds/M2 │ │ │ │ +00067cd0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00067ce0: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ +00067cf0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00067d00: 696f 6e73 2e6d 323a 3236 3030 3a30 2e0a ions.m2:2600:0.. │ │ │ │ +00067d10: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ +00067d20: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00067d30: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ +00067d40: 6465 3a20 5332 2c20 4e65 7874 3a20 5368 de: S2, Next: Sh │ │ │ │ +00067d50: 616d 6173 682c 2050 7265 763a 2072 6567 amash, Prev: reg │ │ │ │ +00067d60: 756c 6172 6974 7953 6571 7565 6e63 652c ularitySequence, │ │ │ │ +00067d70: 2055 703a 2054 6f70 0a0a 5332 202d 2d20 Up: Top..S2 -- │ │ │ │ +00067d80: 556e 6976 6572 7361 6c20 6d61 7020 746f Universal map to │ │ │ │ +00067d90: 2061 206d 6f64 756c 6520 7361 7469 7366 a module satisf │ │ │ │ +00067da0: 7969 6e67 2053 6572 7265 2773 2063 6f6e ying Serre's con │ │ │ │ +00067db0: 6469 7469 6f6e 2053 320a 2a2a 2a2a 2a2a dition S2.****** │ │ │ │ +00067dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067de0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00067df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00067e00: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -00067e10: 200a 2020 2020 2020 2020 6620 3d20 5332 . f = S2 │ │ │ │ -00067e20: 2862 2c4d 290a 2020 2a20 496e 7075 7473 (b,M). * Inputs │ │ │ │ -00067e30: 3a0a 2020 2020 2020 2a20 622c 2061 6e20 :. * b, an │ │ │ │ -00067e40: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ -00067e50: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ -00067e60: 2c20 6465 6772 6565 2062 6f75 6e64 2074 , degree bound t │ │ │ │ -00067e70: 6f20 7768 6963 6820 746f 2063 6172 7279 o which to carry │ │ │ │ -00067e80: 0a20 2020 2020 2020 2074 6865 2063 6f6d . the com │ │ │ │ -00067e90: 7075 7461 7469 6f6e 0a20 2020 2020 202a putation. * │ │ │ │ -00067ea0: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -00067eb0: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -00067ec0: 6329 4d6f 6475 6c65 2c2c 200a 2020 2a20 c)Module,, . * │ │ │ │ -00067ed0: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -00067ee0: 2066 2c20 6120 2a6e 6f74 6520 6d61 7472 f, a *note matr │ │ │ │ -00067ef0: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -00067f00: 6329 4d61 7472 6978 2c2c 2064 6566 696e c)Matrix,, defin │ │ │ │ -00067f10: 696e 6720 6120 6d61 7020 4d2d 2d3e 4d27 ing a map M-->M' │ │ │ │ -00067f20: 2074 6861 740a 2020 2020 2020 2020 6167 that. ag │ │ │ │ -00067f30: 7265 6573 2077 6974 6820 7468 6520 5332 rees with the S2 │ │ │ │ -00067f40: 2d69 6669 6361 7469 6f6e 206f 6620 4d20 -ification of M │ │ │ │ -00067f50: 696e 2064 6567 7265 6573 2024 5c67 6571 in degrees $\geq │ │ │ │ -00067f60: 2062 240a 0a44 6573 6372 6970 7469 6f6e b$..Description │ │ │ │ -00067f70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 .===========..If │ │ │ │ -00067f80: 204d 2069 7320 6120 6772 6164 6564 206d M is a graded m │ │ │ │ -00067f90: 6f64 756c 6520 6f76 6572 2061 2072 696e odule over a rin │ │ │ │ -00067fa0: 6720 532c 2074 6865 6e20 7468 6520 5332 g S, then the S2 │ │ │ │ -00067fb0: 2d69 6669 6361 7469 6f6e 206f 6620 4d20 -ification of M │ │ │ │ -00067fc0: 6973 205c 7375 6d5f 7b64 0a5c 696e 205a is \sum_{d.\in Z │ │ │ │ -00067fd0: 5a7d 2048 5e30 2828 7368 6561 6620 4d29 Z} H^0((sheaf M) │ │ │ │ -00067fe0: 2864 2929 2c20 7768 6963 6820 6d61 7920 (d)), which may │ │ │ │ -00067ff0: 6265 2063 6f6d 7075 7465 6420 6173 206c be computed as l │ │ │ │ -00068000: 696d 5f7b 642d 3e5c 696e 6674 797d 2048 im_{d->\infty} H │ │ │ │ -00068010: 6f6d 2849 5f64 2c4d 292c 0a77 6865 7265 om(I_d,M),.where │ │ │ │ -00068020: 2049 5f64 2069 7320 616e 7920 7365 7175 I_d is any sequ │ │ │ │ -00068030: 656e 6365 206f 6620 6964 6561 6c73 2063 ence of ideals c │ │ │ │ -00068040: 6f6e 7461 696e 6564 2069 6e20 6869 6768 ontained in high │ │ │ │ -00068050: 6572 2061 6e64 2068 6967 6865 7220 706f er and higher po │ │ │ │ -00068060: 7765 7273 206f 660a 535f 2b2e 2054 6865 wers of.S_+. The │ │ │ │ -00068070: 7265 2069 7320 6120 6e61 7475 7261 6c20 re is a natural │ │ │ │ -00068080: 7265 7374 7269 6374 696f 6e20 6d61 7020 restriction map │ │ │ │ -00068090: 663a 204d 203d 2048 6f6d 2853 2c4d 2920 f: M = Hom(S,M) │ │ │ │ -000680a0: 5c74 6f20 486f 6d28 495f 642c 4d29 2e20 \to Hom(I_d,M). │ │ │ │ -000680b0: 5765 0a63 6f6d 7075 7465 2061 6c6c 2074 We.compute all t │ │ │ │ -000680c0: 6869 7320 7573 696e 6720 7468 6520 6964 his using the id │ │ │ │ -000680d0: 6561 6c73 2049 5f64 2067 656e 6572 6174 eals I_d generat │ │ │ │ -000680e0: 6564 2062 7920 7468 6520 642d 7468 2070 ed by the d-th p │ │ │ │ -000680f0: 6f77 6572 7320 6f66 2074 6865 0a76 6172 owers of the.var │ │ │ │ -00068100: 6961 626c 6573 2069 6e20 532e 0a0a 5369 iables in S...Si │ │ │ │ -00068110: 6e63 6520 7468 6520 7265 7375 6c74 206d nce the result m │ │ │ │ -00068120: 6179 206e 6f74 2062 6520 6669 6e69 7465 ay not be finite │ │ │ │ -00068130: 6c79 2067 656e 6572 6174 6564 2028 7468 ly generated (th │ │ │ │ -00068140: 6973 2068 6170 7065 6e73 2069 6620 616e is happens if an │ │ │ │ -00068150: 6420 6f6e 6c79 2069 6620 4d0a 6861 7320 d only if M.has │ │ │ │ -00068160: 616e 2061 7373 6f63 6961 7465 6420 7072 an associated pr │ │ │ │ -00068170: 696d 6520 6f66 2064 696d 656e 7369 6f6e ime of dimension │ │ │ │ -00068180: 2031 292c 2077 6520 636f 6d70 7574 6520 1), we compute │ │ │ │ -00068190: 6f6e 6c79 2075 7020 746f 2061 2073 7065 only up to a spe │ │ │ │ -000681a0: 6369 6669 6564 0a64 6567 7265 6520 626f cified.degree bo │ │ │ │ -000681b0: 756e 6420 622e 2046 6f72 2074 6865 2072 und b. For the r │ │ │ │ -000681c0: 6573 756c 7420 746f 2062 6520 636f 7272 esult to be corr │ │ │ │ -000681d0: 6563 7420 646f 776e 2074 6f20 6465 6772 ect down to degr │ │ │ │ -000681e0: 6565 2062 2c20 6974 2069 7320 7375 6666 ee b, it is suff │ │ │ │ -000681f0: 6963 6965 6e74 0a74 6f20 636f 6d70 7574 icient.to comput │ │ │ │ -00068200: 6520 486f 6d28 492c 4d29 2077 6865 7265 e Hom(I,M) where │ │ │ │ -00068210: 2049 205c 7375 6273 6574 2028 535f 2b29 I \subset (S_+) │ │ │ │ -00068220: 5e7b 722d 627d 2e0a 0a2b 2d2d 2d2d 2d2d ^{r-b}...+------ │ │ │ │ +00067df0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00067e00: 7361 6765 3a20 0a20 2020 2020 2020 2066 sage: . f │ │ │ │ +00067e10: 203d 2053 3228 622c 4d29 0a20 202a 2049 = S2(b,M). * I │ │ │ │ +00067e20: 6e70 7574 733a 0a20 2020 2020 202a 2062 nputs:. * b │ │ │ │ +00067e30: 2c20 616e 202a 6e6f 7465 2069 6e74 6567 , an *note integ │ │ │ │ +00067e40: 6572 3a20 284d 6163 6175 6c61 7932 446f er: (Macaulay2Do │ │ │ │ +00067e50: 6329 5a5a 2c2c 2064 6567 7265 6520 626f c)ZZ,, degree bo │ │ │ │ +00067e60: 756e 6420 746f 2077 6869 6368 2074 6f20 und to which to │ │ │ │ +00067e70: 6361 7272 790a 2020 2020 2020 2020 7468 carry. th │ │ │ │ +00067e80: 6520 636f 6d70 7574 6174 696f 6e0a 2020 e computation. │ │ │ │ +00067e90: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +00067ea0: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +00067eb0: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +00067ec0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00067ed0: 2020 2020 2a20 662c 2061 202a 6e6f 7465 * f, a *note │ │ │ │ +00067ee0: 206d 6174 7269 783a 2028 4d61 6361 756c matrix: (Macaul │ │ │ │ +00067ef0: 6179 3244 6f63 294d 6174 7269 782c 2c20 ay2Doc)Matrix,, │ │ │ │ +00067f00: 6465 6669 6e69 6e67 2061 206d 6170 204d defining a map M │ │ │ │ +00067f10: 2d2d 3e4d 2720 7468 6174 0a20 2020 2020 -->M' that. │ │ │ │ +00067f20: 2020 2061 6772 6565 7320 7769 7468 2074 agrees with t │ │ │ │ +00067f30: 6865 2053 322d 6966 6963 6174 696f 6e20 he S2-ification │ │ │ │ +00067f40: 6f66 204d 2069 6e20 6465 6772 6565 7320 of M in degrees │ │ │ │ +00067f50: 245c 6765 7120 6224 0a0a 4465 7363 7269 $\geq b$..Descri │ │ │ │ +00067f60: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +00067f70: 3d0a 0a49 6620 4d20 6973 2061 2067 7261 =..If M is a gra │ │ │ │ +00067f80: 6465 6420 6d6f 6475 6c65 206f 7665 7220 ded module over │ │ │ │ +00067f90: 6120 7269 6e67 2053 2c20 7468 656e 2074 a ring S, then t │ │ │ │ +00067fa0: 6865 2053 322d 6966 6963 6174 696f 6e20 he S2-ification │ │ │ │ +00067fb0: 6f66 204d 2069 7320 5c73 756d 5f7b 640a of M is \sum_{d. │ │ │ │ +00067fc0: 5c69 6e20 5a5a 7d20 485e 3028 2873 6865 \in ZZ} H^0((she │ │ │ │ +00067fd0: 6166 204d 2928 6429 292c 2077 6869 6368 af M)(d)), which │ │ │ │ +00067fe0: 206d 6179 2062 6520 636f 6d70 7574 6564 may be computed │ │ │ │ +00067ff0: 2061 7320 6c69 6d5f 7b64 2d3e 5c69 6e66 as lim_{d->\inf │ │ │ │ +00068000: 7479 7d20 486f 6d28 495f 642c 4d29 2c0a ty} Hom(I_d,M),. │ │ │ │ +00068010: 7768 6572 6520 495f 6420 6973 2061 6e79 where I_d is any │ │ │ │ +00068020: 2073 6571 7565 6e63 6520 6f66 2069 6465 sequence of ide │ │ │ │ +00068030: 616c 7320 636f 6e74 6169 6e65 6420 696e als contained in │ │ │ │ +00068040: 2068 6967 6865 7220 616e 6420 6869 6768 higher and high │ │ │ │ +00068050: 6572 2070 6f77 6572 7320 6f66 0a53 5f2b er powers of.S_+ │ │ │ │ +00068060: 2e20 5468 6572 6520 6973 2061 206e 6174 . There is a nat │ │ │ │ +00068070: 7572 616c 2072 6573 7472 6963 7469 6f6e ural restriction │ │ │ │ +00068080: 206d 6170 2066 3a20 4d20 3d20 486f 6d28 map f: M = Hom( │ │ │ │ +00068090: 532c 4d29 205c 746f 2048 6f6d 2849 5f64 S,M) \to Hom(I_d │ │ │ │ +000680a0: 2c4d 292e 2057 650a 636f 6d70 7574 6520 ,M). We.compute │ │ │ │ +000680b0: 616c 6c20 7468 6973 2075 7369 6e67 2074 all this using t │ │ │ │ +000680c0: 6865 2069 6465 616c 7320 495f 6420 6765 he ideals I_d ge │ │ │ │ +000680d0: 6e65 7261 7465 6420 6279 2074 6865 2064 nerated by the d │ │ │ │ +000680e0: 2d74 6820 706f 7765 7273 206f 6620 7468 -th powers of th │ │ │ │ +000680f0: 650a 7661 7269 6162 6c65 7320 696e 2053 e.variables in S │ │ │ │ +00068100: 2e0a 0a53 696e 6365 2074 6865 2072 6573 ...Since the res │ │ │ │ +00068110: 756c 7420 6d61 7920 6e6f 7420 6265 2066 ult may not be f │ │ │ │ +00068120: 696e 6974 656c 7920 6765 6e65 7261 7465 initely generate │ │ │ │ +00068130: 6420 2874 6869 7320 6861 7070 656e 7320 d (this happens │ │ │ │ +00068140: 6966 2061 6e64 206f 6e6c 7920 6966 204d if and only if M │ │ │ │ +00068150: 0a68 6173 2061 6e20 6173 736f 6369 6174 .has an associat │ │ │ │ +00068160: 6564 2070 7269 6d65 206f 6620 6469 6d65 ed prime of dime │ │ │ │ +00068170: 6e73 696f 6e20 3129 2c20 7765 2063 6f6d nsion 1), we com │ │ │ │ +00068180: 7075 7465 206f 6e6c 7920 7570 2074 6f20 pute only up to │ │ │ │ +00068190: 6120 7370 6563 6966 6965 640a 6465 6772 a specified.degr │ │ │ │ +000681a0: 6565 2062 6f75 6e64 2062 2e20 466f 7220 ee bound b. For │ │ │ │ +000681b0: 7468 6520 7265 7375 6c74 2074 6f20 6265 the result to be │ │ │ │ +000681c0: 2063 6f72 7265 6374 2064 6f77 6e20 746f correct down to │ │ │ │ +000681d0: 2064 6567 7265 6520 622c 2069 7420 6973 degree b, it is │ │ │ │ +000681e0: 2073 7566 6669 6369 656e 740a 746f 2063 sufficient.to c │ │ │ │ +000681f0: 6f6d 7075 7465 2048 6f6d 2849 2c4d 2920 ompute Hom(I,M) │ │ │ │ +00068200: 7768 6572 6520 4920 5c73 7562 7365 7420 where I \subset │ │ │ │ +00068210: 2853 5f2b 295e 7b72 2d62 7d2e 0a0a 2b2d (S_+)^{r-b}...+- │ │ │ │ +00068220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068270: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ -00068280: 6b3d 5a5a 2f31 3031 2020 2020 2020 2020 k=ZZ/101 │ │ │ │ +00068260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068270: 3120 3a20 6b6b 3d5a 5a2f 3130 3120 2020 1 : kk=ZZ/101 │ │ │ │ +00068280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000682b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000682c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000682b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000682c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068310: 2020 2020 2020 207c 0a7c 6f31 203d 206b |.|o1 = k │ │ │ │ -00068320: 6b20 2020 2020 2020 2020 2020 2020 2020 k │ │ │ │ +00068300: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068310: 3120 3d20 6b6b 2020 2020 2020 2020 2020 1 = kk │ │ │ │ +00068320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068360: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068350: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000683a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000683b0: 2020 2020 2020 207c 0a7c 6f31 203a 2051 |.|o1 : Q │ │ │ │ -000683c0: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +000683a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000683b0: 3120 3a20 5175 6f74 6965 6e74 5269 6e67 1 : QuotientRing │ │ │ │ +000683c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000683d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000683e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000683f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068400: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000683f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00068400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068450: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 -------+.|i2 : S │ │ │ │ -00068460: 203d 206b 6b5b 612c 622c 632c 645d 2020 = kk[a,b,c,d] │ │ │ │ +00068440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068450: 3220 3a20 5320 3d20 6b6b 5b61 2c62 2c63 2 : S = kk[a,b,c │ │ │ │ +00068460: 2c64 5d20 2020 2020 2020 2020 2020 2020 ,d] │ │ │ │ 00068470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000684a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068490: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000684a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000684b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000684c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000684d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000684e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000684f0: 2020 2020 2020 207c 0a7c 6f32 203d 2053 |.|o2 = S │ │ │ │ +000684e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000684f0: 3220 3d20 5320 2020 2020 2020 2020 2020 2 = S │ │ │ │ 00068500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068540: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068530: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068590: 2020 2020 2020 207c 0a7c 6f32 203a 2050 |.|o2 : P │ │ │ │ -000685a0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +00068580: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068590: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ +000685a0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 000685b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000685c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000685d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000685e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000685d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000685e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000685f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068630: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 204d -------+.|i3 : M │ │ │ │ -00068640: 203d 2074 7275 6e63 6174 6528 332c 535e = truncate(3,S^ │ │ │ │ -00068650: 3129 2020 2020 2020 2020 2020 2020 2020 1) │ │ │ │ +00068620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068630: 3320 3a20 4d20 3d20 7472 756e 6361 7465 3 : M = truncate │ │ │ │ +00068640: 2833 2c53 5e31 2920 2020 2020 2020 2020 (3,S^1) │ │ │ │ +00068650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068680: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068670: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000686a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000686b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000686c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000686d0: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ -000686e0: 6d61 6765 207c 2064 3320 6364 3220 6264 mage | d3 cd2 bd │ │ │ │ -000686f0: 3220 6164 3220 6332 6420 6263 6420 6163 2 ad2 c2d bcd ac │ │ │ │ -00068700: 6420 6232 6420 6162 6420 6132 6420 6333 d b2d abd a2d c3 │ │ │ │ -00068710: 2062 6332 2061 6332 2062 3263 2061 6263 bc2 ac2 b2c abc │ │ │ │ -00068720: 2061 3263 2062 337c 0a7c 2020 2020 2020 a2c b3|.| │ │ │ │ +000686c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000686d0: 3320 3d20 696d 6167 6520 7c20 6433 2063 3 = image | d3 c │ │ │ │ +000686e0: 6432 2062 6432 2061 6432 2063 3264 2062 d2 bd2 ad2 c2d b │ │ │ │ +000686f0: 6364 2061 6364 2062 3264 2061 6264 2061 cd acd b2d abd a │ │ │ │ +00068700: 3264 2063 3320 6263 3220 6163 3220 6232 2d c3 bc2 ac2 b2 │ │ │ │ +00068710: 6320 6162 6320 6132 6320 6233 7c0a 7c20 c abc a2c b3|.| │ │ │ │ +00068720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068770: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068790: 2020 2020 2020 2031 2020 2020 2020 2020 1 │ │ │ │ +00068760: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00068780: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +00068790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000687a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000687b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000687c0: 2020 2020 2020 207c 0a7c 6f33 203a 2053 |.|o3 : S │ │ │ │ -000687d0: 2d6d 6f64 756c 652c 2073 7562 6d6f 6475 -module, submodu │ │ │ │ -000687e0: 6c65 206f 6620 5320 2020 2020 2020 2020 le of S │ │ │ │ +000687b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000687c0: 3320 3a20 532d 6d6f 6475 6c65 2c20 7375 3 : S-module, su │ │ │ │ +000687d0: 626d 6f64 756c 6520 6f66 2053 2020 2020 bmodule of S │ │ │ │ +000687e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000687f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068810: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d |.|------ │ │ │ │ +00068800: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ +00068810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068860: 2d2d 2d2d 2d2d 2d7c 0a7c 6162 3220 6132 -------|.|ab2 a2 │ │ │ │ -00068870: 6220 6133 207c 2020 2020 2020 2020 2020 b a3 | │ │ │ │ +00068850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c61 ------------|.|a │ │ │ │ +00068860: 6232 2061 3262 2061 3320 7c20 2020 2020 b2 a2b a3 | │ │ │ │ +00068870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000688a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000688b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000688a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000688b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000688c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000688d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000688e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000688f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068900: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2062 -------+.|i4 : b │ │ │ │ -00068910: 6574 7469 206d 6174 7269 7820 5332 2830 etti matrix S2(0 │ │ │ │ -00068920: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +000688f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068900: 3420 3a20 6265 7474 6920 6d61 7472 6978 4 : betti matrix │ │ │ │ +00068910: 2053 3228 302c 4d29 2020 2020 2020 2020 S2(0,M) │ │ │ │ +00068920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068950: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068940: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000689a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000689b0: 2020 2020 2020 3020 2031 2020 2020 2020 0 1 │ │ │ │ +00068990: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000689a0: 2020 2020 2020 2020 2020 2030 2020 3120 0 1 │ │ │ │ +000689b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000689c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000689d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000689e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000689f0: 2020 2020 2020 207c 0a7c 6f34 203d 2074 |.|o4 = t │ │ │ │ -00068a00: 6f74 616c 3a20 3120 3230 2020 2020 2020 otal: 1 20 │ │ │ │ +000689e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000689f0: 3420 3d20 746f 7461 6c3a 2031 2032 3020 4 = total: 1 20 │ │ │ │ +00068a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068a50: 2020 2030 3a20 3120 202e 2020 2020 2020 0: 1 . │ │ │ │ +00068a30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068a40: 2020 2020 2020 2020 303a 2031 2020 2e20 0: 1 . │ │ │ │ +00068a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068aa0: 2020 2031 3a20 2e20 202e 2020 2020 2020 1: . . │ │ │ │ +00068a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068a90: 2020 2020 2020 2020 313a 202e 2020 2e20 1: . . │ │ │ │ +00068aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068ae0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068af0: 2020 2032 3a20 2e20 3230 2020 2020 2020 2: . 20 │ │ │ │ +00068ad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068ae0: 2020 2020 2020 2020 323a 202e 2032 3020 2: . 20 │ │ │ │ +00068af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b80: 2020 2020 2020 207c 0a7c 6f34 203a 2042 |.|o4 : B │ │ │ │ -00068b90: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +00068b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068b80: 3420 3a20 4265 7474 6954 616c 6c79 2020 4 : BettiTally │ │ │ │ +00068b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00068bc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00068bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068c20: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2062 -------+.|i5 : b │ │ │ │ -00068c30: 6574 7469 206d 6174 7269 7820 5332 2831 etti matrix S2(1 │ │ │ │ -00068c40: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +00068c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068c20: 3520 3a20 6265 7474 6920 6d61 7472 6978 5 : betti matrix │ │ │ │ +00068c30: 2053 3228 312c 4d29 2020 2020 2020 2020 S2(1,M) │ │ │ │ +00068c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068c70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068cd0: 2020 2020 2020 3020 2031 2020 2020 2020 0 1 │ │ │ │ +00068cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068cc0: 2020 2020 2020 2020 2020 2030 2020 3120 0 1 │ │ │ │ +00068cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d10: 2020 2020 2020 207c 0a7c 6f35 203d 2074 |.|o5 = t │ │ │ │ -00068d20: 6f74 616c 3a20 3120 3230 2020 2020 2020 otal: 1 20 │ │ │ │ +00068d00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068d10: 3520 3d20 746f 7461 6c3a 2031 2032 3020 5 = total: 1 20 │ │ │ │ +00068d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068d70: 2020 2030 3a20 3120 202e 2020 2020 2020 0: 1 . │ │ │ │ +00068d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068d60: 2020 2020 2020 2020 303a 2031 2020 2e20 0: 1 . │ │ │ │ +00068d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068db0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068dc0: 2020 2031 3a20 2e20 202e 2020 2020 2020 1: . . │ │ │ │ +00068da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068db0: 2020 2020 2020 2020 313a 202e 2020 2e20 1: . . │ │ │ │ +00068dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068e00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068e10: 2020 2032 3a20 2e20 3230 2020 2020 2020 2: . 20 │ │ │ │ +00068df0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068e00: 2020 2020 2020 2020 323a 202e 2032 3020 2: . 20 │ │ │ │ +00068e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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+00068ee0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00068ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068f40: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d -------+.|i6 : M │ │ │ │ -00068f50: 203d 2053 5e31 2f69 6e74 6572 7365 6374 = S^1/intersect │ │ │ │ -00068f60: 2869 6465 616c 2261 2c62 2c63 222c 2069 (ideal"a,b,c", i │ │ │ │ -00068f70: 6465 616c 2262 2c63 2c64 222c 6964 6561 deal"b,c,d",idea │ │ │ │ -00068f80: 6c22 632c 642c 6122 2c69 6465 616c 2264 l"c,d,a",ideal"d │ │ │ │ -00068f90: 2c61 2c62 2229 207c 0a7c 2020 2020 2020 ,a,b") |.| │ │ │ │ +00068f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068f40: 3620 3a20 4d20 3d20 535e 312f 696e 7465 6 : M = S^1/inte │ │ │ │ +00068f50: 7273 6563 7428 6964 6561 6c22 612c 622c rsect(ideal"a,b, │ │ │ │ +00068f60: 6322 2c20 6964 6561 6c22 622c 632c 6422 c", ideal"b,c,d" │ │ │ │ +00068f70: 2c69 6465 616c 2263 2c64 2c61 222c 6964 ,ideal"c,d,a",id │ │ │ │ +00068f80: 6561 6c22 642c 612c 6222 2920 7c0a 7c20 eal"d,a,b") |.| │ │ │ │ +00068f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068fe0: 2020 2020 2020 207c 0a7c 6f36 203d 2063 |.|o6 = c │ │ │ │ -00068ff0: 6f6b 6572 6e65 6c20 7c20 6364 2062 6420 okernel | cd bd │ │ │ │ -00069000: 6164 2062 6320 6163 2061 6220 7c20 2020 ad bc ac ab | │ │ │ │ +00068fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068fe0: 3620 3d20 636f 6b65 726e 656c 207c 2063 6 = cokernel | c │ │ │ │ +00068ff0: 6420 6264 2061 6420 6263 2061 6320 6162 d bd ad bc ac ab │ │ │ │ +00069000: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00069010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069030: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00069020: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069080: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00069090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000690a0: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +00069070: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069090: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +000690a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000690b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000690c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000690d0: 2020 2020 2020 207c 0a7c 6f36 203a 2053 |.|o6 : S │ │ │ │ -000690e0: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -000690f0: 7420 6f66 2053 2020 2020 2020 2020 2020 t of S │ │ │ │ +000690c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000690d0: 3620 3a20 532d 6d6f 6475 6c65 2c20 7175 6 : S-module, qu │ │ │ │ +000690e0: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +000690f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069120: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069110: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069170: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2070 -------+.|i7 : p │ │ │ │ -00069180: 7275 6e65 2073 6f75 7263 6520 5332 2830 rune source S2(0 │ │ │ │ -00069190: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +00069160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00069170: 3720 3a20 7072 756e 6520 736f 7572 6365 7 : prune source │ │ │ │ +00069180: 2053 3228 302c 4d29 2020 2020 2020 2020 S2(0,M) │ │ │ │ +00069190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000691a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000691b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000691c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 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0a7c 2020 2020 2020 |.| │ │ │ │ +00069250: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000692c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692d0: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +000692a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000692b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000692c0: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +000692d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000692e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069300: 2020 2020 2020 207c 0a7c 6f37 203a 2053 |.|o7 : S │ │ │ │ -00069310: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -00069320: 7420 6f66 2053 2020 2020 2020 2020 2020 t of S │ │ │ │ +000692f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00069300: 3720 3a20 532d 6d6f 6475 6c65 2c20 7175 7 : S-module, qu │ │ │ │ +00069310: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +00069320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069350: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069340: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000693a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2070 -------+.|i8 : p │ │ │ │ -000693b0: 7275 6e65 2074 6172 6765 7420 5332 2830 rune target S2(0 │ │ │ │ -000693c0: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +00069390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000693a0: 3820 3a20 7072 756e 6520 7461 7267 6574 8 : prune target │ │ │ │ +000693b0: 2053 3228 302c 4d29 2020 2020 2020 2020 S2(0,M) │ │ │ │ +000693c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000693d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000693e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000693f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000693e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000693f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069440: 2020 2020 2020 207c 0a7c 6f38 203d 2063 |.|o8 = c │ │ │ │ -00069450: 6f6b 6572 6e65 6c20 7b2d 317d 207c 2064 okernel {-1} | d │ │ │ │ -00069460: 2063 2062 2030 2030 2030 2030 2030 2030 c b 0 0 0 0 0 0 │ │ │ │ -00069470: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -00069480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069490: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000694a0: 2020 2020 2020 2020 7b2d 317d 207c 2030 {-1} | 0 │ │ │ │ -000694b0: 2030 2030 2064 2063 2061 2030 2030 2030 0 0 d c a 0 0 0 │ │ │ │ -000694c0: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -000694d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000694e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000694f0: 2020 2020 2020 2020 7b2d 317d 207c 2030 {-1} | 0 │ │ │ │ -00069500: 2030 2030 2030 2030 2030 2064 2062 2061 0 0 0 0 0 d b a │ │ │ │ -00069510: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -00069520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069530: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00069540: 2020 2020 2020 2020 7b2d 317d 207c 2030 {-1} | 0 │ │ │ │ -00069550: 2030 2030 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 0 │ │ │ │ -00069560: 2063 2062 2061 207c 2020 2020 2020 2020 c b a | │ │ │ │ -00069570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069580: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00069430: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00069440: 3820 3d20 636f 6b65 726e 656c 207b 2d31 8 = cokernel {-1 │ │ │ │ +00069450: 7d20 7c20 6420 6320 6220 3020 3020 3020 } | d c b 0 0 0 │ │ │ │ +00069460: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +00069470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069480: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069490: 2020 2020 2020 2020 2020 2020 207b 2d31 {-1 │ │ │ │ +000694a0: 7d20 7c20 3020 3020 3020 6420 6320 6120 } | 0 0 0 d c a │ │ │ │ +000694b0: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +000694c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000694d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000694e0: 2020 2020 2020 2020 2020 2020 207b 2d31 {-1 │ │ │ │ +000694f0: 7d20 7c20 3020 3020 3020 3020 3020 3020 } | 0 0 0 0 0 0 │ │ │ │ +00069500: 6420 6220 6120 3020 3020 3020 7c20 2020 d b a 0 0 0 | │ │ │ │ +00069510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069520: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069530: 2020 2020 2020 2020 2020 2020 207b 2d31 {-1 │ │ │ │ +00069540: 7d20 7c20 3020 3020 3020 3020 3020 3020 } | 0 0 0 0 0 0 │ │ │ │ +00069550: 3020 3020 3020 6320 6220 6120 7c20 2020 0 0 0 c b a | │ │ │ │ +00069560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069570: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000695a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000695b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000695c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000695d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000695e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000695f0: 2020 2020 2020 3420 2020 2020 2020 2020 4 │ │ │ │ +000695c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000695d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000695e0: 2020 2020 2020 2020 2020 2034 2020 2020 4 │ │ │ │ +000695f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069620: 2020 2020 2020 207c 0a7c 6f38 203a 2053 |.|o8 : S │ │ │ │ -00069630: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -00069640: 7420 6f66 2053 2020 2020 2020 2020 2020 t of S │ │ │ │ +00069610: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00069620: 3820 3a20 532d 6d6f 6475 6c65 2c20 7175 8 : S-module, qu │ │ │ │ +00069630: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +00069640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069670: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069660: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000696a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000696b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000696c0: 2d2d 2d2d 2d2d 2d2b 0a0a 4174 206f 6e65 -------+..At one │ │ │ │ -000696d0: 2074 696d 6520 4445 2068 6f70 6564 2074 time DE hoped t │ │ │ │ -000696e0: 6861 742c 2069 6620 4d20 7765 7265 2061 hat, if M were a │ │ │ │ -000696f0: 206d 6f64 756c 6520 6f76 6572 2074 6865 module over the │ │ │ │ -00069700: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -00069710: 6563 7469 6f6e 2052 0a77 6974 6820 7265 ection R.with re │ │ │ │ -00069720: 7369 6475 6520 6669 656c 6420 6b2c 2074 sidue field k, t │ │ │ │ -00069730: 6865 6e20 7468 6520 6e61 7475 7261 6c20 hen the natural │ │ │ │ -00069740: 6d61 7020 6672 6f6d 2022 636f 6d70 6c65 map from "comple │ │ │ │ -00069750: 7465 2220 4578 7420 6d6f 6475 6c65 2022 te" Ext module " │ │ │ │ -00069760: 2877 6964 6568 6174 0a45 7874 295f 5228 (widehat.Ext)_R( │ │ │ │ -00069770: 4d2c 6b29 2220 746f 2074 6865 2053 322d M,k)" to the S2- │ │ │ │ -00069780: 6966 6963 6174 696f 6e20 6f66 2045 7874 ification of Ext │ │ │ │ -00069790: 5f52 284d 2c6b 2920 776f 756c 6420 6265 _R(M,k) would be │ │ │ │ -000697a0: 2073 7572 6a65 6374 6976 653b 0a65 7175 surjective;.equ │ │ │ │ -000697b0: 6976 616c 656e 746c 792c 2069 6620 4e20 ivalently, if N │ │ │ │ -000697c0: 7765 7265 2061 2073 7566 6669 6369 656e were a sufficien │ │ │ │ -000697d0: 746c 7920 6e65 6761 7469 7665 2073 797a tly negative syz │ │ │ │ -000697e0: 7967 7920 6f66 204d 2c20 7468 656e 2074 ygy of M, then t │ │ │ │ -000697f0: 6865 2066 6972 7374 0a6c 6f63 616c 2063 he first.local c │ │ │ │ -00069800: 6f68 6f6d 6f6c 6f67 7920 6d6f 6475 6c65 ohomology module │ │ │ │ -00069810: 206f 6620 4578 745f 5228 4d2c 6b29 2077 of Ext_R(M,k) w │ │ │ │ -00069820: 6f75 6c64 2062 6520 7a65 726f 2e20 5468 ould be zero. Th │ │ │ │ -00069830: 6973 2069 7320 6661 6c73 652c 2061 7320 is is false, as │ │ │ │ -00069840: 7368 6f77 6e20 6279 0a74 6865 2066 6f6c shown by.the fol │ │ │ │ -00069850: 6c6f 7769 6e67 2065 7861 6d70 6c65 3a0a lowing example:. │ │ │ │ -00069860: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000696b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 ------------+..A │ │ │ │ +000696c0: 7420 6f6e 6520 7469 6d65 2044 4520 686f t one time DE ho │ │ │ │ +000696d0: 7065 6420 7468 6174 2c20 6966 204d 2077 ped that, if M w │ │ │ │ +000696e0: 6572 6520 6120 6d6f 6475 6c65 206f 7665 ere a module ove │ │ │ │ +000696f0: 7220 7468 6520 636f 6d70 6c65 7465 2069 r the complete i │ │ │ │ +00069700: 6e74 6572 7365 6374 696f 6e20 520a 7769 ntersection R.wi │ │ │ │ +00069710: 7468 2072 6573 6964 7565 2066 6965 6c64 th residue field │ │ │ │ +00069720: 206b 2c20 7468 656e 2074 6865 206e 6174 k, then the nat │ │ │ │ +00069730: 7572 616c 206d 6170 2066 726f 6d20 2263 ural map from "c │ │ │ │ +00069740: 6f6d 706c 6574 6522 2045 7874 206d 6f64 omplete" Ext mod │ │ │ │ +00069750: 756c 6520 2228 7769 6465 6861 740a 4578 ule "(widehat.Ex │ │ │ │ +00069760: 7429 5f52 284d 2c6b 2922 2074 6f20 7468 t)_R(M,k)" to th │ │ │ │ +00069770: 6520 5332 2d69 6669 6361 7469 6f6e 206f e S2-ification o │ │ │ │ +00069780: 6620 4578 745f 5228 4d2c 6b29 2077 6f75 f Ext_R(M,k) wou │ │ │ │ +00069790: 6c64 2062 6520 7375 726a 6563 7469 7665 ld be surjective │ │ │ │ +000697a0: 3b0a 6571 7569 7661 6c65 6e74 6c79 2c20 ;.equivalently, │ │ │ │ +000697b0: 6966 204e 2077 6572 6520 6120 7375 6666 if N were a suff │ │ │ │ +000697c0: 6963 6965 6e74 6c79 206e 6567 6174 6976 iciently negativ │ │ │ │ +000697d0: 6520 7379 7a79 6779 206f 6620 4d2c 2074 e syzygy of M, t │ │ │ │ +000697e0: 6865 6e20 7468 6520 6669 7273 740a 6c6f hen the first.lo │ │ │ │ +000697f0: 6361 6c20 636f 686f 6d6f 6c6f 6779 206d cal cohomology m │ │ │ │ +00069800: 6f64 756c 6520 6f66 2045 7874 5f52 284d odule of Ext_R(M │ │ │ │ +00069810: 2c6b 2920 776f 756c 6420 6265 207a 6572 ,k) would be zer │ │ │ │ +00069820: 6f2e 2054 6869 7320 6973 2066 616c 7365 o. This is false │ │ │ │ +00069830: 2c20 6173 2073 686f 776e 2062 790a 7468 , as shown by.th │ │ │ │ +00069840: 6520 666f 6c6c 6f77 696e 6720 6578 616d e following exam │ │ │ │ +00069850: 706c 653a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ple:..+--------- │ │ │ │ +00069860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069890: 2d2d 2d2b 0a7c 6939 203a 2053 203d 205a ---+.|i9 : S = Z │ │ │ │ -000698a0: 5a2f 3130 315b 785f 302e 2e78 5f32 5d3b Z/101[x_0..x_2]; │ │ │ │ -000698b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000698c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069880: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 --------+.|i9 : │ │ │ │ +00069890: 5320 3d20 5a5a 2f31 3031 5b78 5f30 2e2e S = ZZ/101[x_0.. │ │ │ │ +000698a0: 785f 325d 3b20 2020 2020 2020 2020 2020 x_2]; │ │ │ │ +000698b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000698c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000698d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000698e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000698f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069900: 3020 3a20 6666 203d 2061 7070 6c79 2833 0 : ff = apply(3 │ │ │ │ -00069910: 2c20 692d 3e78 5f69 5e32 293b 2020 2020 , i->x_i^2); │ │ │ │ -00069920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069930: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000698f0: 2b0a 7c69 3130 203a 2066 6620 3d20 6170 +.|i10 : ff = ap │ │ │ │ +00069900: 706c 7928 332c 2069 2d3e 785f 695e 3229 ply(3, i->x_i^2) │ │ │ │ +00069910: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +00069920: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069960: 2d2d 2d2b 0a7c 6931 3120 3a20 5220 3d20 ---+.|i11 : R = │ │ │ │ -00069970: 532f 6964 6561 6c20 6666 3b20 2020 2020 S/ideal ff; │ │ │ │ -00069980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069990: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069950: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a --------+.|i11 : │ │ │ │ +00069960: 2052 203d 2053 2f69 6465 616c 2066 663b R = S/ideal ff; │ │ │ │ +00069970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069980: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000699a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000699b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000699c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -000699d0: 3220 3a20 4d20 3d20 636f 6b65 726e 656c 2 : M = cokernel │ │ │ │ -000699e0: 206d 6174 7269 7820 7b7b 785f 302c 2078 matrix {{x_0, x │ │ │ │ -000699f0: 5f31 2a78 5f32 7d7d 3b20 2020 2020 207c _1*x_2}}; | │ │ │ │ -00069a00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000699c0: 2b0a 7c69 3132 203a 204d 203d 2063 6f6b +.|i12 : M = cok │ │ │ │ +000699d0: 6572 6e65 6c20 6d61 7472 6978 207b 7b78 ernel matrix {{x │ │ │ │ +000699e0: 5f30 2c20 785f 312a 785f 327d 7d3b 2020 _0, x_1*x_2}}; │ │ │ │ +000699f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069a30: 2d2d 2d2b 0a7c 6931 3320 3a20 6220 3d20 ---+.|i13 : b = │ │ │ │ -00069a40: 353b 2020 2020 2020 2020 2020 2020 2020 5; │ │ │ │ -00069a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069a60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069a20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a --------+.|i13 : │ │ │ │ +00069a30: 2062 203d 2035 3b20 2020 2020 2020 2020 b = 5; │ │ │ │ +00069a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069a50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069aa0: 3420 3a20 4d62 203d 2070 7275 6e65 2073 4 : Mb = prune s │ │ │ │ -00069ab0: 797a 7967 794d 6f64 756c 6528 2d62 2c4d yzygyModule(-b,M │ │ │ │ -00069ac0: 293b 2020 2020 2020 2020 2020 2020 207c ); | │ │ │ │ -00069ad0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00069a90: 2b0a 7c69 3134 203a 204d 6220 3d20 7072 +.|i14 : Mb = pr │ │ │ │ +00069aa0: 756e 6520 7379 7a79 6779 4d6f 6475 6c65 une syzygyModule │ │ │ │ +00069ab0: 282d 622c 4d29 3b20 2020 2020 2020 2020 (-b,M); │ │ │ │ +00069ac0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069b00: 2d2d 2d2b 0a7c 6931 3520 3a20 4520 3d20 ---+.|i15 : E = │ │ │ │ -00069b10: 7072 756e 6520 6576 656e 4578 744d 6f64 prune evenExtMod │ │ │ │ -00069b20: 756c 6520 4d62 3b20 2020 2020 2020 2020 ule Mb; │ │ │ │ -00069b30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069af0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a --------+.|i15 : │ │ │ │ +00069b00: 2045 203d 2070 7275 6e65 2065 7665 6e45 E = prune evenE │ │ │ │ +00069b10: 7874 4d6f 6475 6c65 204d 623b 2020 2020 xtModule Mb; │ │ │ │ +00069b20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069b70: 3620 3a20 5332 6d61 7020 3d20 5332 2830 6 : S2map = S2(0 │ │ │ │ -00069b80: 2c45 293b 2020 2020 2020 2020 2020 2020 ,E); │ │ │ │ -00069b90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00069b60: 2b0a 7c69 3136 203a 2053 326d 6170 203d +.|i16 : S2map = │ │ │ │ +00069b70: 2053 3228 302c 4529 3b20 2020 2020 2020 S2(0,E); │ │ │ │ +00069b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069b90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00069ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069bd0: 2020 207c 0a7c 6f31 3620 3a20 4d61 7472 |.|o16 : Matr │ │ │ │ -00069be0: 6978 2020 2020 2020 2020 2020 2020 2020 ix │ │ │ │ -00069bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069c00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069bc0: 2020 2020 2020 2020 7c0a 7c6f 3136 203a |.|o16 : │ │ │ │ +00069bd0: 204d 6174 7269 7820 2020 2020 2020 2020 Matrix │ │ │ │ +00069be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069bf0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069c40: 3720 3a20 5345 203d 2070 7275 6e65 2074 7 : SE = prune t │ │ │ │ -00069c50: 6172 6765 7420 5332 6d61 703b 2020 2020 arget S2map; │ │ │ │ -00069c60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069c70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00069c30: 2b0a 7c69 3137 203a 2053 4520 3d20 7072 +.|i17 : SE = pr │ │ │ │ +00069c40: 756e 6520 7461 7267 6574 2053 326d 6170 une target S2map │ │ │ │ +00069c50: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +00069c60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069ca0: 2d2d 2d2b 0a7c 6931 3820 3a20 6578 7472 ---+.|i18 : extr │ │ │ │ -00069cb0: 6120 3d20 7072 756e 6520 636f 6b65 7220 a = prune coker │ │ │ │ -00069cc0: 5332 6d61 703b 2020 2020 2020 2020 2020 S2map; │ │ │ │ -00069cd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069c90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a --------+.|i18 : │ │ │ │ +00069ca0: 2065 7874 7261 203d 2070 7275 6e65 2063 extra = prune c │ │ │ │ +00069cb0: 6f6b 6572 2053 326d 6170 3b20 2020 2020 oker S2map; │ │ │ │ +00069cc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069d10: 3920 3a20 4b45 203d 2070 7275 6e65 206b 9 : KE = prune k │ │ │ │ -00069d20: 6572 2053 326d 6170 3b20 2020 2020 2020 er S2map; │ │ │ │ -00069d30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069d40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00069d00: 2b0a 7c69 3139 203a 204b 4520 3d20 7072 +.|i19 : KE = pr │ │ │ │ +00069d10: 756e 6520 6b65 7220 5332 6d61 703b 2020 une ker S2map; │ │ │ │ +00069d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069d30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069d70: 2d2d 2d2b 0a7c 6932 3020 3a20 6265 7474 ---+.|i20 : bett │ │ │ │ -00069d80: 6920 6672 6565 5265 736f 6c75 7469 6f6e i freeResolution │ │ │ │ -00069d90: 284d 622c 204c 656e 6774 684c 696d 6974 (Mb, LengthLimit │ │ │ │ -00069da0: 203d 3e20 3130 297c 0a7c 2020 2020 2020 => 10)|.| │ │ │ │ +00069d60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a --------+.|i20 : │ │ │ │ +00069d70: 2062 6574 7469 2066 7265 6552 6573 6f6c betti freeResol │ │ │ │ +00069d80: 7574 696f 6e28 4d62 2c20 4c65 6e67 7468 ution(Mb, Length │ │ │ │ +00069d90: 4c69 6d69 7420 3d3e 2031 3029 7c0a 7c20 Limit => 10)|.| │ │ │ │ +00069da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00069de0: 2020 2020 2020 2020 2020 2020 3020 2031 0 1 │ │ │ │ -00069df0: 2032 2033 2034 2035 2036 2037 2038 2020 2 3 4 5 6 7 8 │ │ │ │ -00069e00: 3920 3130 2020 2020 2020 2020 2020 207c 9 10 | │ │ │ │ -00069e10: 0a7c 6f32 3020 3d20 746f 7461 6c3a 2032 .|o20 = total: 2 │ │ │ │ -00069e20: 3020 3134 2039 2035 2032 2031 2032 2034 0 14 9 5 2 1 2 4 │ │ │ │ -00069e30: 2037 2031 3120 3136 2020 2020 2020 2020 7 11 16 │ │ │ │ -00069e40: 2020 207c 0a7c 2020 2020 2020 2020 202d |.| - │ │ │ │ -00069e50: 363a 2032 3020 3134 2039 2035 2032 202e 6: 20 14 9 5 2 . │ │ │ │ -00069e60: 202e 202e 202e 2020 2e20 202e 2020 2020 . . . . . │ │ │ │ -00069e70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00069e80: 2020 202d 353a 2020 2e20 202e 202e 202e -5: . . . . │ │ │ │ -00069e90: 202e 2031 2031 2031 2031 2020 3120 2031 . 1 1 1 1 1 1 │ │ │ │ -00069ea0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00069eb0: 2020 2020 2020 202d 343a 2020 2e20 202e -4: . . │ │ │ │ -00069ec0: 202e 202e 202e 202e 2031 2033 2036 2031 . . . . 1 3 6 1 │ │ │ │ -00069ed0: 3020 3135 2020 2020 2020 2020 2020 207c 0 15 | │ │ │ │ -00069ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00069dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00069de0: 2030 2020 3120 3220 3320 3420 3520 3620 0 1 2 3 4 5 6 │ │ │ │ +00069df0: 3720 3820 2039 2031 3020 2020 2020 2020 7 8 9 10 │ │ │ │ +00069e00: 2020 2020 7c0a 7c6f 3230 203d 2074 6f74 |.|o20 = tot │ │ │ │ +00069e10: 616c 3a20 3230 2031 3420 3920 3520 3220 al: 20 14 9 5 2 │ │ │ │ +00069e20: 3120 3220 3420 3720 3131 2031 3620 2020 1 2 4 7 11 16 │ │ │ │ +00069e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00069e40: 2020 2020 2d36 3a20 3230 2031 3420 3920 -6: 20 14 9 │ │ │ │ +00069e50: 3520 3220 2e20 2e20 2e20 2e20 202e 2020 5 2 . . . . . │ │ │ │ +00069e60: 2e20 2020 2020 2020 2020 2020 7c0a 7c20 . |.| │ │ │ │ +00069e70: 2020 2020 2020 2020 2d35 3a20 202e 2020 -5: . │ │ │ │ +00069e80: 2e20 2e20 2e20 2e20 3120 3120 3120 3120 . . . . 1 1 1 1 │ │ │ │ +00069e90: 2031 2020 3120 2020 2020 2020 2020 2020 1 1 │ │ │ │ +00069ea0: 7c0a 7c20 2020 2020 2020 2020 2d34 3a20 |.| -4: │ │ │ │ +00069eb0: 202e 2020 2e20 2e20 2e20 2e20 2e20 3120 . . . . . . 1 │ │ │ │ +00069ec0: 3320 3620 3130 2031 3520 2020 2020 2020 3 6 10 15 │ │ │ │ +00069ed0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00069ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069f10: 2020 207c 0a7c 6f32 3020 3a20 4265 7474 |.|o20 : Bett │ │ │ │ -00069f20: 6954 616c 6c79 2020 2020 2020 2020 2020 iTally │ │ │ │ -00069f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069f40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069f00: 2020 2020 2020 2020 7c0a 7c6f 3230 203a |.|o20 : │ │ │ │ +00069f10: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +00069f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069f30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00069f80: 3120 3a20 6170 706c 7920 2835 2c20 692d 1 : apply (5, i- │ │ │ │ -00069f90: 3e20 6869 6c62 6572 7446 756e 6374 696f > hilbertFunctio │ │ │ │ -00069fa0: 6e28 692c 204b 4529 2920 2020 2020 207c n(i, KE)) | │ │ │ │ -00069fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00069f70: 2b0a 7c69 3231 203a 2061 7070 6c79 2028 +.|i21 : apply ( │ │ │ │ +00069f80: 352c 2069 2d3e 2068 696c 6265 7274 4675 5, i-> hilbertFu │ │ │ │ +00069f90: 6e63 7469 6f6e 2869 2c20 4b45 2929 2020 nction(i, KE)) │ │ │ │ +00069fa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00069fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069fe0: 2020 207c 0a7c 6f32 3120 3d20 7b32 302c |.|o21 = {20, │ │ │ │ -00069ff0: 2039 2c20 322c 2030 2c20 307d 2020 2020 9, 2, 0, 0} │ │ │ │ -0006a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a010: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00069fd0: 2020 2020 2020 2020 7c0a 7c6f 3231 203d |.|o21 = │ │ │ │ +00069fe0: 207b 3230 2c20 392c 2032 2c20 302c 2030 {20, 9, 2, 0, 0 │ │ │ │ +00069ff0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0006a000: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a040: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a050: 3120 3a20 4c69 7374 2020 2020 2020 2020 1 : List │ │ │ │ +0006a040: 7c0a 7c6f 3231 203a 204c 6973 7420 2020 |.|o21 : List │ │ │ │ +0006a050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a070: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a080: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0006a070: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0006a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a0b0: 2d2d 2d2b 0a7c 6932 3220 3a20 6170 706c ---+.|i22 : appl │ │ │ │ -0006a0c0: 7920 2835 2c20 692d 3e20 6869 6c62 6572 y (5, i-> hilber │ │ │ │ -0006a0d0: 7446 756e 6374 696f 6e28 692c 2045 2929 tFunction(i, E)) │ │ │ │ -0006a0e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006a0a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a --------+.|i22 : │ │ │ │ +0006a0b0: 2061 7070 6c79 2028 352c 2069 2d3e 2068 apply (5, i-> h │ │ │ │ +0006a0c0: 696c 6265 7274 4675 6e63 7469 6f6e 2869 ilbertFunction(i │ │ │ │ +0006a0d0: 2c20 4529 2920 2020 2020 2020 7c0a 7c20 , E)) |.| │ │ │ │ +0006a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a110: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a120: 3220 3d20 7b32 302c 2039 2c20 322c 2032 2 = {20, 9, 2, 2 │ │ │ │ -0006a130: 2c20 377d 2020 2020 2020 2020 2020 2020 , 7} │ │ │ │ -0006a140: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a150: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006a110: 7c0a 7c6f 3232 203d 207b 3230 2c20 392c |.|o22 = {20, 9, │ │ │ │ +0006a120: 2032 2c20 322c 2037 7d20 2020 2020 2020 2, 2, 7} │ │ │ │ +0006a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006a140: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a180: 2020 207c 0a7c 6f32 3220 3a20 4c69 7374 |.|o22 : List │ │ │ │ +0006a170: 2020 2020 2020 2020 7c0a 7c6f 3232 203a |.|o22 : │ │ │ │ +0006a180: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 0006a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a1b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006a1a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0006a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -0006a1f0: 3320 3a20 6170 706c 7920 2835 2c20 692d 3 : apply (5, i- │ │ │ │ -0006a200: 3e20 6869 6c62 6572 7446 756e 6374 696f > hilbertFunctio │ │ │ │ -0006a210: 6e28 692c 2053 4529 2920 2020 2020 207c n(i, SE)) | │ │ │ │ -0006a220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006a1e0: 2b0a 7c69 3233 203a 2061 7070 6c79 2028 +.|i23 : apply ( │ │ │ │ +0006a1f0: 352c 2069 2d3e 2068 696c 6265 7274 4675 5, i-> hilbertFu │ │ │ │ +0006a200: 6e63 7469 6f6e 2869 2c20 5345 2929 2020 nction(i, SE)) │ │ │ │ +0006a210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006a220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a250: 2020 207c 0a7c 6f32 3320 3d20 7b31 2c20 |.|o23 = {1, │ │ │ │ -0006a260: 312c 2031 2c20 322c 2037 7d20 2020 2020 1, 1, 2, 7} │ │ │ │ -0006a270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a280: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006a240: 2020 2020 2020 2020 7c0a 7c6f 3233 203d |.|o23 = │ │ │ │ +0006a250: 207b 312c 2031 2c20 312c 2032 2c20 377d {1, 1, 1, 2, 7} │ │ │ │ +0006a260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006a270: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006a280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a2b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a2c0: 3320 3a20 4c69 7374 2020 2020 2020 2020 3 : List │ │ │ │ +0006a2b0: 7c0a 7c6f 3233 203a 204c 6973 7420 2020 |.|o23 : List │ │ │ │ +0006a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a2e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a2f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0006a2e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0006a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a320: 2d2d 2d2b 0a7c 6932 3420 3a20 6170 706c ---+.|i24 : appl │ │ │ │ -0006a330: 7920 2835 2c20 692d 3e20 6869 6c62 6572 y (5, i-> hilber │ │ │ │ -0006a340: 7446 756e 6374 696f 6e28 692c 2065 7874 tFunction(i, ext │ │ │ │ -0006a350: 7261 2929 2020 207c 0a7c 2020 2020 2020 ra)) |.| │ │ │ │ +0006a310: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a --------+.|i24 : │ │ │ │ +0006a320: 2061 7070 6c79 2028 352c 2069 2d3e 2068 apply (5, i-> h │ │ │ │ +0006a330: 696c 6265 7274 4675 6e63 7469 6f6e 2869 ilbertFunction(i │ │ │ │ +0006a340: 2c20 6578 7472 6129 2920 2020 7c0a 7c20 , extra)) |.| │ │ │ │ +0006a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a380: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a390: 3420 3d20 7b31 2c20 312c 2031 2c20 302c 4 = {1, 1, 1, 0, │ │ │ │ -0006a3a0: 2030 7d20 2020 2020 2020 2020 2020 2020 0} │ │ │ │ -0006a3b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a3c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006a380: 7c0a 7c6f 3234 203d 207b 312c 2031 2c20 |.|o24 = {1, 1, │ │ │ │ +0006a390: 312c 2030 2c20 307d 2020 2020 2020 2020 1, 0, 0} │ │ │ │ +0006a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006a3b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a3f0: 2020 207c 0a7c 6f32 3420 3a20 4c69 7374 |.|o24 : List │ │ │ │ +0006a3e0: 2020 2020 2020 2020 7c0a 7c6f 3234 203a |.|o24 : │ │ │ │ +0006a3f0: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 0006a400: 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546f 702c 202d 2d20 4265 (BGG)Top, -- Be │ │ │ │ +0006a650: 726e 7374 6569 6e2d 4765 6c27 6661 6e64 rnstein-Gel'fand │ │ │ │ +0006a660: 2d47 656c 2766 616e 6420 636f 7272 6573 -Gel'fand corres │ │ │ │ +0006a670: 706f 6e64 656e 6365 0a20 202a 202a 6e6f pondence. * *no │ │ │ │ +0006a680: 7465 2063 6f68 6f6d 6f6c 6f67 793a 2028 te cohomology: ( │ │ │ │ +0006a690: 4d61 6361 756c 6179 3244 6f63 2963 6f68 Macaulay2Doc)coh │ │ │ │ +0006a6a0: 6f6d 6f6c 6f67 792c 202d 2d20 6765 6e65 omology, -- gene │ │ │ │ +0006a6b0: 7261 6c20 636f 686f 6d6f 6c6f 6779 2066 ral cohomology f │ │ │ │ +0006a6c0: 756e 6374 6f72 0a20 202a 2048 485e 5a5a unctor. * HH^ZZ │ │ │ │ +0006a6d0: 2053 756d 4f66 5477 6973 7473 2028 6d69 SumOfTwists (mi │ │ │ │ +0006a6e0: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ +0006a6f0: 696f 6e29 0a0a 5761 7973 2074 6f20 7573 ion)..Ways to us │ │ │ │ +0006a700: 6520 5332 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d e S2:.========== │ │ │ │ +0006a710: 3d3d 3d3d 3d0a 0a20 202a 2022 5332 285a =====.. * "S2(Z │ │ │ │ +0006a720: 5a2c 4d6f 6475 6c65 2922 0a0a 466f 7220 Z,Module)"..For │ │ │ │ +0006a730: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +0006a740: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006a750: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +0006a760: 6f74 6520 5332 3a20 5332 2c20 6973 2061 ote S2: S2, is a │ │ │ │ +0006a770: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ +0006a780: 6e63 7469 6f6e 3a0a 284d 6163 6175 6c61 nction:.(Macaula │ │ │ │ +0006a790: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +0006a7a0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +0006a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a800: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0006a810: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0006a820: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0006a830: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0006a840: 6163 6175 6c61 7932 2d31 2e32 352e 3131 acaulay2-1.25.11 │ │ │ │ -0006a850: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0006a860: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -0006a870: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -0006a880: 6573 6f6c 7574 696f 6e73 2e6d 323a 3338 esolutions.m2:38 │ │ │ │ -0006a890: 3833 3a30 2e0a 1f0a 4669 6c65 3a20 436f 83:0....File: Co │ │ │ │ -0006a8a0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -0006a8b0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e 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ulay2/packages/. │ │ │ │ +0006a860: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +0006a870: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +0006a880: 6d32 3a33 3838 333a 302e 0a1f 0a46 696c m2:3883:0....Fil │ │ │ │ +0006a890: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ +0006a8a0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +0006a8b0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2053 ns.info, Node: S │ │ │ │ +0006a8c0: 6861 6d61 7368 2c20 4e65 7874 3a20 7370 hamash, Next: sp │ │ │ │ +0006a8d0: 6c69 7474 696e 6773 2c20 5072 6576 3a20 littings, Prev: │ │ │ │ +0006a8e0: 5332 2c20 5570 3a20 546f 700a 0a53 6861 S2, Up: Top..Sha │ │ │ │ +0006a8f0: 6d61 7368 202d 2d20 436f 6d70 7574 6573 mash -- Computes │ │ │ │ +0006a900: 2074 6865 2053 6861 6d61 7368 2043 6f6d the Shamash Com │ │ │ │ +0006a910: 706c 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a plex.*********** │ │ │ │ 0006a920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006a930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006a940: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -0006a950: 6765 3a20 0a20 2020 2020 2020 2046 4620 ge: . FF │ │ │ │ -0006a960: 3d20 5368 616d 6173 6828 6666 2c46 2c6c = Shamash(ff,F,l │ │ │ │ -0006a970: 656e 290a 2020 2020 2020 2020 4646 203d en). FF = │ │ │ │ -0006a980: 2053 6861 6d61 7368 2852 6261 722c 462c Shamash(Rbar,F, │ │ │ │ -0006a990: 6c65 6e29 0a20 202a 2049 6e70 7574 733a len). * Inputs: │ │ │ │ -0006a9a0: 0a20 2020 2020 202a 2066 662c 2061 202a . * ff, a * │ │ │ │ -0006a9b0: 6e6f 7465 206d 6174 7269 783a 2028 4d61 note matrix: (Ma │ │ │ │ -0006a9c0: 6361 756c 6179 3244 6f63 294d 6174 7269 caulay2Doc)Matri │ │ │ │ -0006a9d0: 782c 2c20 3120 7820 3120 4d61 7472 6978 x,, 1 x 1 Matrix │ │ │ │ -0006a9e0: 206f 7665 7220 7269 6e67 2046 2e0a 2020 over ring F.. │ │ │ │ -0006a9f0: 2020 2020 2a20 5262 6172 2c20 6120 2a6e * Rbar, a *n │ │ │ │ -0006aa00: 6f74 6520 7269 6e67 3a20 284d 6163 6175 ote ring: (Macau │ │ │ │ -0006aa10: 6c61 7932 446f 6329 5269 6e67 2c2c 2072 lay2Doc)Ring,, r │ │ │ │ -0006aa20: 696e 6720 4620 6d6f 6420 6964 6561 6c20 ing F mod ideal │ │ │ │ -0006aa30: 6666 0a20 2020 2020 202a 2046 2c20 6120 ff. * F, a │ │ │ │ -0006aa40: 2a6e 6f74 6520 636f 6d70 6c65 783a 2028 *note complex: ( │ │ │ │ -0006aa50: 436f 6d70 6c65 7865 7329 436f 6d70 6c65 Complexes)Comple │ │ │ │ -0006aa60: 782c 2c20 6465 6669 6e65 6420 6f76 6572 x,, defined over │ │ │ │ -0006aa70: 2072 696e 6720 6666 0a20 2020 2020 202a ring ff. * │ │ │ │ -0006aa80: 206c 656e 2c20 616e 202a 6e6f 7465 2069 len, an *note i │ │ │ │ -0006aa90: 6e74 6567 6572 3a20 284d 6163 6175 6c61 nteger: (Macaula │ │ │ │ -0006aaa0: 7932 446f 6329 5a5a 2c2c 200a 2020 2a20 y2Doc)ZZ,, . * │ │ │ │ -0006aab0: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -0006aac0: 2046 462c 2061 202a 6e6f 7465 2063 6f6d FF, a *note com │ │ │ │ -0006aad0: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ -0006aae0: 2943 6f6d 706c 6578 2c2c 2063 6861 696e )Complex,, chain │ │ │ │ -0006aaf0: 2063 6f6d 706c 6578 206f 7665 7220 2872 complex over (r │ │ │ │ -0006ab00: 696e 670a 2020 2020 2020 2020 4629 2f28 ing. F)/( │ │ │ │ -0006ab10: 6964 6561 6c20 6666 290a 0a44 6573 6372 ideal ff)..Descr │ │ │ │ -0006ab20: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -0006ab30: 3d3d 0a0a 4c65 7420 5220 3d20 7269 6e67 ==..Let R = ring │ │ │ │ -0006ab40: 2046 203d 2072 696e 6720 6666 2c20 616e F = ring ff, an │ │ │ │ -0006ab50: 6420 5262 6172 203d 2052 2f28 6964 6561 d Rbar = R/(idea │ │ │ │ -0006ab60: 6c20 6629 2c20 7768 6572 6520 6666 203d l f), where ff = │ │ │ │ -0006ab70: 206d 6174 7269 787b 7b66 7d7d 2069 7320 matrix{{f}} is │ │ │ │ -0006ab80: 610a 3178 3120 6d61 7472 6978 2077 686f a.1x1 matrix who │ │ │ │ -0006ab90: 7365 2065 6e74 7279 2069 7320 6120 6e6f se entry is a no │ │ │ │ -0006aba0: 6e7a 6572 6f64 6976 6973 6f72 2069 6e20 nzerodivisor in │ │ │ │ -0006abb0: 522e 2054 6865 2063 6f6d 706c 6578 2046 R. The complex F │ │ │ │ -0006abc0: 2073 686f 756c 6420 6164 6d69 7420 610a should admit a. │ │ │ │ -0006abd0: 7379 7374 656d 206f 6620 6869 6768 6572 system of higher │ │ │ │ -0006abe0: 2068 6f6d 6f74 6f70 6965 7320 666f 7220 homotopies for │ │ │ │ -0006abf0: 7468 6520 656e 7472 7920 6f66 2066 662c the entry of ff, │ │ │ │ -0006ac00: 2072 6574 7572 6e65 6420 6279 2074 6865 returned by the │ │ │ │ -0006ac10: 2063 616c 6c0a 6d61 6b65 486f 6d6f 746f call.makeHomoto │ │ │ │ -0006ac20: 7069 6573 2866 662c 4629 2e0a 0a54 6865 pies(ff,F)...The │ │ │ │ -0006ac30: 2063 6f6d 706c 6578 2046 4620 6861 7320 complex FF has │ │ │ │ -0006ac40: 7465 726d 730a 0a46 465f 7b32 2a69 7d20 terms..FF_{2*i} │ │ │ │ -0006ac50: 3d20 5262 6172 2a2a 2846 5f30 202b 2b20 = Rbar**(F_0 ++ │ │ │ │ -0006ac60: 465f 3220 2b2b 202e 2e20 2b2b 2046 5f69 F_2 ++ .. ++ F_i │ │ │ │ -0006ac70: 290a 0a46 465f 7b32 2a69 2b31 7d20 3d20 )..FF_{2*i+1} = │ │ │ │ -0006ac80: 5262 6172 2a2a 2846 5f31 202b 2b20 465f Rbar**(F_1 ++ F_ │ │ │ │ -0006ac90: 3320 2b2b 2e2e 2b2b 465f 7b32 2a69 2b31 3 ++..++F_{2*i+1 │ │ │ │ -0006aca0: 7d29 0a0a 616e 6420 6d61 7073 206d 6164 })..and maps mad │ │ │ │ -0006acb0: 6520 6672 6f6d 2074 6865 2068 6967 6865 e from the highe │ │ │ │ -0006acc0: 7220 686f 6d6f 746f 7069 6573 2e0a 0a46 r homotopies...F │ │ │ │ -0006acd0: 6f72 2074 6865 2063 6173 6520 6f66 2061 or the case of a │ │ │ │ -0006ace0: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -0006acf0: 6563 7469 6f6e 206f 6620 6869 6768 6572 ection of higher │ │ │ │ -0006ad00: 2063 6f64 696d 656e 7369 6f6e 2c20 6f72 codimension, or │ │ │ │ -0006ad10: 2074 6f20 7365 6520 7468 650a 636f 6d70 to see the.comp │ │ │ │ -0006ad20: 6f6e 656e 7473 206f 6620 7468 6520 7265 onents of the re │ │ │ │ -0006ad30: 736f 6c75 7469 6f6e 2061 7320 7375 6d6d solution as summ │ │ │ │ -0006ad40: 616e 6473 206f 6620 4646 5f6a 2c20 7573 ands of FF_j, us │ │ │ │ -0006ad50: 6520 7468 6520 726f 7574 696e 650a 4569 e the routine.Ei │ │ │ │ -0006ad60: 7365 6e62 7564 5368 616d 6173 6820 696e senbudShamash in │ │ │ │ -0006ad70: 7374 6561 642e 0a0a 2b2d 2d2d 2d2d 2d2d stead...+------- │ │ │ │ +0006a930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +0006a940: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +0006a950: 2020 4646 203d 2053 6861 6d61 7368 2866 FF = Shamash(f │ │ │ │ +0006a960: 662c 462c 6c65 6e29 0a20 2020 2020 2020 f,F,len). │ │ │ │ +0006a970: 2046 4620 3d20 5368 616d 6173 6828 5262 FF = Shamash(Rb │ │ │ │ +0006a980: 6172 2c46 2c6c 656e 290a 2020 2a20 496e ar,F,len). * In │ │ │ │ +0006a990: 7075 7473 3a0a 2020 2020 2020 2a20 6666 puts:. * ff │ │ │ │ +0006a9a0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978 , a *note matrix │ │ │ │ +0006a9b0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0006a9c0: 4d61 7472 6978 2c2c 2031 2078 2031 204d Matrix,, 1 x 1 M │ │ │ │ +0006a9d0: 6174 7269 7820 6f76 6572 2072 696e 6720 atrix over ring │ │ │ │ +0006a9e0: 462e 0a20 2020 2020 202a 2052 6261 722c F.. * Rbar, │ │ │ │ +0006a9f0: 2061 202a 6e6f 7465 2072 696e 673a 2028 a *note ring: ( │ │ │ │ +0006aa00: 4d61 6361 756c 6179 3244 6f63 2952 696e Macaulay2Doc)Rin │ │ │ │ +0006aa10: 672c 2c20 7269 6e67 2046 206d 6f64 2069 g,, ring F mod i │ │ │ │ +0006aa20: 6465 616c 2066 660a 2020 2020 2020 2a20 deal ff. * │ │ │ │ +0006aa30: 462c 2061 202a 6e6f 7465 2063 6f6d 706c F, a *note compl │ │ │ │ +0006aa40: 6578 3a20 2843 6f6d 706c 6578 6573 2943 ex: (Complexes)C │ │ │ │ +0006aa50: 6f6d 706c 6578 2c2c 2064 6566 696e 6564 omplex,, defined │ │ │ │ +0006aa60: 206f 7665 7220 7269 6e67 2066 660a 2020 over ring ff. │ │ │ │ +0006aa70: 2020 2020 2a20 6c65 6e2c 2061 6e20 2a6e * len, an *n │ │ │ │ +0006aa80: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ +0006aa90: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ +0006aaa0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +0006aab0: 2020 2020 2a20 4646 2c20 6120 2a6e 6f74 * FF, a *not │ │ │ │ +0006aac0: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ +0006aad0: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ +0006aae0: 6368 6169 6e20 636f 6d70 6c65 7820 6f76 chain complex ov │ │ │ │ +0006aaf0: 6572 2028 7269 6e67 0a20 2020 2020 2020 er (ring. │ │ │ │ +0006ab00: 2046 292f 2869 6465 616c 2066 6629 0a0a F)/(ideal ff).. │ │ │ │ +0006ab10: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +0006ab20: 3d3d 3d3d 3d3d 3d0a 0a4c 6574 2052 203d =======..Let R = │ │ │ │ +0006ab30: 2072 696e 6720 4620 3d20 7269 6e67 2066 ring F = ring f │ │ │ │ +0006ab40: 662c 2061 6e64 2052 6261 7220 3d20 522f f, and Rbar = R/ │ │ │ │ +0006ab50: 2869 6465 616c 2066 292c 2077 6865 7265 (ideal f), where │ │ │ │ +0006ab60: 2066 6620 3d20 6d61 7472 6978 7b7b 667d ff = matrix{{f} │ │ │ │ +0006ab70: 7d20 6973 2061 0a31 7831 206d 6174 7269 } is a.1x1 matri │ │ │ │ +0006ab80: 7820 7768 6f73 6520 656e 7472 7920 6973 x whose entry is │ │ │ │ +0006ab90: 2061 206e 6f6e 7a65 726f 6469 7669 736f a nonzerodiviso │ │ │ │ +0006aba0: 7220 696e 2052 2e20 5468 6520 636f 6d70 r in R. The comp │ │ │ │ +0006abb0: 6c65 7820 4620 7368 6f75 6c64 2061 646d lex F should adm │ │ │ │ +0006abc0: 6974 2061 0a73 7973 7465 6d20 6f66 2068 it a.system of h │ │ │ │ +0006abd0: 6967 6865 7220 686f 6d6f 746f 7069 6573 igher homotopies │ │ │ │ +0006abe0: 2066 6f72 2074 6865 2065 6e74 7279 206f for the entry o │ │ │ │ +0006abf0: 6620 6666 2c20 7265 7475 726e 6564 2062 f ff, returned b │ │ │ │ +0006ac00: 7920 7468 6520 6361 6c6c 0a6d 616b 6548 y the call.makeH │ │ │ │ +0006ac10: 6f6d 6f74 6f70 6965 7328 6666 2c46 292e omotopies(ff,F). │ │ │ │ +0006ac20: 0a0a 5468 6520 636f 6d70 6c65 7820 4646 ..The complex FF │ │ │ │ +0006ac30: 2068 6173 2074 6572 6d73 0a0a 4646 5f7b has terms..FF_{ │ │ │ │ +0006ac40: 322a 697d 203d 2052 6261 722a 2a28 465f 2*i} = Rbar**(F_ │ │ │ │ +0006ac50: 3020 2b2b 2046 5f32 202b 2b20 2e2e 202b 0 ++ F_2 ++ .. + │ │ │ │ +0006ac60: 2b20 465f 6929 0a0a 4646 5f7b 322a 692b + F_i)..FF_{2*i+ │ │ │ │ +0006ac70: 317d 203d 2052 6261 722a 2a28 465f 3120 1} = Rbar**(F_1 │ │ │ │ +0006ac80: 2b2b 2046 5f33 202b 2b2e 2e2b 2b46 5f7b ++ F_3 ++..++F_{ │ │ │ │ +0006ac90: 322a 692b 317d 290a 0a61 6e64 206d 6170 2*i+1})..and map │ │ │ │ +0006aca0: 7320 6d61 6465 2066 726f 6d20 7468 6520 s made from the │ │ │ │ +0006acb0: 6869 6768 6572 2068 6f6d 6f74 6f70 6965 higher homotopie │ │ │ │ +0006acc0: 732e 0a0a 466f 7220 7468 6520 6361 7365 s...For the case │ │ │ │ +0006acd0: 206f 6620 6120 636f 6d70 6c65 7465 2069 of a complete i │ │ │ │ +0006ace0: 6e74 6572 7365 6374 696f 6e20 6f66 2068 ntersection of h │ │ │ │ +0006acf0: 6967 6865 7220 636f 6469 6d65 6e73 696f igher codimensio │ │ │ │ +0006ad00: 6e2c 206f 7220 746f 2073 6565 2074 6865 n, or to see the │ │ │ │ +0006ad10: 0a63 6f6d 706f 6e65 6e74 7320 6f66 2074 .components of t │ │ │ │ +0006ad20: 6865 2072 6573 6f6c 7574 696f 6e20 6173 he resolution as │ │ │ │ +0006ad30: 2073 756d 6d61 6e64 7320 6f66 2046 465f summands of FF_ │ │ │ │ +0006ad40: 6a2c 2075 7365 2074 6865 2072 6f75 7469 j, use the routi │ │ │ │ +0006ad50: 6e65 0a45 6973 656e 6275 6453 6861 6d61 ne.EisenbudShama │ │ │ │ +0006ad60: 7368 2069 6e73 7465 6164 2e0a 0a2b 2d2d sh instead...+-- │ │ │ │ +0006ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ad80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0006adb0: 6931 203a 2053 203d 205a 5a2f 3130 315b i1 : S = ZZ/101[ │ │ │ │ -0006adc0: 782c 792c 7a5d 2020 2020 2020 2020 2020 x,y,z] │ │ │ │ -0006add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ade0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006ada0: 2d2d 2b0a 7c69 3120 3a20 5320 3d20 5a5a --+.|i1 : S = ZZ │ │ │ │ +0006adb0: 2f31 3031 5b78 2c79 2c7a 5d20 2020 2020 /101[x,y,z] │ │ │ │ +0006adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006add0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006adf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae10: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0006ae20: 203d 2053 2020 2020 2020 2020 2020 2020 = S │ │ │ │ +0006ae10: 7c0a 7c6f 3120 3d20 5320 2020 2020 2020 |.|o1 = S │ │ │ │ +0006ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006ae40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006ae50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae80: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ -0006ae90: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ +0006ae70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006ae80: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c |o1 : Polynomial │ │ │ │ +0006ae90: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 0006aea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006aeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006aec0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0006aeb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0006aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006aed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006aef0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 -------+.|i2 : R │ │ │ │ -0006af00: 203d 2053 2f69 6465 616c 2278 332c 7933 = S/ideal"x3,y3 │ │ │ │ -0006af10: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -0006af20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006af30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0006aef0: 3220 3a20 5220 3d20 532f 6964 6561 6c22 2 : R = S/ideal" │ │ │ │ +0006af00: 7833 2c79 3322 2020 2020 2020 2020 2020 x3,y3" │ │ │ │ +0006af10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006af20: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006af30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006af40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006af60: 2020 2020 207c 0a7c 6f32 203d 2052 2020 |.|o2 = R │ │ │ │ +0006af50: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0006af60: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 0006af70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006af90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006af90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006afd0: 2020 207c 0a7c 6f32 203a 2051 756f 7469 |.|o2 : Quoti │ │ │ │ -0006afe0: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ -0006aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b000: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006afc0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ +0006afd0: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +0006afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006aff0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006b000: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0006b010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b040: 2d2b 0a7c 6933 203a 204d 203d 2052 5e31 -+.|i3 : M = R^1 │ │ │ │ -0006b050: 2f69 6465 616c 2878 2c79 2c7a 2920 2020 /ideal(x,y,z) │ │ │ │ -0006b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b070: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006b030: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 4d20 ------+.|i3 : M │ │ │ │ +0006b040: 3d20 525e 312f 6964 6561 6c28 782c 792c = R^1/ideal(x,y, │ │ │ │ +0006b050: 7a29 2020 2020 2020 2020 2020 2020 2020 z) │ │ │ │ +0006b060: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b0a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006b0b0: 0a7c 6f33 203d 2063 6f6b 6572 6e65 6c20 .|o3 = cokernel │ │ │ │ -0006b0c0: 7c20 7820 7920 7a20 7c20 2020 2020 2020 | x y z | │ │ │ │ -0006b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b0e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006b0a0: 2020 2020 7c0a 7c6f 3320 3d20 636f 6b65 |.|o3 = coke │ │ │ │ +0006b0b0: 726e 656c 207c 2078 2079 207a 207c 2020 rnel | x y z | │ │ │ │ +0006b0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b0d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006b0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b110: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b110: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b130: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -0006b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b150: 2020 2020 7c0a 7c6f 3320 3a20 522d 6d6f |.|o3 : R-mo │ │ │ │ -0006b160: 6475 6c65 2c20 7175 6f74 6965 6e74 206f dule, quotient o │ │ │ │ -0006b170: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ -0006b180: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0006b130: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0006b140: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ +0006b150: 2052 2d6d 6f64 756c 652c 2071 756f 7469 R-module, quoti │ │ │ │ +0006b160: 656e 7420 6f66 2052 2020 2020 2020 2020 ent of R │ │ │ │ +0006b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b180: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0006b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b1c0: 2d2d 2b0a 7c69 3420 3a20 4620 3d20 6672 --+.|i4 : F = fr │ │ │ │ -0006b1d0: 6565 5265 736f 6c75 7469 6f6e 284d 2c20 eeResolution(M, │ │ │ │ -0006b1e0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 2034 LengthLimit => 4 │ │ │ │ -0006b1f0: 2920 2020 2020 2020 207c 0a7c 2020 2020 ) |.| │ │ │ │ +0006b1b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2046 -------+.|i4 : F │ │ │ │ +0006b1c0: 203d 2066 7265 6552 6573 6f6c 7574 696f = freeResolutio │ │ │ │ +0006b1d0: 6e28 4d2c 204c 656e 6774 684c 696d 6974 n(M, LengthLimit │ │ │ │ +0006b1e0: 203d 3e20 3429 2020 2020 2020 2020 7c0a => 4) |. │ │ │ │ +0006b1f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b230: 7c0a 7c20 2020 2020 2031 2020 2020 2020 |.| 1 │ │ │ │ -0006b240: 3320 2020 2020 2035 2020 2020 2020 3720 3 5 7 │ │ │ │ -0006b250: 2020 2020 2039 2020 2020 2020 2020 2020 9 │ │ │ │ -0006b260: 2020 2020 2020 207c 0a7c 6f34 203d 2052 |.|o4 = R │ │ │ │ -0006b270: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020 <-- R <-- R │ │ │ │ -0006b280: 3c2d 2d20 5220 203c 2d2d 2052 2020 2020 <-- R <-- R │ │ │ │ -0006b290: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006b2a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006b220: 2020 2020 207c 0a7c 2020 2020 2020 3120 |.| 1 │ │ │ │ +0006b230: 2020 2020 2033 2020 2020 2020 3520 2020 3 5 │ │ │ │ +0006b240: 2020 2037 2020 2020 2020 3920 2020 2020 7 9 │ │ │ │ +0006b250: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0006b260: 3420 3d20 5220 203c 2d2d 2052 2020 3c2d 4 = R <-- R <- │ │ │ │ +0006b270: 2d20 5220 203c 2d2d 2052 2020 3c2d 2d20 - R <-- R <-- │ │ │ │ +0006b280: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0006b290: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006b2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b2d0: 2020 2020 207c 0a7c 2020 2020 2030 2020 |.| 0 │ │ │ │ -0006b2e0: 2020 2020 3120 2020 2020 2032 2020 2020 1 2 │ │ │ │ -0006b2f0: 2020 3320 2020 2020 2034 2020 2020 2020 3 4 │ │ │ │ -0006b300: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006b2c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006b2d0: 2020 3020 2020 2020 2031 2020 2020 2020 0 1 │ │ │ │ +0006b2e0: 3220 2020 2020 2033 2020 2020 2020 3420 2 3 4 │ │ │ │ +0006b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b300: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b340: 2020 207c 0a7c 6f34 203a 2043 6f6d 706c |.|o4 : Compl │ │ │ │ -0006b350: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ -0006b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b370: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006b330: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ +0006b340: 436f 6d70 6c65 7820 2020 2020 2020 2020 Complex │ │ │ │ +0006b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b360: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006b370: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0006b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b3b0: 2d2b 0a7c 6935 203a 2066 6620 3d20 6d61 -+.|i5 : ff = ma │ │ │ │ -0006b3c0: 7472 6978 7b7b 7a5e 337d 7d20 2020 2020 trix{{z^3}} │ │ │ │ -0006b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006b3a0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6666 ------+.|i5 : ff │ │ │ │ +0006b3b0: 203d 206d 6174 7269 787b 7b7a 5e33 7d7d = matrix{{z^3}} │ │ │ │ +0006b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b3d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b410: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006b420: 0a7c 6f35 203d 207c 207a 3320 7c20 2020 .|o5 = | z3 | │ │ │ │ +0006b410: 2020 2020 7c0a 7c6f 3520 3d20 7c20 7a33 |.|o5 = | z3 │ │ │ │ +0006b420: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b450: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006b440: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006b450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b480: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0006b490: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -0006b4a0: 2020 2020 3120 2020 2020 2020 2020 2020 1 │ │ │ │ -0006b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b4c0: 2020 2020 7c0a 7c6f 3520 3a20 4d61 7472 |.|o5 : Matr │ │ │ │ -0006b4d0: 6978 2052 2020 3c2d 2d20 5220 2020 2020 ix R <-- R │ │ │ │ +0006b480: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006b490: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ +0006b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b4b0: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ +0006b4c0: 204d 6174 7269 7820 5220 203c 2d2d 2052 Matrix R <-- R │ │ │ │ +0006b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b4f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0006b4f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0006b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b530: 2d2d 2b0a 7c69 3620 3a20 5231 203d 2052 --+.|i6 : R1 = R │ │ │ │ -0006b540: 2f69 6465 616c 2066 6620 2020 2020 2020 /ideal ff │ │ │ │ -0006b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b560: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006b520: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2052 -------+.|i6 : R │ │ │ │ +0006b530: 3120 3d20 522f 6964 6561 6c20 6666 2020 1 = R/ideal ff │ │ │ │ +0006b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b550: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006b560: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b5a0: 7c0a 7c6f 3620 3d20 5231 2020 2020 2020 |.|o6 = R1 │ │ │ │ +0006b590: 2020 2020 207c 0a7c 6f36 203d 2052 3120 |.|o6 = R1 │ │ │ │ +0006b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b5d0: 2020 2020 2020 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---------------- │ │ │ │ 0006b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0006b680: 3720 3a20 6265 7474 6920 4620 2020 2020 7 : betti F │ │ │ │ +0006b670: 2d2b 0a7c 6937 203a 2062 6574 7469 2046 -+.|i7 : betti F │ │ │ │ +0006b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006b6a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006b6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0006b6f0: 2020 2020 2020 2020 2030 2031 2032 2033 0 1 2 3 │ │ │ │ -0006b700: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0006b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b720: 207c 0a7c 6f37 203d 2074 6f74 616c 3a20 |.|o7 = total: │ │ │ │ -0006b730: 3120 3320 3520 3720 3920 2020 2020 2020 1 3 5 7 9 │ │ │ │ -0006b740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b750: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006b760: 2020 2020 303a 2031 2033 2033 2031 202e 0: 1 3 3 1 . │ │ │ │ +0006b6d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006b6e0: 0a7c 2020 2020 2020 2020 2020 2020 3020 .| 0 │ │ │ │ +0006b6f0: 3120 3220 3320 3420 2020 2020 2020 2020 1 2 3 4 │ │ │ │ +0006b700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b710: 2020 2020 2020 7c0a 7c6f 3720 3d20 746f |.|o7 = to │ │ │ │ +0006b720: 7461 6c3a 2031 2033 2035 2037 2039 2020 tal: 1 3 5 7 9 │ │ │ │ +0006b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b740: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b750: 2020 2020 2020 2020 2030 3a20 3120 3320 0: 1 3 │ │ │ │ +0006b760: 3320 3120 2e20 2020 2020 2020 2020 2020 3 1 . │ │ │ │ 0006b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b780: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006b790: 0a7c 2020 2020 2020 2020 2031 3a20 2e20 .| 1: . │ │ │ │ -0006b7a0: 2e20 3220 3620 3620 2020 2020 2020 2020 . 2 6 6 │ │ │ │ -0006b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b7c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0006b7d0: 2020 323a 202e 202e 202e 202e 2033 2020 2: . . . . 3 │ │ │ │ +0006b780: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006b790: 313a 202e 202e 2032 2036 2036 2020 2020 1: . . 2 6 6 │ │ │ │ +0006b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b7b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006b7c0: 2020 2020 2020 2032 3a20 2e20 2e20 2e20 2: . . . │ │ │ │ +0006b7d0: 2e20 3320 2020 2020 2020 2020 2020 2020 . 3 │ │ │ │ 0006b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b7f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b830: 2020 2020 7c0a 7c6f 3720 3a20 4265 7474 |.|o7 : Bett │ │ │ │ -0006b840: 6954 616c 6c79 2020 2020 2020 2020 2020 iTally │ │ │ │ +0006b820: 2020 2020 2020 2020 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ +0006b830: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0006b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b860: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0006b860: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0006b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b8a0: 2d2d 2b0a 7c69 3820 3a20 4646 203d 2053 --+.|i8 : FF = S │ │ │ │ -0006b8b0: 6861 6d61 7368 2866 662c 462c 3429 2020 hamash(ff,F,4) │ │ │ │ -0006b8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b8d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006b890: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2046 -------+.|i8 : F │ │ │ │ +0006b8a0: 4620 3d20 5368 616d 6173 6828 6666 2c46 F = Shamash(ff,F │ │ │ │ +0006b8b0: 2c34 2920 2020 2020 2020 2020 2020 2020 ,4) │ │ │ │ +0006b8c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006b8d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b910: 7c0a 7c20 2020 2020 2f20 525c 3120 2020 |.| / R\1 │ │ │ │ -0006b920: 2020 2f20 525c 3320 2020 2020 2f20 525c / R\3 / R\ │ │ │ │ -0006b930: 3620 2020 2020 2f20 525c 3130 2020 2020 6 / R\10 │ │ │ │ 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2020 2f20 525c 3135 7c0a 7c6f 0 / R\15|.|o │ │ │ │ +0006b940: 3820 3d20 7c2d 2d7c 2020 3c2d 2d20 7c2d 8 = |--| <-- |- │ │ │ │ +0006b950: 2d7c 2020 3c2d 2d20 7c2d 2d7c 2020 3c2d -| <-- |--| <- │ │ │ │ +0006b960: 2d20 7c2d 2d7c 2020 203c 2d2d 207c 2d2d - |--| <-- |-- │ │ │ │ +0006b970: 7c20 207c 0a7c 2020 2020 207c 2033 7c20 | |.| | 3| │ │ │ │ +0006b980: 2020 2020 207c 2033 7c20 2020 2020 207c | 3| | │ │ │ │ +0006b990: 2033 7c20 2020 2020 207c 2033 7c20 2020 3| | 3| │ │ │ │ +0006b9a0: 2020 2020 7c20 337c 2020 7c0a 7c20 2020 | 3| |.| │ │ │ │ +0006b9b0: 2020 5c7a 202f 2020 2020 2020 5c7a 202f \z / \z / │ │ │ │ +0006b9c0: 2020 2020 2020 5c7a 202f 2020 2020 2020 \z / │ │ │ │ +0006b9d0: 5c7a 202f 2020 2020 2020 205c 7a20 2f20 \z / \z / │ │ │ │ +0006b9e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba20: 2020 207c 0a7c 2020 2020 2030 2020 2020 |.| 0 │ │ │ │ -0006ba30: 2020 2020 2031 2020 2020 2020 2020 2032 1 2 │ │ │ │ -0006ba40: 2020 2020 2020 2020 2033 2020 2020 2020 3 │ │ │ │ -0006ba50: 2020 2020 3420 2020 2020 7c0a 7c20 2020 4 |.| │ │ │ │ +0006ba10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006ba20: 3020 2020 2020 2020 2020 3120 2020 2020 0 1 │ │ │ │ +0006ba30: 2020 2020 3220 2020 2020 2020 2020 3320 2 3 │ │ │ │ +0006ba40: 2020 2020 2020 2020 2034 2020 2020 207c 4 | │ │ │ │ +0006ba50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0006ba60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ba70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba90: 207c 0a7c 6f38 203a 2043 6f6d 706c 6578 |.|o8 : Complex │ │ │ │ +0006ba80: 2020 2020 2020 7c0a 7c6f 3820 3a20 436f |.|o8 : Co │ │ │ │ +0006ba90: 6d70 6c65 7820 2020 2020 2020 2020 2020 mplex │ │ │ │ 0006baa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bac0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0006bab0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0006bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006baf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0006bb00: 0a7c 6939 203a 2047 4720 3d20 5368 616d .|i9 : GG = Sham │ │ │ │ -0006bb10: 6173 6828 5231 2c46 2c34 2920 2020 2020 ash(R1,F,4) │ │ │ │ -0006bb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bb30: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006baf0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4747 203d ----+.|i9 : GG = │ │ │ │ +0006bb00: 2053 6861 6d61 7368 2852 312c 462c 3429 Shamash(R1,F,4) │ │ │ │ +0006bb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bb20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006bb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bb60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0006bb70: 2020 2020 2020 2031 2020 2020 2020 2033 1 3 │ │ │ │ -0006bb80: 2020 2020 2020 2036 2020 2020 2020 2031 6 1 │ │ │ │ -0006bb90: 3020 2020 2020 2020 3135 2020 2020 2020 0 15 │ │ │ │ -0006bba0: 2020 2020 7c0a 7c6f 3920 3d20 5231 2020 |.|o9 = R1 │ │ │ │ -0006bbb0: 3c2d 2d20 5231 2020 3c2d 2d20 5231 2020 <-- R1 <-- R1 │ │ │ │ -0006bbc0: 3c2d 2d20 5231 2020 203c 2d2d 2052 3120 <-- R1 <-- R1 │ │ │ │ -0006bbd0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006bb60: 2020 7c0a 7c20 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ +0006bb70: 2020 2020 3320 2020 2020 2020 3620 2020 3 6 │ │ │ │ +0006bb80: 2020 2020 3130 2020 2020 2020 2031 3520 10 15 │ │ │ │ +0006bb90: 2020 2020 2020 2020 207c 0a7c 6f39 203d |.|o9 = │ │ │ │ +0006bba0: 2052 3120 203c 2d2d 2052 3120 203c 2d2d R1 <-- R1 <-- │ │ │ │ +0006bbb0: 2052 3120 203c 2d2d 2052 3120 2020 3c2d R1 <-- R1 <- │ │ │ │ +0006bbc0: 2d20 5231 2020 2020 2020 2020 2020 2020 - R1 │ │ │ │ +0006bbd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc10: 2020 7c0a 7c20 2020 2020 3020 2020 2020 |.| 0 │ │ │ │ -0006bc20: 2020 3120 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -0006bc30: 2020 3320 2020 2020 2020 2034 2020 2020 3 4 │ │ │ │ -0006bc40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006bc00: 2020 2020 2020 207c 0a7c 2020 2020 2030 |.| 0 │ │ │ │ +0006bc10: 2020 2020 2020 2031 2020 2020 2020 2032 1 2 │ │ │ │ +0006bc20: 2020 2020 2020 2033 2020 2020 2020 2020 3 │ │ │ │ +0006bc30: 3420 2020 2020 2020 2020 2020 2020 7c0a 4 |. │ │ │ │ +0006bc40: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006bc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc80: 7c0a 7c6f 3920 3a20 436f 6d70 6c65 7820 |.|o9 : Complex │ │ │ │ +0006bc70: 2020 2020 207c 0a7c 6f39 203a 2043 6f6d |.|o9 : Com │ │ │ │ +0006bc80: 706c 6578 2020 2020 2020 2020 2020 2020 plex │ │ │ │ 0006bc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bcb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006bca0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0006bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0006bcf0: 7c69 3130 203a 2062 6574 7469 2046 4620 |i10 : betti FF │ │ │ │ +0006bce0: 2d2d 2d2b 0a7c 6931 3020 3a20 6265 7474 ---+.|i10 : bett │ │ │ │ +0006bcf0: 6920 4646 2020 2020 2020 2020 2020 2020 i FF │ │ │ │ 0006bd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006bd10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006bd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0006bd60: 2020 2020 2020 2020 2020 2020 3020 3120 0 1 │ │ │ │ -0006bd70: 3220 2033 2020 3420 2020 2020 2020 2020 2 3 4 │ │ │ │ -0006bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd90: 2020 207c 0a7c 6f31 3020 3d20 746f 7461 |.|o10 = tota │ │ │ │ -0006bda0: 6c3a 2031 2033 2036 2031 3020 3135 2020 l: 1 3 6 10 15 │ │ │ │ -0006bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bdc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0006bdd0: 2020 2020 2020 2030 3a20 3120 3320 3320 0: 1 3 3 │ │ │ │ -0006bde0: 2031 2020 2e20 2020 2020 2020 2020 2020 1 . │ │ │ │ -0006bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006be00: 207c 0a7c 2020 2020 2020 2020 2020 313a |.| 1: │ │ │ │ -0006be10: 202e 202e 2033 2020 3920 2039 2020 2020 . . 3 9 9 │ │ │ │ -0006be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006be30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006be40: 2020 2020 2032 3a20 2e20 2e20 2e20 202e 2: . . . . │ │ │ │ -0006be50: 2020 3620 2020 2020 2020 2020 2020 2020 6 │ │ │ │ -0006be60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006be70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006bd50: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006bd60: 2030 2031 2032 2020 3320 2034 2020 2020 0 1 2 3 4 │ │ │ │ +0006bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bd80: 2020 2020 2020 2020 7c0a 7c6f 3130 203d |.|o10 = │ │ │ │ +0006bd90: 2074 6f74 616c 3a20 3120 3320 3620 3130 total: 1 3 6 10 │ │ │ │ +0006bda0: 2031 3520 2020 2020 2020 2020 2020 2020 15 │ │ │ │ +0006bdb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006bdc0: 0a7c 2020 2020 2020 2020 2020 303a 2031 .| 0: 1 │ │ │ │ +0006bdd0: 2033 2033 2020 3120 202e 2020 2020 2020 3 3 1 . │ │ │ │ +0006bde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bdf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006be00: 2020 2031 3a20 2e20 2e20 3320 2039 2020 1: . . 3 9 │ │ │ │ +0006be10: 3920 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ +0006be20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006be30: 2020 2020 2020 2020 2020 323a 202e 202e 2: . . │ │ │ │ +0006be40: 202e 2020 2e20 2036 2020 2020 2020 2020 . . 6 │ │ │ │ +0006be50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006be60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bea0: 2020 2020 2020 7c0a 7c6f 3130 203a 2042 |.|o10 : B │ │ │ │ -0006beb0: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +0006be90: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0006bea0: 3020 3a20 4265 7474 6954 616c 6c79 2020 0 : BettiTally │ │ │ │ +0006beb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bed0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0006bed0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0006bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006bf10: 2d2d 2d2d 2b0a 7c69 3131 203a 2062 6574 ----+.|i11 : bet │ │ │ │ -0006bf20: 7469 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│ │ │ │ -0006bfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bff0: 7c0a 7c20 2020 2020 2020 2020 2030 3a20 |.| 0: │ │ │ │ -0006c000: 3120 3320 3320 2031 2020 2e20 2020 2020 1 3 3 1 . │ │ │ │ -0006c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c020: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0006c030: 2020 2020 313a 202e 202e 2033 2020 3920 1: . . 3 9 │ │ │ │ -0006c040: 2039 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ -0006c050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006c060: 7c20 2020 2020 2020 2020 2032 3a20 2e20 | 2: . │ │ │ │ -0006c070: 2e20 2e20 202e 2020 3620 2020 2020 2020 . . . 6 │ │ │ │ -0006c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c090: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006bf70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006bf80: 2020 2020 2020 2030 2031 2032 2020 3320 0 1 2 3 │ │ │ │ +0006bf90: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +0006bfa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006bfb0: 7c6f 3131 203d 2074 6f74 616c 3a20 3120 |o11 = total: 1 │ │ │ │ +0006bfc0: 3320 3620 3130 2031 3520 2020 2020 2020 3 6 10 15 │ │ │ │ +0006bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bfe0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006bff0: 2020 303a 2031 2033 2033 2020 3120 202e 0: 1 3 3 1 . │ │ │ │ +0006c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c010: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006c020: 2020 2020 2020 2020 2031 3a20 2e20 2e20 1: . . │ │ │ │ +0006c030: 3320 2039 2020 3920 2020 2020 2020 2020 3 9 9 │ │ │ │ +0006c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c050: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006c060: 323a 202e 202e 202e 2020 2e20 2036 2020 2: . . . . 6 │ │ │ │ +0006c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c080: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0006c0d0: 3131 203a 2042 6574 7469 5461 6c6c 7920 11 : BettiTally │ │ │ │ +0006c0c0: 207c 0a7c 6f31 3120 3a20 4265 7474 6954 |.|o11 : BettiT │ │ │ │ +0006c0d0: 616c 6c79 2020 2020 2020 2020 2020 2020 ally │ │ │ │ 0006c0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c100: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0006c0f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0006c100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c130: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 ----------+.|i12 │ │ │ │ -0006c140: 203a 2072 696e 6720 4747 2020 2020 2020 : ring GG │ │ │ │ +0006c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0006c130: 0a7c 6931 3220 3a20 7269 6e67 2047 4720 .|i12 : ring GG │ │ │ │ +0006c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c170: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006c160: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006c170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c1a0: 2020 2020 2020 2020 7c0a 7c6f 3132 203d |.|o12 = │ │ │ │ -0006c1b0: 2052 3120 2020 2020 2020 2020 2020 2020 R1 │ │ │ │ +0006c190: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006c1a0: 6f31 3220 3d20 5231 2020 2020 2020 2020 o12 = R1 │ │ │ │ +0006c1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006c1c0: 2020 2020 2020 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2020 2020 2020 2020 2020 │ │ │ │ -0006c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006c390: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006c380: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0006c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0006c3d0: 0a43 6176 6561 740a 3d3d 3d3d 3d3d 0a0a .Caveat.======.. │ │ │ │ -0006c3e0: 4620 6973 2061 7373 756d 6564 2074 6f20 F is assumed to │ │ │ │ -0006c3f0: 6265 2061 2068 6f6d 6f6c 6f67 6963 616c be a homological │ │ │ │ -0006c400: 2063 6f6d 706c 6578 2073 7461 7274 696e complex startin │ │ │ │ -0006c410: 6720 6672 6f6d 2046 5f30 2e20 5468 6520 g from F_0. The │ │ │ │ -0006c420: 6d61 7472 6978 2066 6620 6d75 7374 0a62 matrix ff must.b │ │ │ │ -0006c430: 6520 3178 312e 0a0a 5365 6520 616c 736f e 1x1...See also │ │ │ │ -0006c440: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ -0006c450: 6e6f 7465 2045 6973 656e 6275 6453 6861 note EisenbudSha │ │ │ │ -0006c460: 6d61 7368 3a20 4569 7365 6e62 7564 5368 mash: EisenbudSh │ │ │ │ -0006c470: 616d 6173 682c 202d 2d20 436f 6d70 7574 amash, -- Comput │ │ │ │ -0006c480: 6573 2074 6865 2045 6973 656e 6275 642d es the Eisenbud- │ │ │ │ -0006c490: 5368 616d 6173 680a 2020 2020 436f 6d70 Shamash. Comp │ │ │ │ -0006c4a0: 6c65 780a 2020 2a20 2a6e 6f74 6520 6d61 lex. * *note ma │ │ │ │ -0006c4b0: 6b65 486f 6d6f 746f 7069 6573 3a20 6d61 keHomotopies: ma │ │ │ │ -0006c4c0: 6b65 486f 6d6f 746f 7069 6573 2c20 2d2d keHomotopies, -- │ │ │ │ -0006c4d0: 2072 6574 7572 6e73 2061 2073 7973 7465 returns a syste │ │ │ │ -0006c4e0: 6d20 6f66 2068 6967 6865 720a 2020 2020 m of higher. │ │ │ │ -0006c4f0: 686f 6d6f 746f 7069 6573 0a0a 5761 7973 homotopies..Ways │ │ │ │ -0006c500: 2074 6f20 7573 6520 5368 616d 6173 683a to use Shamash: │ │ │ │ -0006c510: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0006c520: 3d3d 3d3d 3d0a 0a20 202a 2022 5368 616d =====.. * "Sham │ │ │ │ -0006c530: 6173 6828 4d61 7472 6978 2c43 6f6d 706c ash(Matrix,Compl │ │ │ │ -0006c540: 6578 2c5a 5a29 220a 2020 2a20 2253 6861 ex,ZZ)". * "Sha │ │ │ │ -0006c550: 6d61 7368 2852 696e 672c 436f 6d70 6c65 mash(Ring,Comple │ │ │ │ -0006c560: 782c 5a5a 2922 0a0a 466f 7220 7468 6520 x,ZZ)"..For the │ │ │ │ -0006c570: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0006c580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0006c590: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0006c5a0: 5368 616d 6173 683a 2053 6861 6d61 7368 Shamash: Shamash │ │ │ │ -0006c5b0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -0006c5c0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ -0006c5d0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -0006c5e0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +0006c3c0: 2d2d 2d2b 0a0a 4361 7665 6174 0a3d 3d3d ---+..Caveat.=== │ │ │ │ +0006c3d0: 3d3d 3d0a 0a46 2069 7320 6173 7375 6d65 ===..F is assume │ │ │ │ +0006c3e0: 6420 746f 2062 6520 6120 686f 6d6f 6c6f d to be a homolo │ │ │ │ +0006c3f0: 6769 6361 6c20 636f 6d70 6c65 7820 7374 gical complex st │ │ │ │ +0006c400: 6172 7469 6e67 2066 726f 6d20 465f 302e arting from F_0. │ │ │ │ +0006c410: 2054 6865 206d 6174 7269 7820 6666 206d The matrix ff m │ │ │ │ +0006c420: 7573 740a 6265 2031 7831 2e0a 0a53 6565 ust.be 1x1...See │ │ │ │ +0006c430: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ +0006c440: 2020 2a20 2a6e 6f74 6520 4569 7365 6e62 * *note Eisenb │ │ │ │ +0006c450: 7564 5368 616d 6173 683a 2045 6973 656e udShamash: Eisen │ │ │ │ +0006c460: 6275 6453 6861 6d61 7368 2c20 2d2d 2043 budShamash, -- C │ │ │ │ +0006c470: 6f6d 7075 7465 7320 7468 6520 4569 7365 omputes the Eise │ │ │ │ +0006c480: 6e62 7564 2d53 6861 6d61 7368 0a20 2020 nbud-Shamash. │ │ │ │ +0006c490: 2043 6f6d 706c 6578 0a20 202a 202a 6e6f Complex. * *no │ │ │ │ +0006c4a0: 7465 206d 616b 6548 6f6d 6f74 6f70 6965 te makeHomotopie │ │ │ │ +0006c4b0: 733a 206d 616b 6548 6f6d 6f74 6f70 6965 s: makeHomotopie │ │ │ │ +0006c4c0: 732c 202d 2d20 7265 7475 726e 7320 6120 s, -- returns a │ │ │ │ +0006c4d0: 7379 7374 656d 206f 6620 6869 6768 6572 system of higher │ │ │ │ +0006c4e0: 0a20 2020 2068 6f6d 6f74 6f70 6965 730a . homotopies. │ │ │ │ +0006c4f0: 0a57 6179 7320 746f 2075 7365 2053 6861 .Ways to use Sha │ │ │ │ +0006c500: 6d61 7368 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d mash:.========== │ │ │ │ +0006c510: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0006c520: 2253 6861 6d61 7368 284d 6174 7269 782c "Shamash(Matrix, │ │ │ │ +0006c530: 436f 6d70 6c65 782c 5a5a 2922 0a20 202a Complex,ZZ)". * │ │ │ │ +0006c540: 2022 5368 616d 6173 6828 5269 6e67 2c43 "Shamash(Ring,C │ │ │ │ +0006c550: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72 omplex,ZZ)"..For │ │ │ │ +0006c560: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0006c570: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006c580: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0006c590: 6e6f 7465 2053 6861 6d61 7368 3a20 5368 note Shamash: Sh │ │ │ │ +0006c5a0: 616d 6173 682c 2069 7320 6120 2a6e 6f74 amash, is a *not │ │ │ │ +0006c5b0: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +0006c5c0: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +0006c5d0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +0006c5e0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0006c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0006c640: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0006c650: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0006c660: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -0006c670: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -0006c680: 6179 322d 312e 3235 2e31 312b 6473 2f4d ay2-1.25.11+ds/M │ │ │ │ -0006c690: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -0006c6a0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -0006c6b0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -0006c6c0: 7469 6f6e 732e 6d32 3a34 3736 303a 302e tions.m2:4760:0. │ │ │ │ -0006c6d0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -0006c6e0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -0006c6f0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -0006c700: 6f64 653a 2073 706c 6974 7469 6e67 732c ode: splittings, │ │ │ │ -0006c710: 204e 6578 743a 2073 7461 626c 6548 6f6d Next: stableHom │ │ │ │ -0006c720: 2c20 5072 6576 3a20 5368 616d 6173 682c , Prev: Shamash, │ │ │ │ -0006c730: 2055 703a 2054 6f70 0a0a 7370 6c69 7474 Up: Top..splitt │ │ │ │ -0006c740: 696e 6773 202d 2d20 636f 6d70 7574 6520 ings -- compute │ │ │ │ -0006c750: 7468 6520 7370 6c69 7474 696e 6773 206f the splittings o │ │ │ │ -0006c760: 6620 6120 7370 6c69 7420 7269 6768 7420 f a split right │ │ │ │ -0006c770: 6578 6163 7420 7365 7175 656e 6365 0a2a exact sequence.* │ │ │ │ +0006c630: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0006c640: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0006c650: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0006c660: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0006c670: 6163 6175 6c61 7932 2d31 2e32 352e 3131 acaulay2-1.25.11 │ │ │ │ +0006c680: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0006c690: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +0006c6a0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0006c6b0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3437 esolutions.m2:47 │ │ │ │ +0006c6c0: 3630 3a30 2e0a 1f0a 4669 6c65 3a20 436f 60:0....File: Co │ │ │ │ +0006c6d0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +0006c6e0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +0006c6f0: 666f 2c20 4e6f 6465 3a20 7370 6c69 7474 fo, Node: splitt │ │ │ │ +0006c700: 696e 6773 2c20 4e65 7874 3a20 7374 6162 ings, Next: stab │ │ │ │ +0006c710: 6c65 486f 6d2c 2050 7265 763a 2053 6861 leHom, Prev: Sha │ │ │ │ +0006c720: 6d61 7368 2c20 5570 3a20 546f 700a 0a73 mash, Up: Top..s │ │ │ │ +0006c730: 706c 6974 7469 6e67 7320 2d2d 2063 6f6d plittings -- com │ │ │ │ +0006c740: 7075 7465 2074 6865 2073 706c 6974 7469 pute the splitti │ │ │ │ +0006c750: 6e67 7320 6f66 2061 2073 706c 6974 2072 ngs of a split r │ │ │ │ +0006c760: 6967 6874 2065 7861 6374 2073 6571 7565 ight exact seque │ │ │ │ +0006c770: 6e63 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a nce.************ │ │ │ │ 0006c780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006c790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006c7a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006c7b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006c7c0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -0006c7d0: 0a20 2020 2020 2020 2078 203d 2073 706c . x = spl │ │ │ │ -0006c7e0: 6974 7469 6e67 7328 612c 6229 0a20 202a ittings(a,b). * │ │ │ │ -0006c7f0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -0006c800: 2061 2c20 6120 2a6e 6f74 6520 6d61 7472 a, a *note matr │ │ │ │ -0006c810: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -0006c820: 6329 4d61 7472 6978 2c2c 206d 6170 7320 c)Matrix,, maps │ │ │ │ -0006c830: 696e 746f 2074 6865 206b 6572 6e65 6c20 into the kernel │ │ │ │ -0006c840: 6f66 2062 0a20 2020 2020 202a 2062 2c20 of b. * b, │ │ │ │ -0006c850: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ -0006c860: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ -0006c870: 7472 6978 2c2c 2072 6570 7265 7365 6e74 trix,, represent │ │ │ │ -0006c880: 696e 6720 6120 7375 726a 6563 7469 6f6e ing a surjection │ │ │ │ -0006c890: 0a20 2020 2020 2020 2066 726f 6d20 7461 . from ta │ │ │ │ -0006c8a0: 7267 6574 2061 2074 6f20 6120 6672 6565 rget a to a free │ │ │ │ -0006c8b0: 206d 6f64 756c 650a 2020 2a20 4f75 7470 module. * Outp │ │ │ │ -0006c8c0: 7574 733a 0a20 2020 2020 202a 204c 2c20 uts:. * L, │ │ │ │ -0006c8d0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ -0006c8e0: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ -0006c8f0: 2c2c 204c 203d 205c 7b73 6967 6d61 2c74 ,, L = \{sigma,t │ │ │ │ -0006c900: 6175 5c7d 2c20 7370 6c69 7474 696e 6773 au\}, splittings │ │ │ │ -0006c910: 206f 660a 2020 2020 2020 2020 612c 6220 of. a,b │ │ │ │ -0006c920: 7265 7370 6563 7469 7665 6c79 0a0a 4465 respectively..De │ │ │ │ -0006c930: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -0006c940: 3d3d 3d3d 3d0a 0a41 7373 756d 696e 6720 =====..Assuming │ │ │ │ -0006c950: 7468 6174 2028 612c 6229 2061 7265 2074 that (a,b) are t │ │ │ │ -0006c960: 6865 206d 6170 7320 6f66 2061 2072 6967 he maps of a rig │ │ │ │ -0006c970: 6874 2065 7861 6374 2073 6571 7565 6e63 ht exact sequenc │ │ │ │ -0006c980: 650a 0a24 305c 746f 2041 5c74 6f20 425c e..$0\to A\to B\ │ │ │ │ -0006c990: 746f 2043 205c 746f 2030 240a 0a77 6974 to C \to 0$..wit │ │ │ │ -0006c9a0: 6820 422c 2043 2066 7265 652c 2074 6865 h B, C free, the │ │ │ │ -0006c9b0: 2073 6372 6970 7420 7072 6f64 7563 6573 script produces │ │ │ │ -0006c9c0: 2061 2070 6169 7220 6f66 206d 6170 7320 a pair of maps │ │ │ │ -0006c9d0: 7369 676d 612c 2074 6175 2077 6974 6820 sigma, tau with │ │ │ │ -0006c9e0: 2474 6175 3a20 4320 5c74 6f0a 4224 2061 $tau: C \to.B$ a │ │ │ │ -0006c9f0: 2073 706c 6974 7469 6e67 206f 6620 6220 splitting of b │ │ │ │ -0006ca00: 616e 6420 2473 6967 6d61 3a20 4220 5c74 and $sigma: B \t │ │ │ │ -0006ca10: 6f20 4124 2061 2073 706c 6974 7469 6e67 o A$ a splitting │ │ │ │ -0006ca20: 206f 6620 613b 2074 6861 7420 6973 2c0a of a; that is,. │ │ │ │ -0006ca30: 0a24 612a 7369 676d 612b 7461 752a 6220 .$a*sigma+tau*b │ │ │ │ -0006ca40: 3d20 315f 4224 0a0a 2473 6967 6d61 2a61 = 1_B$..$sigma*a │ │ │ │ -0006ca50: 203d 2031 5f41 240a 0a24 622a 7461 7520 = 1_A$..$b*tau │ │ │ │ -0006ca60: 3d20 315f 4324 0a0a 2b2d 2d2d 2d2d 2d2d = 1_C$..+------- │ │ │ │ +0006c7b0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +0006c7c0: 6167 653a 200a 2020 2020 2020 2020 7820 age: . x │ │ │ │ +0006c7d0: 3d20 7370 6c69 7474 696e 6773 2861 2c62 = splittings(a,b │ │ │ │ +0006c7e0: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +0006c7f0: 2020 2020 2a20 612c 2061 202a 6e6f 7465 * a, a *note │ │ │ │ +0006c800: 206d 6174 7269 783a 2028 4d61 6361 756c matrix: (Macaul │ │ │ │ +0006c810: 6179 3244 6f63 294d 6174 7269 782c 2c20 ay2Doc)Matrix,, │ │ │ │ +0006c820: 6d61 7073 2069 6e74 6f20 7468 6520 6b65 maps into the ke │ │ │ │ +0006c830: 726e 656c 206f 6620 620a 2020 2020 2020 rnel of b. │ │ │ │ +0006c840: 2a20 622c 2061 202a 6e6f 7465 206d 6174 * b, a *note mat │ │ │ │ +0006c850: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ +0006c860: 6f63 294d 6174 7269 782c 2c20 7265 7072 oc)Matrix,, repr │ │ │ │ +0006c870: 6573 656e 7469 6e67 2061 2073 7572 6a65 esenting a surje │ │ │ │ +0006c880: 6374 696f 6e0a 2020 2020 2020 2020 6672 ction. fr │ │ │ │ +0006c890: 6f6d 2074 6172 6765 7420 6120 746f 2061 om target a to a │ │ │ │ +0006c8a0: 2066 7265 6520 6d6f 6475 6c65 0a20 202a free module. * │ │ │ │ +0006c8b0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +0006c8c0: 2a20 4c2c 2061 202a 6e6f 7465 206c 6973 * L, a *note lis │ │ │ │ +0006c8d0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +0006c8e0: 294c 6973 742c 2c20 4c20 3d20 5c7b 7369 )List,, L = \{si │ │ │ │ +0006c8f0: 676d 612c 7461 755c 7d2c 2073 706c 6974 gma,tau\}, split │ │ │ │ +0006c900: 7469 6e67 7320 6f66 0a20 2020 2020 2020 tings of. │ │ │ │ +0006c910: 2061 2c62 2072 6573 7065 6374 6976 656c a,b respectivel │ │ │ │ +0006c920: 790a 0a44 6573 6372 6970 7469 6f6e 0a3d y..Description.= │ │ │ │ +0006c930: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4173 7375 ==========..Assu │ │ │ │ +0006c940: 6d69 6e67 2074 6861 7420 2861 2c62 2920 ming that (a,b) │ │ │ │ +0006c950: 6172 6520 7468 6520 6d61 7073 206f 6620 are the maps of │ │ │ │ +0006c960: 6120 7269 6768 7420 6578 6163 7420 7365 a right exact se │ │ │ │ +0006c970: 7175 656e 6365 0a0a 2430 5c74 6f20 415c quence..$0\to A\ │ │ │ │ +0006c980: 746f 2042 5c74 6f20 4320 5c74 6f20 3024 to B\to C \to 0$ │ │ │ │ +0006c990: 0a0a 7769 7468 2042 2c20 4320 6672 6565 ..with B, C free │ │ │ │ +0006c9a0: 2c20 7468 6520 7363 7269 7074 2070 726f , the script pro │ │ │ │ +0006c9b0: 6475 6365 7320 6120 7061 6972 206f 6620 duces a pair of │ │ │ │ +0006c9c0: 6d61 7073 2073 6967 6d61 2c20 7461 7520 maps sigma, tau │ │ │ │ +0006c9d0: 7769 7468 2024 7461 753a 2043 205c 746f with $tau: C \to │ │ │ │ +0006c9e0: 0a42 2420 6120 7370 6c69 7474 696e 6720 .B$ a splitting │ │ │ │ +0006c9f0: 6f66 2062 2061 6e64 2024 7369 676d 613a of b and $sigma: │ │ │ │ +0006ca00: 2042 205c 746f 2041 2420 6120 7370 6c69 B \to A$ a spli │ │ │ │ +0006ca10: 7474 696e 6720 6f66 2061 3b20 7468 6174 tting of a; that │ │ │ │ +0006ca20: 2069 732c 0a0a 2461 2a73 6967 6d61 2b74 is,..$a*sigma+t │ │ │ │ +0006ca30: 6175 2a62 203d 2031 5f42 240a 0a24 7369 au*b = 1_B$..$si │ │ │ │ +0006ca40: 676d 612a 6120 3d20 315f 4124 0a0a 2462 gma*a = 1_A$..$b │ │ │ │ +0006ca50: 2a74 6175 203d 2031 5f43 240a 0a2b 2d2d *tau = 1_C$..+-- │ │ │ │ +0006ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006cab0: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a ---+.|i1 : kk= Z │ │ │ │ -0006cac0: 5a2f 3130 3120 2020 2020 2020 2020 2020 Z/101 │ │ │ │ +0006caa0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +0006cab0: 6b6b 3d20 5a5a 2f31 3031 2020 2020 2020 kk= ZZ/101 │ │ │ │ +0006cac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006caf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cb00: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006caf0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006cb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cb40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0006cb50: 6f31 203d 206b 6b20 2020 2020 2020 2020 o1 = kk │ │ │ │ +0006cb40: 2020 7c0a 7c6f 3120 3d20 6b6b 2020 2020 |.|o1 = kk │ │ │ │ +0006cb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cb90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006cb80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006cb90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0006cba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cbe0: 2020 2020 2020 207c 0a7c 6f31 203a 2051 |.|o1 : Q │ │ │ │ -0006cbf0: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +0006cbd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0006cbe0: 3120 3a20 5175 6f74 6965 6e74 5269 6e67 1 : QuotientRing │ │ │ │ +0006cbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cc30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0006cc20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0006cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006cc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006cc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006cc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006cc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006cc80: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b -+.|i2 : S = kk[ │ │ │ │ -0006cc90: 782c 792c 7a5d 2020 2020 2020 2020 2020 x,y,z] │ │ │ │ +0006cc70: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 ------+.|i2 : S │ │ │ │ +0006cc80: 3d20 6b6b 5b78 2c79 2c7a 5d20 2020 2020 = kk[x,y,z] │ │ │ │ +0006cc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ccc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006ccd0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006ccc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006ccd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ccf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cd10: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006cd20: 203d 2053 2020 2020 2020 2020 2020 2020 = S │ │ │ │ +0006cd10: 7c0a 7c6f 3220 3d20 5320 2020 2020 2020 |.|o2 = S │ │ │ │ +0006cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cd60: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006cd50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006cd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cdb0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c |.|o2 : Pol │ │ │ │ -0006cdc0: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +0006cda0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0006cdb0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +0006cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ce00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0006cdf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0006ce50: 0a7c 6933 203a 2073 6574 5261 6e64 6f6d .|i3 : setRandom │ │ │ │ -0006ce60: 5365 6564 2030 2020 2020 2020 2020 2020 Seed 0 │ │ │ │ +0006ce40: 2d2d 2d2d 2b0a 7c69 3320 3a20 7365 7452 ----+.|i3 : setR │ │ │ │ +0006ce50: 616e 646f 6d53 6565 6420 3020 2020 2020 andomSeed 0 │ │ │ │ +0006ce60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ce70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ce80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ce90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0006cea0: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ -0006ceb0: 6d20 7365 6564 2074 6f20 3020 2020 2020 m seed to 0 │ │ │ │ +0006ce90: 207c 0a7c 202d 2d20 7365 7474 696e 6720 |.| -- setting │ │ │ │ +0006cea0: 7261 6e64 6f6d 2073 6565 6420 746f 2030 random seed to 0 │ │ │ │ +0006ceb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ced0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cee0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006ced0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006cee0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cf30: 2020 2020 2020 7c0a 7c6f 3320 3d20 3020 |.|o3 = 0 │ │ │ │ +0006cf20: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0006cf30: 203d 2030 2020 2020 2020 2020 2020 2020 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+0006cfc0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 203d -----+.|i4 : t = │ │ │ │ +0006cfd0: 2072 616e 646f 6d28 535e 7b32 3a2d 312c random(S^{2:-1, │ │ │ │ +0006cfe0: 323a 2d32 7d2c 2053 5e7b 333a 2d31 2c34 2:-2}, S^{3:-1,4 │ │ │ │ +0006cff0: 3a2d 327d 2920 2020 2020 2020 2020 2020 :-2}) │ │ │ │ 0006d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d010: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006d010: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d060: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -0006d070: 3d20 7b31 7d20 7c20 3234 2020 2d33 3620 = {1} | 24 -36 │ │ │ │ -0006d080: 2d33 3020 3339 782d 3433 792b 3435 7a20 -30 39x-43y+45z │ │ │ │ -0006d090: 2032 3178 2d31 3579 2d33 347a 2033 3478 21x-15y-34z 34x │ │ │ │ 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│ │ │ │ -0006d320: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006d310: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006d320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d360: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0006d370: 3520 3d20 7b7b 317d 207c 2030 2030 2031 5 = {{1} | 0 0 1 │ │ │ │ -0006d380: 2030 2030 2030 2020 2030 2020 7c2c 207b 0 0 0 0 |, { │ │ │ │ -0006d390: 317d 207c 202d 3237 2032 2020 3133 782d 1} | -27 2 13x- │ │ │ │ -0006d3a0: 3130 792b 3433 7a20 3530 782d 3334 792d 10y+43z 50x-34y- │ │ │ │ -0006d3b0: 3530 7a20 7c7d 2020 207c 0a7c 2020 2020 50z |} |.| │ │ │ │ -0006d3c0: 2020 7b32 7d20 7c20 3020 3020 3020 3020 {2} | 0 0 0 0 │ │ │ │ -0006d3d0: 3020 2d33 3120 2d36 207c 2020 7b31 7d20 0 -31 -6 | {1} │ │ │ │ -0006d3e0: 7c20 2d34 2020 3335 2032 3278 2b33 3279 | 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2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d830: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0006d840: 2020 2020 2020 2020 2032 3a20 3220 2e20 2: 2 . │ │ │ │ -0006d850: 2020 2020 2032 3a20 3420 2e20 2020 2020 2: 4 . │ │ │ │ +0006d830: 207c 0a7c 2020 2020 2020 2020 2020 323a |.| 2: │ │ │ │ +0006d840: 2032 202e 2020 2020 2020 323a 2034 202e 2 . 2: 4 . │ │ │ │ +0006d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d880: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006d870: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006d880: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d8d0: 2020 2020 2020 7c0a 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0a57 6179 7320 746f 2075 7365 2073 +..Ways to use s │ │ │ │ -0006d980: 706c 6974 7469 6e67 733a 0a3d 3d3d 3d3d plittings:.===== │ │ │ │ -0006d990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006d9a0: 3d3d 0a0a 2020 2a20 2273 706c 6974 7469 ==.. * "splitti │ │ │ │ -0006d9b0: 6e67 7328 4d61 7472 6978 2c4d 6174 7269 ngs(Matrix,Matri │ │ │ │ -0006d9c0: 7829 220a 0a46 6f72 2074 6865 2070 726f x)"..For the pro │ │ │ │ -0006d9d0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -0006d9e0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -0006d9f0: 6f62 6a65 6374 202a 6e6f 7465 2073 706c object *note spl │ │ │ │ -0006da00: 6974 7469 6e67 733a 2073 706c 6974 7469 ittings: splitti │ │ │ │ -0006da10: 6e67 732c 2069 7320 6120 2a6e 6f74 6520 ngs, is a *note │ │ │ │ -0006da20: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ -0006da30: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ -0006da40: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ -0006da50: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +0006d960: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20 -----+..Ways to │ │ │ │ +0006d970: 7573 6520 7370 6c69 7474 696e 6773 3a0a use splittings:. │ │ │ │ +0006d980: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006d990: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7370 =======.. * "sp │ │ │ │ +0006d9a0: 6c69 7474 696e 6773 284d 6174 7269 782c littings(Matrix, │ │ │ │ +0006d9b0: 4d61 7472 6978 2922 0a0a 466f 7220 7468 Matrix)"..For th │ │ │ │ +0006d9c0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0006d9d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006d9e0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0006d9f0: 6520 7370 6c69 7474 696e 6773 3a20 7370 e splittings: sp │ │ │ │ +0006da00: 6c69 7474 696e 6773 2c20 6973 2061 202a littings, is a * │ │ │ │ +0006da10: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ +0006da20: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ +0006da30: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ +0006da40: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0006da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006daa0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -0006dab0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -0006dac0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -0006dad0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -0006dae0: 6175 6c61 7932 2d31 2e32 352e 3131 2b64 aulay2-1.25.11+d │ │ │ │ -0006daf0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -0006db00: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ -0006db10: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -0006db20: 6f6c 7574 696f 6e73 2e6d 323a 3339 3235 olutions.m2:3925 │ │ │ │ -0006db30: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ -0006db40: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -0006db50: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ -0006db60: 2c20 4e6f 6465 3a20 7374 6162 6c65 486f , Node: stableHo │ │ │ │ -0006db70: 6d2c 204e 6578 743a 2073 756d 5477 6f4d m, Next: sumTwoM │ │ │ │ -0006db80: 6f6e 6f6d 6961 6c73 2c20 5072 6576 3a20 onomials, Prev: │ │ │ │ -0006db90: 7370 6c69 7474 696e 6773 2c20 5570 3a20 splittings, Up: │ │ │ │ -0006dba0: 546f 700a 0a73 7461 626c 6548 6f6d 202d Top..stableHom - │ │ │ │ -0006dbb0: 2d20 6d61 7020 6672 6f6d 2048 6f6d 284d - map from Hom(M │ │ │ │ -0006dbc0: 2c4e 2920 746f 2074 6865 2073 7461 626c ,N) to the stabl │ │ │ │ -0006dbd0: 6520 486f 6d20 6d6f 6475 6c65 0a2a 2a2a e Hom module.*** │ │ │ │ +0006da90: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0006daa0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0006dab0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0006dac0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0006dad0: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ +0006dae0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+ds/M2/Macaul │ │ │ │ +0006daf0: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ +0006db00: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +0006db10: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ +0006db20: 3a33 3932 353a 302e 0a1f 0a46 696c 653a :3925:0....File: │ │ │ │ +0006db30: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ +0006db40: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +0006db50: 2e69 6e66 6f2c 204e 6f64 653a 2073 7461 .info, Node: sta │ │ │ │ +0006db60: 626c 6548 6f6d 2c20 4e65 7874 3a20 7375 bleHom, Next: su │ │ │ │ +0006db70: 6d54 776f 4d6f 6e6f 6d69 616c 732c 2050 mTwoMonomials, P │ │ │ │ +0006db80: 7265 763a 2073 706c 6974 7469 6e67 732c rev: splittings, │ │ │ │ +0006db90: 2055 703a 2054 6f70 0a0a 7374 6162 6c65 Up: Top..stable │ │ │ │ +0006dba0: 486f 6d20 2d2d 206d 6170 2066 726f 6d20 Hom -- map from │ │ │ │ +0006dbb0: 486f 6d28 4d2c 4e29 2074 6f20 7468 6520 Hom(M,N) to the │ │ │ │ +0006dbc0: 7374 6162 6c65 2048 6f6d 206d 6f64 756c stable Hom modul │ │ │ │ +0006dbd0: 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a e.************** │ │ │ │ 0006dbe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006dbf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006dc00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006dc10: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -0006dc20: 200a 2020 2020 2020 2020 7020 3d20 7374 . p = st │ │ │ │ -0006dc30: 6162 6c65 486f 6d28 4d2c 4e29 0a20 202a ableHom(M,N). * │ │ │ │ -0006dc40: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -0006dc50: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -0006dc60: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -0006dc70: 6329 4d6f 6475 6c65 2c2c 200a 2020 2020 c)Module,, . │ │ │ │ -0006dc80: 2020 2a20 4e2c 2061 202a 6e6f 7465 206d * N, a *note m │ │ │ │ -0006dc90: 6f64 756c 653a 2028 4d61 6361 756c 6179 odule: (Macaulay │ │ │ │ -0006dca0: 3244 6f63 294d 6f64 756c 652c 2c20 0a20 2Doc)Module,, . │ │ │ │ -0006dcb0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -0006dcc0: 2020 2a20 702c 2061 202a 6e6f 7465 206d * p, a *note m │ │ │ │ -0006dcd0: 6174 7269 783a 2028 4d61 6361 756c 6179 atrix: (Macaulay │ │ │ │ -0006dce0: 3244 6f63 294d 6174 7269 782c 2c20 7072 2Doc)Matrix,, pr │ │ │ │ -0006dcf0: 6f6a 6563 7469 6f6e 2066 726f 6d20 486f ojection from Ho │ │ │ │ -0006dd00: 6d28 4d2c 4e29 2074 6f0a 2020 2020 2020 m(M,N) to. │ │ │ │ -0006dd10: 2020 7468 6520 7374 6162 6c65 2048 6f6d the stable Hom │ │ │ │ -0006dd20: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0006dd30: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2073 =========..The s │ │ │ │ -0006dd40: 7461 626c 6520 486f 6d20 6973 2048 6f6d table Hom is Hom │ │ │ │ -0006dd50: 284d 2c4e 292f 5420 7768 6572 6520 5420 (M,N)/T where T │ │ │ │ -0006dd60: 6973 2074 6865 2073 7562 6d6f 6475 6c65 is the submodule │ │ │ │ -0006dd70: 206f 6620 686f 6d6f 6d6f 7270 6869 736d of homomorphism │ │ │ │ -0006dd80: 7320 7468 6174 0a66 6163 746f 7220 7468 s that.factor th │ │ │ │ -0006dd90: 726f 7567 6820 6120 6672 6565 2063 6f76 rough a free cov │ │ │ │ -0006dda0: 6572 206f 6620 4e20 286f 722c 2065 7175 er of N (or, equ │ │ │ │ -0006ddb0: 6976 616c 656e 746c 792c 2074 6872 6f75 ivalently, throu │ │ │ │ -0006ddc0: 6768 2061 6e79 2070 726f 6a65 6374 6976 gh any projectiv │ │ │ │ -0006ddd0: 6529 0a0a 5365 6520 616c 736f 0a3d 3d3d e)..See also.=== │ │ │ │ -0006dde0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -0006ddf0: 2069 7353 7461 626c 7954 7269 7669 616c isStablyTrivial │ │ │ │ -0006de00: 3a20 6973 5374 6162 6c79 5472 6976 6961 : isStablyTrivia │ │ │ │ -0006de10: 6c2c 202d 2d20 7265 7475 726e 7320 7472 l, -- returns tr │ │ │ │ -0006de20: 7565 2069 6620 7468 6520 6d61 7020 676f ue if the map go │ │ │ │ -0006de30: 6573 2074 6f0a 2020 2020 3020 756e 6465 es to. 0 unde │ │ │ │ -0006de40: 7220 7374 6162 6c65 486f 6d0a 0a57 6179 r stableHom..Way │ │ │ │ -0006de50: 7320 746f 2075 7365 2073 7461 626c 6548 s to use stableH │ │ │ │ -0006de60: 6f6d 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d om:.============ │ │ │ │ -0006de70: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -0006de80: 2273 7461 626c 6548 6f6d 284d 6f64 756c "stableHom(Modul │ │ │ │ -0006de90: 652c 4d6f 6475 6c65 2922 0a0a 466f 7220 e,Module)"..For │ │ │ │ -0006dea0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0006deb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006dec0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0006ded0: 6f74 6520 7374 6162 6c65 486f 6d3a 2073 ote stableHom: s │ │ │ │ -0006dee0: 7461 626c 6548 6f6d 2c20 6973 2061 202a tableHom, is a * │ │ │ │ -0006def0: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ -0006df00: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ -0006df10: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -0006df20: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0006dc00: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +0006dc10: 7361 6765 3a20 0a20 2020 2020 2020 2070 sage: . p │ │ │ │ +0006dc20: 203d 2073 7461 626c 6548 6f6d 284d 2c4e = stableHom(M,N │ │ │ │ +0006dc30: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +0006dc40: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +0006dc50: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +0006dc60: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +0006dc70: 0a20 2020 2020 202a 204e 2c20 6120 2a6e . * N, a *n │ │ │ │ +0006dc80: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163 ote module: (Mac │ │ │ │ +0006dc90: 6175 6c61 7932 446f 6329 4d6f 6475 6c65 aulay2Doc)Module │ │ │ │ +0006dca0: 2c2c 200a 2020 2a20 4f75 7470 7574 733a ,, . * Outputs: │ │ │ │ +0006dcb0: 0a20 2020 2020 202a 2070 2c20 6120 2a6e . * p, a *n │ │ │ │ +0006dcc0: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ +0006dcd0: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ +0006dce0: 2c2c 2070 726f 6a65 6374 696f 6e20 6672 ,, projection fr │ │ │ │ +0006dcf0: 6f6d 2048 6f6d 284d 2c4e 2920 746f 0a20 om Hom(M,N) to. │ │ │ │ +0006dd00: 2020 2020 2020 2074 6865 2073 7461 626c the stabl │ │ │ │ +0006dd10: 6520 486f 6d0a 0a44 6573 6372 6970 7469 e Hom..Descripti │ │ │ │ +0006dd20: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0006dd30: 5468 6520 7374 6162 6c65 2048 6f6d 2069 The stable Hom i │ │ │ │ +0006dd40: 7320 486f 6d28 4d2c 4e29 2f54 2077 6865 s Hom(M,N)/T whe │ │ │ │ +0006dd50: 7265 2054 2069 7320 7468 6520 7375 626d re T is the subm │ │ │ │ +0006dd60: 6f64 756c 6520 6f66 2068 6f6d 6f6d 6f72 odule of homomor │ │ │ │ +0006dd70: 7068 6973 6d73 2074 6861 740a 6661 6374 phisms that.fact │ │ │ │ +0006dd80: 6f72 2074 6872 6f75 6768 2061 2066 7265 or through a fre │ │ │ │ +0006dd90: 6520 636f 7665 7220 6f66 204e 2028 6f72 e cover of N (or │ │ │ │ +0006dda0: 2c20 6571 7569 7661 6c65 6e74 6c79 2c20 , equivalently, │ │ │ │ +0006ddb0: 7468 726f 7567 6820 616e 7920 7072 6f6a through any proj │ │ │ │ +0006ddc0: 6563 7469 7665 290a 0a53 6565 2061 6c73 ective)..See als │ │ │ │ +0006ddd0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +0006dde0: 2a6e 6f74 6520 6973 5374 6162 6c79 5472 *note isStablyTr │ │ │ │ +0006ddf0: 6976 6961 6c3a 2069 7353 7461 626c 7954 ivial: isStablyT │ │ │ │ +0006de00: 7269 7669 616c 2c20 2d2d 2072 6574 7572 rivial, -- retur │ │ │ │ +0006de10: 6e73 2074 7275 6520 6966 2074 6865 206d ns true if the m │ │ │ │ +0006de20: 6170 2067 6f65 7320 746f 0a20 2020 2030 ap goes to. 0 │ │ │ │ +0006de30: 2075 6e64 6572 2073 7461 626c 6548 6f6d under stableHom │ │ │ │ +0006de40: 0a0a 5761 7973 2074 6f20 7573 6520 7374 ..Ways to use st │ │ │ │ +0006de50: 6162 6c65 486f 6d3a 0a3d 3d3d 3d3d 3d3d ableHom:.======= │ │ │ │ +0006de60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006de70: 0a20 202a 2022 7374 6162 6c65 486f 6d28 . * "stableHom( │ │ │ │ +0006de80: 4d6f 6475 6c65 2c4d 6f64 756c 6529 220a Module,Module)". │ │ │ │ +0006de90: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0006dea0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0006deb0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0006dec0: 6374 202a 6e6f 7465 2073 7461 626c 6548 ct *note stableH │ │ │ │ +0006ded0: 6f6d 3a20 7374 6162 6c65 486f 6d2c 2069 om: stableHom, i │ │ │ │ +0006dee0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +0006def0: 2066 756e 6374 696f 6e3a 0a28 4d61 6361 function:.(Maca │ │ │ │ +0006df00: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ +0006df10: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +0006df20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006df30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006df40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006df50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006df60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006df70: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -0006df80: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -0006df90: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -0006dfa0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -0006dfb0: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ -0006dfc0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+ds/M2/Macaul │ │ │ │ -0006dfd0: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ -0006dfe0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -0006dff0: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ -0006e000: 3a34 3634 393a 302e 0a1f 0a46 696c 653a :4649:0....File: │ │ │ │ -0006e010: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ -0006e020: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ -0006e030: 2e69 6e66 6f2c 204e 6f64 653a 2073 756d .info, Node: sum │ │ │ │ -0006e040: 5477 6f4d 6f6e 6f6d 6961 6c73 2c20 4e65 TwoMonomials, Ne │ │ │ │ -0006e050: 7874 3a20 5461 7465 5265 736f 6c75 7469 xt: TateResoluti │ │ │ │ -0006e060: 6f6e 2c20 5072 6576 3a20 7374 6162 6c65 on, Prev: stable │ │ │ │ -0006e070: 486f 6d2c 2055 703a 2054 6f70 0a0a 7375 Hom, Up: Top..su │ │ │ │ -0006e080: 6d54 776f 4d6f 6e6f 6d69 616c 7320 2d2d mTwoMonomials -- │ │ │ │ -0006e090: 2074 616c 6c79 2074 6865 2073 6571 7565 tally the seque │ │ │ │ -0006e0a0: 6e63 6573 206f 6620 4252 616e 6b73 2066 nces of BRanks f │ │ │ │ -0006e0b0: 6f72 2063 6572 7461 696e 2065 7861 6d70 or certain examp │ │ │ │ -0006e0c0: 6c65 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a les.************ │ │ │ │ +0006df60: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +0006df70: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +0006df80: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +0006df90: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +0006dfa0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +0006dfb0: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d -1.25.11+ds/M2/M │ │ │ │ +0006dfc0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +0006dfd0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ +0006dfe0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +0006dff0: 6e73 2e6d 323a 3436 3439 3a30 2e0a 1f0a ns.m2:4649:0.... │ │ │ │ +0006e000: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ +0006e010: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +0006e020: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ +0006e030: 3a20 7375 6d54 776f 4d6f 6e6f 6d69 616c : sumTwoMonomial │ │ │ │ +0006e040: 732c 204e 6578 743a 2054 6174 6552 6573 s, Next: TateRes │ │ │ │ +0006e050: 6f6c 7574 696f 6e2c 2050 7265 763a 2073 olution, Prev: s │ │ │ │ +0006e060: 7461 626c 6548 6f6d 2c20 5570 3a20 546f tableHom, Up: To │ │ │ │ +0006e070: 700a 0a73 756d 5477 6f4d 6f6e 6f6d 6961 p..sumTwoMonomia │ │ │ │ +0006e080: 6c73 202d 2d20 7461 6c6c 7920 7468 6520 ls -- tally the │ │ │ │ +0006e090: 7365 7175 656e 6365 7320 6f66 2042 5261 sequences of BRa │ │ │ │ +0006e0a0: 6e6b 7320 666f 7220 6365 7274 6169 6e20 nks for certain │ │ │ │ +0006e0b0: 6578 616d 706c 6573 0a2a 2a2a 2a2a 2a2a examples.******* │ │ │ │ +0006e0c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006e0d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006e0e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006e0f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006e100: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -0006e110: 7361 6765 3a20 0a20 2020 2020 2020 2073 sage: . s │ │ │ │ -0006e120: 756d 5477 6f4d 6f6e 6f6d 6961 6c73 2863 umTwoMonomials(c │ │ │ │ -0006e130: 2c64 290a 2020 2a20 496e 7075 7473 3a0a ,d). * Inputs:. │ │ │ │ -0006e140: 2020 2020 2020 2a20 632c 2061 6e20 2a6e * c, an *n │ │ │ │ -0006e150: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ -0006e160: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ -0006e170: 636f 6469 6d65 6e73 696f 6e20 696e 2077 codimension in w │ │ │ │ -0006e180: 6869 6368 2074 6f20 776f 726b 0a20 2020 hich to work. │ │ │ │ -0006e190: 2020 202a 2064 2c20 616e 202a 6e6f 7465 * d, an *note │ │ │ │ -0006e1a0: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ -0006e1b0: 6c61 7932 446f 6329 5a5a 2c2c 2064 6567 lay2Doc)ZZ,, deg │ │ │ │ -0006e1c0: 7265 6520 6f66 2074 6865 206d 6f6e 6f6d ree of the monom │ │ │ │ -0006e1d0: 6961 6c73 2074 6f20 7461 6b65 0a20 202a ials to take. * │ │ │ │ -0006e1e0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -0006e1f0: 2a20 542c 2061 202a 6e6f 7465 2074 616c * T, a *note tal │ │ │ │ -0006e200: 6c79 3a20 284d 6163 6175 6c61 7932 446f ly: (Macaulay2Do │ │ │ │ -0006e210: 6329 5461 6c6c 792c 2c20 0a0a 4465 7363 c)Tally,, ..Desc │ │ │ │ -0006e220: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -0006e230: 3d3d 3d0a 0a74 616c 6c69 6573 2074 6865 ===..tallies the │ │ │ │ -0006e240: 2073 6571 7565 6e63 6573 206f 6620 422d sequences of B- │ │ │ │ -0006e250: 7261 6e6b 7320 7468 6174 206f 6363 7572 ranks that occur │ │ │ │ -0006e260: 2066 6f72 2073 756d 7320 6f66 2070 6169 for sums of pai │ │ │ │ -0006e270: 7273 206f 6620 6d6f 6e6f 6d69 616c 7320 rs of monomials │ │ │ │ -0006e280: 696e 2052 0a3d 2053 2f28 642d 7468 2070 in R.= S/(d-th p │ │ │ │ -0006e290: 6f77 6572 7320 6f66 2074 6865 2076 6172 owers of the var │ │ │ │ -0006e2a0: 6961 626c 6573 292c 2077 6974 6820 6675 iables), with fu │ │ │ │ -0006e2b0: 6c6c 2063 6f6d 706c 6578 6974 7920 283d ll complexity (= │ │ │ │ -0006e2c0: 6329 3b20 7468 6174 2069 732c 2066 6f72 c); that is, for │ │ │ │ -0006e2d0: 2061 6e0a 6170 7072 6f70 7269 6174 6520 an.appropriate │ │ │ │ -0006e2e0: 7379 7a79 6779 204d 206f 6620 4d30 203d syzygy M of M0 = │ │ │ │ -0006e2f0: 2052 2f28 6d31 2b6d 3229 2077 6865 7265 R/(m1+m2) where │ │ │ │ -0006e300: 206d 3120 616e 6420 6d32 2061 7265 206d m1 and m2 are m │ │ │ │ -0006e310: 6f6e 6f6d 6961 6c73 206f 6620 7468 650a onomials of the. │ │ │ │ -0006e320: 7361 6d65 2064 6567 7265 652e 0a0a 2b2d same degree...+- │ │ │ │ +0006e0f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0006e100: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0006e110: 2020 2020 7375 6d54 776f 4d6f 6e6f 6d69 sumTwoMonomi │ │ │ │ +0006e120: 616c 7328 632c 6429 0a20 202a 2049 6e70 als(c,d). * Inp │ │ │ │ +0006e130: 7574 733a 0a20 2020 2020 202a 2063 2c20 uts:. * c, │ │ │ │ +0006e140: 616e 202a 6e6f 7465 2069 6e74 6567 6572 an *note integer │ │ │ │ +0006e150: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0006e160: 5a5a 2c2c 2063 6f64 696d 656e 7369 6f6e ZZ,, codimension │ │ │ │ +0006e170: 2069 6e20 7768 6963 6820 746f 2077 6f72 in which to wor │ │ │ │ +0006e180: 6b0a 2020 2020 2020 2a20 642c 2061 6e20 k. * d, an │ │ │ │ +0006e190: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ +0006e1a0: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ +0006e1b0: 2c20 6465 6772 6565 206f 6620 7468 6520 , degree of the │ │ │ │ +0006e1c0: 6d6f 6e6f 6d69 616c 7320 746f 2074 616b monomials to tak │ │ │ │ +0006e1d0: 650a 2020 2a20 4f75 7470 7574 733a 0a20 e. * Outputs:. │ │ │ │ +0006e1e0: 2020 2020 202a 2054 2c20 6120 2a6e 6f74 * T, a *not │ │ │ │ +0006e1f0: 6520 7461 6c6c 793a 2028 4d61 6361 756c e tally: (Macaul │ │ │ │ +0006e200: 6179 3244 6f63 2954 616c 6c79 2c2c 200a ay2Doc)Tally,, . │ │ │ │ +0006e210: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ +0006e220: 3d3d 3d3d 3d3d 3d3d 0a0a 7461 6c6c 6965 ========..tallie │ │ │ │ +0006e230: 7320 7468 6520 7365 7175 656e 6365 7320 s the sequences │ │ │ │ +0006e240: 6f66 2042 2d72 616e 6b73 2074 6861 7420 of B-ranks that │ │ │ │ +0006e250: 6f63 6375 7220 666f 7220 7375 6d73 206f occur for sums o │ │ │ │ +0006e260: 6620 7061 6972 7320 6f66 206d 6f6e 6f6d f pairs of monom │ │ │ │ +0006e270: 6961 6c73 2069 6e20 520a 3d20 532f 2864 ials in R.= S/(d │ │ │ │ +0006e280: 2d74 6820 706f 7765 7273 206f 6620 7468 -th powers of th │ │ │ │ +0006e290: 6520 7661 7269 6162 6c65 7329 2c20 7769 e variables), wi │ │ │ │ +0006e2a0: 7468 2066 756c 6c20 636f 6d70 6c65 7869 th full complexi │ │ │ │ +0006e2b0: 7479 2028 3d63 293b 2074 6861 7420 6973 ty (=c); that is │ │ │ │ +0006e2c0: 2c20 666f 7220 616e 0a61 7070 726f 7072 , for an.appropr │ │ │ │ +0006e2d0: 6961 7465 2073 797a 7967 7920 4d20 6f66 iate syzygy M of │ │ │ │ +0006e2e0: 204d 3020 3d20 522f 286d 312b 6d32 2920 M0 = R/(m1+m2) │ │ │ │ +0006e2f0: 7768 6572 6520 6d31 2061 6e64 206d 3220 where m1 and m2 │ │ │ │ +0006e300: 6172 6520 6d6f 6e6f 6d69 616c 7320 6f66 are monomials of │ │ │ │ +0006e310: 2074 6865 0a73 616d 6520 6465 6772 6565 the.same degree │ │ │ │ +0006e320: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 0006e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e360: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 -------+.|i1 : s │ │ │ │ -0006e370: 6574 5261 6e64 6f6d 5365 6564 2030 2020 etRandomSeed 0 │ │ │ │ +0006e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0006e360: 203a 2073 6574 5261 6e64 6f6d 5365 6564 : setRandomSeed │ │ │ │ +0006e370: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 0006e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e3a0: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67 |.| -- setting │ │ │ │ -0006e3b0: 2072 616e 646f 6d20 7365 6564 2074 6f20 random seed to │ │ │ │ -0006e3c0: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -0006e3d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006e390: 2020 2020 207c 0a7c 202d 2d20 7365 7474 |.| -- sett │ │ │ │ +0006e3a0: 696e 6720 7261 6e64 6f6d 2073 6565 6420 ing random seed │ │ │ │ +0006e3b0: 746f 2030 2020 2020 2020 2020 2020 2020 to 0 │ │ │ │ +0006e3c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006e3d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0006e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e410: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -0006e420: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +0006e400: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ +0006e410: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +0006e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e450: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0006e440: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0006e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0006e490: 7c69 3220 3a20 7375 6d54 776f 4d6f 6e6f |i2 : sumTwoMono │ │ │ │ -0006e4a0: 6d69 616c 7328 322c 3329 2020 2020 2020 mials(2,3) │ │ │ │ -0006e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e4c0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -0006e4d0: 7573 6564 2030 2e33 3731 3936 3473 2028 used 0.371964s ( │ │ │ │ -0006e4e0: 6370 7529 3b20 302e 3331 3836 3337 7320 cpu); 0.318637s │ │ │ │ -0006e4f0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0006e500: 2920 2020 7c0a 7c32 2020 2020 2020 2020 ) |.|2 │ │ │ │ +0006e470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0006e480: 6932 203a 2073 756d 5477 6f4d 6f6e 6f6d i2 : sumTwoMonom │ │ │ │ +0006e490: 6961 6c73 2832 2c33 2920 2020 2020 2020 ials(2,3) │ │ │ │ +0006e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e4b0: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ +0006e4c0: 6564 2030 2e36 3033 3737 3273 2028 6370 ed 0.603772s (cp │ │ │ │ +0006e4d0: 7529 3b20 302e 3432 3732 3435 7320 2874 u); 0.427245s (t │ │ │ │ +0006e4e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +0006e4f0: 207c 0a7c 3220 2020 2020 2020 2020 2020 |.|2 │ │ │ │ +0006e500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e530: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006e540: 0a7c 5461 6c6c 797b 7b7b 322c 2032 7d2c .|Tally{{{2, 2}, │ │ │ │ -0006e550: 207b 312c 2032 7d7d 203d 3e20 337d 2020 {1, 2}} => 3} │ │ │ │ -0006e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e570: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006e520: 2020 2020 2020 2020 2020 207c 0a7c 5461 |.|Ta │ │ │ │ +0006e530: 6c6c 797b 7b7b 322c 2032 7d2c 207b 312c lly{{{2, 2}, {1, │ │ │ │ +0006e540: 2032 7d7d 203d 3e20 337d 2020 2020 2020 2}} => 3} │ │ │ │ +0006e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e560: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e5b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0006e5c0: 2030 2e31 3538 3039 3473 2028 6370 7529 0.158094s (cpu) │ │ │ │ -0006e5d0: 3b20 302e 3131 3434 3673 2028 7468 7265 ; 0.11446s (thre │ │ │ │ -0006e5e0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ -0006e5f0: 7c0a 7c33 2020 2020 2020 2020 2020 2020 |.|3 │ │ │ │ +0006e590: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006e5a0: 0a7c 202d 2d20 7573 6564 2030 2e33 3334 .| -- used 0.334 │ │ │ │ +0006e5b0: 3534 3473 2028 6370 7529 3b20 302e 3136 544s (cpu); 0.16 │ │ │ │ +0006e5c0: 3932 3438 7320 2874 6872 6561 6429 3b20 9248s (thread); │ │ │ │ +0006e5d0: 3073 2028 6763 2920 207c 0a7c 3320 2020 0s (gc) |.|3 │ │ │ │ +0006e5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e620: 2020 2020 2020 2020 2020 207c 0a7c 5461 |.|Ta │ │ │ │ -0006e630: 6c6c 797b 7b7b 322c 2032 7d2c 207b 312c lly{{{2, 2}, {1, │ │ │ │ -0006e640: 2032 7d7d 203d 3e20 317d 2020 2020 2020 2}} => 1} │ │ │ │ +0006e610: 2020 207c 0a7c 5461 6c6c 797b 7b7b 322c |.|Tally{{{2, │ │ │ │ +0006e620: 2032 7d2c 207b 312c 2032 7d7d 203d 3e20 2}, {1, 2}} => │ │ │ │ +0006e630: 317d 2020 2020 2020 2020 2020 2020 2020 1} │ │ │ │ +0006e640: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0006e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e660: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006e660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e6a0: 207c 0a7c 202d 2d20 7573 6564 2033 2e35 |.| -- used 3.5 │ │ │ │ -0006e6b0: 3736 652d 3036 7320 2863 7075 293b 2031 76e-06s (cpu); 1 │ │ │ │ -0006e6c0: 2e35 3830 3965 2d30 3573 2028 7468 7265 .5809e-05s (thre │ │ │ │ -0006e6d0: 6164 293b 2030 7320 2867 6329 7c0a 7c34 ad); 0s (gc)|.|4 │ │ │ │ +0006e680: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ +0006e690: 6564 2036 2e33 3736 652d 3036 7320 2863 ed 6.376e-06s (c │ │ │ │ +0006e6a0: 7075 293b 2033 2e39 3431 652d 3036 7320 pu); 3.941e-06s │ │ │ │ +0006e6b0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +0006e6c0: 297c 0a7c 3420 2020 2020 2020 2020 2020 )|.|4 │ │ │ │ +0006e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e710: 2020 2020 2020 207c 0a7c 5461 6c6c 797b |.|Tally{ │ │ │ │ -0006e720: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ -0006e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e750: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -0006e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0006e790: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ -0006e7a0: 3d0a 0a20 202a 202a 6e6f 7465 2074 776f =.. * *note two │ │ │ │ -0006e7b0: 4d6f 6e6f 6d69 616c 733a 2074 776f 4d6f Monomials: twoMo │ │ │ │ -0006e7c0: 6e6f 6d69 616c 732c 202d 2d20 7461 6c6c nomials, -- tall │ │ │ │ -0006e7d0: 7920 7468 6520 7365 7175 656e 6365 7320 y the sequences │ │ │ │ -0006e7e0: 6f66 2042 5261 6e6b 7320 666f 720a 2020 of BRanks for. │ │ │ │ -0006e7f0: 2020 6365 7274 6169 6e20 6578 616d 706c certain exampl │ │ │ │ -0006e800: 6573 0a0a 5761 7973 2074 6f20 7573 6520 es..Ways to use │ │ │ │ -0006e810: 7375 6d54 776f 4d6f 6e6f 6d69 616c 733a sumTwoMonomials: │ │ │ │ -0006e820: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0006e830: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -0006e840: 202a 2022 7375 6d54 776f 4d6f 6e6f 6d69 * "sumTwoMonomi │ │ │ │ -0006e850: 616c 7328 5a5a 2c5a 5a29 220a 0a46 6f72 als(ZZ,ZZ)"..For │ │ │ │ -0006e860: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -0006e870: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006e880: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -0006e890: 6e6f 7465 2073 756d 5477 6f4d 6f6e 6f6d note sumTwoMonom │ │ │ │ -0006e8a0: 6961 6c73 3a20 7375 6d54 776f 4d6f 6e6f ials: sumTwoMono │ │ │ │ -0006e8b0: 6d69 616c 732c 2069 7320 6120 2a6e 6f74 mials, is a *not │ │ │ │ -0006e8c0: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ -0006e8d0: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ -0006e8e0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ -0006e8f0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0006e6f0: 2020 2020 2020 2020 2020 207c 0a7c 5461 |.|Ta │ │ │ │ +0006e700: 6c6c 797b 7d20 2020 2020 2020 2020 2020 lly{} │ │ │ │ +0006e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006e730: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0006e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0006e770: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +0006e780: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2074 ===.. * *note t │ │ │ │ +0006e790: 776f 4d6f 6e6f 6d69 616c 733a 2074 776f woMonomials: two │ │ │ │ +0006e7a0: 4d6f 6e6f 6d69 616c 732c 202d 2d20 7461 Monomials, -- ta │ │ │ │ +0006e7b0: 6c6c 7920 7468 6520 7365 7175 656e 6365 lly the sequence │ │ │ │ +0006e7c0: 7320 6f66 2042 5261 6e6b 7320 666f 720a s of BRanks for. │ │ │ │ +0006e7d0: 2020 2020 6365 7274 6169 6e20 6578 616d certain exam │ │ │ │ +0006e7e0: 706c 6573 0a0a 5761 7973 2074 6f20 7573 ples..Ways to us │ │ │ │ +0006e7f0: 6520 7375 6d54 776f 4d6f 6e6f 6d69 616c e sumTwoMonomial │ │ │ │ +0006e800: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d s:.============= │ │ │ │ +0006e810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006e820: 0a20 202a 2022 7375 6d54 776f 4d6f 6e6f . * "sumTwoMono │ │ │ │ +0006e830: 6d69 616c 7328 5a5a 2c5a 5a29 220a 0a46 mials(ZZ,ZZ)"..F │ │ │ │ +0006e840: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +0006e850: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +0006e860: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +0006e870: 202a 6e6f 7465 2073 756d 5477 6f4d 6f6e *note sumTwoMon │ │ │ │ +0006e880: 6f6d 6961 6c73 3a20 7375 6d54 776f 4d6f omials: sumTwoMo │ │ │ │ +0006e890: 6e6f 6d69 616c 732c 2069 7320 6120 2a6e nomials, is a *n │ │ │ │ +0006e8a0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ +0006e8b0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ +0006e8c0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +0006e8d0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +0006e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e940: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0006e950: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0006e960: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0006e970: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0006e980: 6163 6175 6c61 7932 2d31 2e32 352e 3131 acaulay2-1.25.11 │ │ │ │ -0006e990: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0006e9a0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -0006e9b0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -0006e9c0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3435 esolutions.m2:45 │ │ │ │ -0006e9d0: 3132 3a30 2e0a 1f0a 4669 6c65 3a20 436f 12:0....File: Co │ │ │ │ -0006e9e0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -0006e9f0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ -0006ea00: 666f 2c20 4e6f 6465 3a20 5461 7465 5265 fo, Node: TateRe │ │ │ │ -0006ea10: 736f 6c75 7469 6f6e 2c20 4e65 7874 3a20 solution, Next: │ │ │ │ -0006ea20: 7465 6e73 6f72 5769 7468 436f 6d70 6f6e tensorWithCompon │ │ │ │ -0006ea30: 656e 7473 2c20 5072 6576 3a20 7375 6d54 ents, Prev: sumT │ │ │ │ -0006ea40: 776f 4d6f 6e6f 6d69 616c 732c 2055 703a woMonomials, Up: │ │ │ │ -0006ea50: 2054 6f70 0a0a 5461 7465 5265 736f 6c75 Top..TateResolu │ │ │ │ -0006ea60: 7469 6f6e 202d 2d20 5461 7465 5265 736f tion -- TateReso │ │ │ │ -0006ea70: 6c75 7469 6f6e 206f 6620 6120 6d6f 6475 lution of a modu │ │ │ │ -0006ea80: 6c65 206f 7665 7220 616e 2065 7874 6572 le over an exter │ │ │ │ -0006ea90: 696f 7220 616c 6765 6272 610a 2a2a 2a2a ior algebra.**** │ │ │ │ +0006e920: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +0006e930: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +0006e940: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +0006e950: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +0006e960: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ +0006e970: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61 11+ds/M2/Macaula │ │ │ │ +0006e980: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ +0006e990: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ +0006e9a0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ +0006e9b0: 3435 3132 3a30 2e0a 1f0a 4669 6c65 3a20 4512:0....File: │ │ │ │ +0006e9c0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +0006e9d0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +0006e9e0: 696e 666f 2c20 4e6f 6465 3a20 5461 7465 info, Node: Tate │ │ │ │ +0006e9f0: 5265 736f 6c75 7469 6f6e 2c20 4e65 7874 Resolution, Next │ │ │ │ +0006ea00: 3a20 7465 6e73 6f72 5769 7468 436f 6d70 : tensorWithComp │ │ │ │ +0006ea10: 6f6e 656e 7473 2c20 5072 6576 3a20 7375 onents, Prev: su │ │ │ │ +0006ea20: 6d54 776f 4d6f 6e6f 6d69 616c 732c 2055 mTwoMonomials, U │ │ │ │ +0006ea30: 703a 2054 6f70 0a0a 5461 7465 5265 736f p: Top..TateReso │ │ │ │ +0006ea40: 6c75 7469 6f6e 202d 2d20 5461 7465 5265 lution -- TateRe │ │ │ │ +0006ea50: 736f 6c75 7469 6f6e 206f 6620 6120 6d6f solution of a mo │ │ │ │ +0006ea60: 6475 6c65 206f 7665 7220 616e 2065 7874 dule over an ext │ │ │ │ +0006ea70: 6572 696f 7220 616c 6765 6272 610a 2a2a erior algebra.** │ │ │ │ +0006ea80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0006ea90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006eaa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006eab0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006eac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006ead0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006eae0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ -0006eaf0: 2020 2020 2020 2046 203d 2054 6174 6552 F = TateR │ │ │ │ -0006eb00: 6573 6f6c 7574 696f 6e28 4d2c 6c6f 7765 esolution(M,lowe │ │ │ │ -0006eb10: 722c 7570 7065 7229 0a20 202a 2049 6e70 r,upper). * Inp │ │ │ │ -0006eb20: 7574 733a 0a20 2020 2020 202a 204d 2c20 uts:. * M, │ │ │ │ -0006eb30: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ -0006eb40: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ -0006eb50: 6475 6c65 2c2c 200a 2020 2020 2020 2a20 dule,, . * │ │ │ │ -0006eb60: 6c6f 7765 722c 2061 6e20 2a6e 6f74 6520 lower, an *note │ │ │ │ -0006eb70: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ -0006eb80: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020 ay2Doc)ZZ,, . │ │ │ │ -0006eb90: 2020 202a 2075 7070 6572 2c20 616e 202a * upper, an * │ │ │ │ -0006eba0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ -0006ebb0: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ -0006ebc0: 206c 6f77 6572 2061 6e64 2075 7070 6572 lower and upper │ │ │ │ -0006ebd0: 2062 6f75 6e64 7320 666f 720a 2020 2020 bounds for. │ │ │ │ -0006ebe0: 2020 2020 7468 6520 7265 736f 6c75 7469 the resoluti │ │ │ │ -0006ebf0: 6f6e 0a20 202a 204f 7574 7075 7473 3a0a on. * Outputs:. │ │ │ │ -0006ec00: 2020 2020 2020 2a20 462c 2061 202a 6e6f * F, a *no │ │ │ │ -0006ec10: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d te complex: (Com │ │ │ │ -0006ec20: 706c 6578 6573 2943 6f6d 706c 6578 2c2c plexes)Complex,, │ │ │ │ -0006ec30: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d ..Description.= │ │ │ │ -0006ec40: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 466f 726d ==========..Form │ │ │ │ -0006ec50: 7320 616e 2069 6e74 6572 7661 6c2c 206c s an interval, l │ │ │ │ -0006ec60: 6f77 6572 2e2e 7570 7065 722c 206f 6620 ower..upper, of │ │ │ │ -0006ec70: 6120 646f 7562 6c79 2069 6e66 696e 6974 a doubly infinit │ │ │ │ -0006ec80: 6520 6672 6565 2072 6573 6f6c 7574 696f e free resolutio │ │ │ │ -0006ec90: 6e20 6f66 2061 2061 0a43 6f68 656e 2d4d n of a a.Cohen-M │ │ │ │ -0006eca0: 6163 6175 6c61 7920 6d6f 6475 6c65 206f acaulay module o │ │ │ │ -0006ecb0: 7665 7220 6120 476f 7265 6e73 7465 696e ver a Gorenstein │ │ │ │ -0006ecc0: 2072 696e 672c 2073 7563 6820 6173 2061 ring, such as a │ │ │ │ -0006ecd0: 6e79 206d 6f64 756c 6520 6f76 6572 2061 ny module over a │ │ │ │ -0006ece0: 6e0a 6578 7465 7269 6f72 2061 6c67 6562 n.exterior algeb │ │ │ │ -0006ecf0: 7261 2028 6163 7475 616c 6c79 2c20 616e ra (actually, an │ │ │ │ -0006ed00: 7920 6d6f 6475 6c65 206f 7665 7220 616e y module over an │ │ │ │ -0006ed10: 7920 7269 6e67 2e29 0a0a 2b2d 2d2d 2d2d y ring.)..+----- │ │ │ │ +0006eac0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +0006ead0: 0a20 2020 2020 2020 2046 203d 2054 6174 . F = Tat │ │ │ │ +0006eae0: 6552 6573 6f6c 7574 696f 6e28 4d2c 6c6f eResolution(M,lo │ │ │ │ +0006eaf0: 7765 722c 7570 7065 7229 0a20 202a 2049 wer,upper). * I │ │ │ │ +0006eb00: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ +0006eb10: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +0006eb20: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0006eb30: 4d6f 6475 6c65 2c2c 200a 2020 2020 2020 Module,, . │ │ │ │ +0006eb40: 2a20 6c6f 7765 722c 2061 6e20 2a6e 6f74 * lower, an *not │ │ │ │ +0006eb50: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +0006eb60: 756c 6179 3244 6f63 295a 5a2c 2c20 0a20 ulay2Doc)ZZ,, . │ │ │ │ +0006eb70: 2020 2020 202a 2075 7070 6572 2c20 616e * upper, an │ │ │ │ +0006eb80: 202a 6e6f 7465 2069 6e74 6567 6572 3a20 *note integer: │ │ │ │ +0006eb90: 284d 6163 6175 6c61 7932 446f 6329 5a5a (Macaulay2Doc)ZZ │ │ │ │ +0006eba0: 2c2c 206c 6f77 6572 2061 6e64 2075 7070 ,, lower and upp │ │ │ │ +0006ebb0: 6572 2062 6f75 6e64 7320 666f 720a 2020 er bounds for. │ │ │ │ +0006ebc0: 2020 2020 2020 7468 6520 7265 736f 6c75 the resolu │ │ │ │ +0006ebd0: 7469 6f6e 0a20 202a 204f 7574 7075 7473 tion. * Outputs │ │ │ │ +0006ebe0: 3a0a 2020 2020 2020 2a20 462c 2061 202a :. * F, a * │ │ │ │ +0006ebf0: 6e6f 7465 2063 6f6d 706c 6578 3a20 2843 note complex: (C │ │ │ │ +0006ec00: 6f6d 706c 6578 6573 2943 6f6d 706c 6578 omplexes)Complex │ │ │ │ +0006ec10: 2c2c 200a 0a44 6573 6372 6970 7469 6f6e ,, ..Description │ │ │ │ +0006ec20: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 466f .===========..Fo │ │ │ │ +0006ec30: 726d 7320 616e 2069 6e74 6572 7661 6c2c rms an interval, │ │ │ │ +0006ec40: 206c 6f77 6572 2e2e 7570 7065 722c 206f lower..upper, o │ │ │ │ +0006ec50: 6620 6120 646f 7562 6c79 2069 6e66 696e f a doubly infin │ │ │ │ +0006ec60: 6974 6520 6672 6565 2072 6573 6f6c 7574 ite free resolut │ │ │ │ +0006ec70: 696f 6e20 6f66 2061 2061 0a43 6f68 656e ion of a a.Cohen │ │ │ │ +0006ec80: 2d4d 6163 6175 6c61 7920 6d6f 6475 6c65 -Macaulay module │ │ │ │ +0006ec90: 206f 7665 7220 6120 476f 7265 6e73 7465 over a Gorenste │ │ │ │ +0006eca0: 696e 2072 696e 672c 2073 7563 6820 6173 in ring, such as │ │ │ │ +0006ecb0: 2061 6e79 206d 6f64 756c 6520 6f76 6572 any module over │ │ │ │ +0006ecc0: 2061 6e0a 6578 7465 7269 6f72 2061 6c67 an.exterior alg │ │ │ │ +0006ecd0: 6562 7261 2028 6163 7475 616c 6c79 2c20 ebra (actually, │ │ │ │ +0006ece0: 616e 7920 6d6f 6475 6c65 206f 7665 7220 any module over │ │ │ │ +0006ecf0: 616e 7920 7269 6e67 2e29 0a0a 2b2d 2d2d any ring.)..+--- │ │ │ │ +0006ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006ed10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ed20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ed30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ed40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ed50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ed60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -0006ed70: 4520 3d20 5a5a 2f31 3031 5b61 2c62 2c63 E = ZZ/101[a,b,c │ │ │ │ -0006ed80: 2c20 536b 6577 436f 6d6d 7574 6174 6976 , SkewCommutativ │ │ │ │ -0006ed90: 653d 3e74 7275 655d 2020 2020 2020 2020 e=>true] │ │ │ │ +0006ed40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +0006ed50: 3a20 4520 3d20 5a5a 2f31 3031 5b61 2c62 : E = ZZ/101[a,b │ │ │ │ +0006ed60: 2c63 2c20 536b 6577 436f 6d6d 7574 6174 ,c, SkewCommutat │ │ │ │ +0006ed70: 6976 653d 3e74 7275 655d 2020 2020 2020 ive=>true] │ │ │ │ +0006ed80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ed90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0006eda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006edb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006edb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006edc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006edd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee00: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -0006ee10: 4520 2020 2020 2020 2020 2020 2020 2020 E │ │ │ │ +0006ede0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +0006edf0: 3d20 4520 2020 2020 2020 2020 2020 2020 = E │ │ │ │ +0006ee00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ee10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ee30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0006ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee50: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006eea0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ -0006eeb0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20 PolynomialRing, │ │ │ │ -0006eec0: 3320 736b 6577 2063 6f6d 6d75 7461 7469 3 skew commutati │ │ │ │ -0006eed0: 7665 2076 6172 6961 626c 6528 7329 2020 ve variable(s) │ │ │ │ -0006eee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006eef0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0006ee80: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +0006ee90: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +0006eea0: 2c20 3320 736b 6577 2063 6f6d 6d75 7461 , 3 skew commuta │ │ │ │ +0006eeb0: 7469 7665 2076 6172 6961 626c 6528 7329 tive variable(s) │ │ │ │ +0006eec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006eed0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ef10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ef40: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -0006ef50: 4d20 3d20 636f 6b65 7220 6d61 7028 455e M = coker map(E^ │ │ │ │ -0006ef60: 322c 2045 5e7b 2d31 7d2c 206d 6174 7269 2, E^{-1}, matri │ │ │ │ -0006ef70: 7822 6162 3b62 6322 2920 2020 2020 2020 x"ab;bc") │ │ │ │ +0006ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +0006ef30: 3a20 4d20 3d20 636f 6b65 7220 6d61 7028 : M = coker map( │ │ │ │ +0006ef40: 455e 322c 2045 5e7b 2d31 7d2c 206d 6174 E^2, E^{-1}, mat │ │ │ │ +0006ef50: 7269 7822 6162 3b62 6322 2920 2020 2020 rix"ab;bc") │ │ │ │ +0006ef60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ef70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0006ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ef90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006efb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006efd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006efe0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -0006eff0: 636f 6b65 726e 656c 207c 2061 6220 7c20 cokernel | ab | │ │ │ │ +0006efc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0006efd0: 3d20 636f 6b65 726e 656c 207c 2061 6220 = cokernel | ab │ │ │ │ +0006efe0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f030: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006f040: 2020 2020 2020 2020 207c 2062 6320 7c20 | bc | │ │ │ │ +0006f010: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006f020: 2020 2020 2020 2020 2020 207c 2062 6320 | bc │ │ │ │ +0006f030: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006f040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f060: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0006f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f080: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f0b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0006f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006f0d0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ 0006f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0f0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -0006f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f120: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ -0006f130: 452d 6d6f 6475 6c65 2c20 7175 6f74 6965 E-module, quotie │ │ │ │ -0006f140: 6e74 206f 6620 4520 2020 2020 2020 2020 nt of E │ │ │ │ -0006f150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f170: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0006f0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f100: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0006f110: 3a20 452d 6d6f 6475 6c65 2c20 7175 6f74 : E-module, quot │ │ │ │ +0006f120: 6965 6e74 206f 6620 4520 2020 2020 2020 ient of E │ │ │ │ +0006f130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f150: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006f1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006f1c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -0006f1d0: 7072 6573 656e 7461 7469 6f6e 204d 2020 presentation M │ │ │ │ +0006f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ +0006f1b0: 3a20 7072 6573 656e 7461 7469 6f6e 204d : presentation M │ │ │ │ +0006f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f1f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0006f200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f210: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006f210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f260: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -0006f270: 7c20 6162 207c 2020 2020 2020 2020 2020 | ab | │ │ │ │ +0006f240: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +0006f250: 3d20 7c20 6162 207c 2020 2020 2020 2020 = | ab | │ │ │ │ +0006f260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f2b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006f2c0: 7c20 6263 207c 2020 2020 2020 2020 2020 | bc | │ │ │ │ +0006f290: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006f2a0: 2020 7c20 6263 207c 2020 2020 2020 2020 | bc | │ │ │ │ +0006f2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f2e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0006f2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f300: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006f300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f350: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006f360: 2020 2020 2020 2020 3220 2020 2020 2031 2 1 │ │ │ │ +0006f330: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006f340: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0006f350: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0006f360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f3a0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ 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6f6e 2220 696e 7374 6561 642e 0a0a tion" instead... │ │ │ │ +0006f8c0: 5761 7973 2074 6f20 7573 6520 5461 7465 Ways to use Tate │ │ │ │ +0006f8d0: 5265 736f 6c75 7469 6f6e 3a0a 3d3d 3d3d Resolution:.==== │ │ │ │ +0006f8e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006f8f0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 5461 =======.. * "Ta │ │ │ │ +0006f900: 7465 5265 736f 6c75 7469 6f6e 284d 6f64 teResolution(Mod │ │ │ │ +0006f910: 756c 6529 220a 2020 2a20 2254 6174 6552 ule)". * "TateR │ │ │ │ +0006f920: 6573 6f6c 7574 696f 6e28 4d6f 6475 6c65 esolution(Module │ │ │ │ +0006f930: 2c5a 5a29 220a 2020 2a20 2254 6174 6552 ,ZZ)". * "TateR │ │ │ │ +0006f940: 6573 6f6c 7574 696f 6e28 4d6f 6475 6c65 esolution(Module │ │ │ │ +0006f950: 2c5a 5a2c 5a5a 2922 0a0a 466f 7220 7468 ,ZZ,ZZ)"..For th │ │ │ │ +0006f960: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0006f970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006f980: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0006f990: 6520 5461 7465 5265 736f 6c75 7469 6f6e e TateResolution │ │ │ │ +0006f9a0: 3a20 5461 7465 5265 736f 6c75 7469 6f6e : TateResolution │ │ │ │ +0006f9b0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +0006f9c0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ +0006f9d0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ +0006f9e0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +0006f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006fa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006fa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006fa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006fa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006fa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 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2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -0006fc20: 2020 2020 5420 3d20 7465 6e73 6f72 284d T = tensor(M │ │ │ │ -0006fc30: 2c4e 290a 2020 2a20 496e 7075 7473 3a0a ,N). * Inputs:. │ │ │ │ -0006fc40: 2020 2020 2020 2a20 4d2c 2061 202a 6e6f * M, a *no │ │ │ │ -0006fc50: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -0006fc60: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -0006fc70: 2c20 0a20 2020 2020 202a 204e 2c20 6120 , . * N, a │ │ │ │ -0006fc80: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -0006fc90: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -0006fca0: 6c65 2c2c 200a 2020 2a20 4f75 7470 7574 le,, . * Output │ │ │ │ -0006fcb0: 733a 0a20 2020 2020 202a 2054 2c20 6120 s:. * T, a │ │ │ │ -0006fcc0: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -0006fcd0: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -0006fce0: 6c65 2c2c 200a 0a44 6573 6372 6970 7469 le,, 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eproducible-path │ │ │ │ +00070020: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ +00070030: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61 11+ds/M2/Macaula │ │ │ │ +00070040: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ +00070050: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ +00070060: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ +00070070: 3236 3637 3a30 2e0a 1f0a 4669 6c65 3a20 2667:0....File: │ │ │ │ +00070080: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +00070090: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +000700a0: 696e 666f 2c20 4e6f 6465 3a20 746f 4172 info, Node: toAr │ │ │ │ +000700b0: 7261 792c 204e 6578 743a 2074 776f 4d6f ray, Next: twoMo │ │ │ │ +000700c0: 6e6f 6d69 616c 732c 2050 7265 763a 2074 nomials, Prev: t │ │ │ │ +000700d0: 656e 736f 7257 6974 6843 6f6d 706f 6e65 ensorWithCompone │ │ │ │ +000700e0: 6e74 732c 2055 703a 2054 6f70 0a0a 746f nts, Up: Top..to │ │ │ │ +000700f0: 4172 7261 7920 2d2d 206d 616b 6573 2061 Array -- makes a │ │ │ │ +00070100: 6e20 6172 7261 7920 6672 6f6d 2061 204c n array from a L │ │ │ │ +00070110: 6973 7420 6f72 2066 726f 6d20 6120 7369 ist or from a si │ │ │ │ +00070120: 6e67 6c65 2069 6e74 6567 6572 0a2a 2a2a ngle integer.*** │ │ │ │ +00070130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00070140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00070150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070170: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070180: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00070190: 7361 6765 3a20 0a20 2020 2020 2020 2061 sage: . a │ │ │ │ -000701a0: 7272 203d 2074 6f41 7272 6179 204c 0a20 rr = toArray L. │ │ │ │ -000701b0: 2020 2020 2020 2061 7272 203d 2074 6f41 arr = toA │ │ │ │ -000701c0: 7272 6179 206e 0a20 202a 2049 6e70 7574 rray n. * Input │ │ │ │ -000701d0: 733a 0a20 2020 2020 202a 204c 2c20 6120 s:. * L, a │ │ │ │ -000701e0: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ -000701f0: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ -00070200: 200a 2020 2020 2020 2a20 6e2c 2061 6e20 . * n, an │ │ │ │ -00070210: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ -00070220: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ -00070230: 2c20 0a20 202a 204f 7574 7075 7473 3a0a , . * Outputs:. │ │ │ │ -00070240: 2020 2020 2020 2a20 6172 722c 2061 6e20 * arr, an │ │ │ │ -00070250: 2a6e 6f74 6520 6172 7261 793a 2028 4d61 *note array: (Ma │ │ │ │ -00070260: 6361 756c 6179 3244 6f63 2941 7272 6179 caulay2Doc)Array │ │ │ │ -00070270: 2c2c 200a 0a57 6179 7320 746f 2075 7365 ,, ..Ways to use │ │ │ │ -00070280: 2074 6f41 7272 6179 3a0a 3d3d 3d3d 3d3d toArray:.====== │ │ │ │ -00070290: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -000702a0: 2020 2a20 2274 6f41 7272 6179 284c 6973 * "toArray(Lis │ │ │ │ -000702b0: 7429 220a 2020 2a20 2274 6f41 7272 6179 t)". * "toArray │ │ │ │ -000702c0: 285a 5a29 220a 0a46 6f72 2074 6865 2070 (ZZ)"..For the p │ │ │ │ -000702d0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -000702e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -000702f0: 6520 6f62 6a65 6374 202a 6e6f 7465 2074 e object *note t │ │ │ │ -00070300: 6f41 7272 6179 3a20 746f 4172 7261 792c oArray: toArray, │ │ │ │ -00070310: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ -00070320: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61 od function:.(Ma │ │ │ │ -00070330: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -00070340: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ +00070160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00070170: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00070180: 2061 7272 203d 2074 6f41 7272 6179 204c arr = toArray L │ │ │ │ +00070190: 0a20 2020 2020 2020 2061 7272 203d 2074 . arr = t │ │ │ │ +000701a0: 6f41 7272 6179 206e 0a20 202a 2049 6e70 oArray n. * Inp │ │ │ │ +000701b0: 7574 733a 0a20 2020 2020 202a 204c 2c20 uts:. * L, │ │ │ │ +000701c0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ +000701d0: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ +000701e0: 2c2c 200a 2020 2020 2020 2a20 6e2c 2061 ,, . * n, a │ │ │ │ +000701f0: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ +00070200: 2028 4d61 6361 756c 6179 3244 6f63 295a (Macaulay2Doc)Z │ │ │ │ +00070210: 5a2c 2c20 0a20 202a 204f 7574 7075 7473 Z,, . * Outputs │ │ │ │ +00070220: 3a0a 2020 2020 2020 2a20 6172 722c 2061 :. * arr, a │ │ │ │ +00070230: 6e20 2a6e 6f74 6520 6172 7261 793a 2028 n *note array: ( │ │ │ │ +00070240: 4d61 6361 756c 6179 3244 6f63 2941 7272 Macaulay2Doc)Arr │ │ │ │ +00070250: 6179 2c2c 200a 0a57 6179 7320 746f 2075 ay,, ..Ways to u │ │ │ │ +00070260: 7365 2074 6f41 7272 6179 3a0a 3d3d 3d3d se toArray:.==== │ │ │ │ +00070270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00070280: 0a0a 2020 2a20 2274 6f41 7272 6179 284c .. * "toArray(L │ │ │ │ +00070290: 6973 7429 220a 2020 2a20 2274 6f41 7272 ist)". * "toArr │ │ │ │ +000702a0: 6179 285a 5a29 220a 0a46 6f72 2074 6865 ay(ZZ)"..For the │ │ │ │ +000702b0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +000702c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +000702d0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ +000702e0: 2074 6f41 7272 6179 3a20 746f 4172 7261 toArray: toArra │ │ │ │ +000702f0: 792c 2069 7320 6120 2a6e 6f74 6520 6d65 y, is a *note me │ │ │ │ +00070300: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ +00070310: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ +00070320: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +00070330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -000703a0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -000703b0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -000703c0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -000703d0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -000703e0: 7932 2d31 2e32 352e 3131 2b64 732f 4d32 y2-1.25.11+ds/M2 │ │ │ │ -000703f0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00070400: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ -00070410: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -00070420: 696f 6e73 2e6d 323a 3339 3433 3a30 2e0a ions.m2:3943:0.. │ │ │ │ -00070430: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ -00070440: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ -00070450: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ -00070460: 6465 3a20 7477 6f4d 6f6e 6f6d 6961 6c73 de: twoMonomials │ │ │ │ -00070470: 2c20 5072 6576 3a20 746f 4172 7261 792c , Prev: toArray, │ │ │ │ -00070480: 2055 703a 2054 6f70 0a0a 7477 6f4d 6f6e Up: Top..twoMon │ │ │ │ -00070490: 6f6d 6961 6c73 202d 2d20 7461 6c6c 7920 omials -- tally │ │ │ │ -000704a0: 7468 6520 7365 7175 656e 6365 7320 6f66 the sequences of │ │ │ │ -000704b0: 2042 5261 6e6b 7320 666f 7220 6365 7274 BRanks for cert │ │ │ │ -000704c0: 6169 6e20 6578 616d 706c 6573 0a2a 2a2a ain examples.*** │ │ │ │ +00070370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00070380: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00070390: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +000703a0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +000703b0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +000703c0: 6c61 7932 2d31 2e32 352e 3131 2b64 732f lay2-1.25.11+ds/ │ │ │ │ +000703d0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +000703e0: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +000703f0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +00070400: 7574 696f 6e73 2e6d 323a 3339 3433 3a30 utions.m2:3943:0 │ │ │ │ +00070410: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +00070420: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +00070430: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +00070440: 4e6f 6465 3a20 7477 6f4d 6f6e 6f6d 6961 Node: twoMonomia │ │ │ │ +00070450: 6c73 2c20 5072 6576 3a20 746f 4172 7261 ls, Prev: toArra │ │ │ │ +00070460: 792c 2055 703a 2054 6f70 0a0a 7477 6f4d y, Up: Top..twoM │ │ │ │ +00070470: 6f6e 6f6d 6961 6c73 202d 2d20 7461 6c6c onomials -- tall │ │ │ │ +00070480: 7920 7468 6520 7365 7175 656e 6365 7320 y the sequences │ │ │ │ +00070490: 6f66 2042 5261 6e6b 7320 666f 7220 6365 of BRanks for ce │ │ │ │ +000704a0: 7274 6169 6e20 6578 616d 706c 6573 0a2a rtain examples.* │ │ │ │ +000704b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000704c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000704d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000704e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000704f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -00070510: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -00070520: 2020 2020 2054 203d 2054 776f 4d6f 6e6f T = TwoMono │ │ │ │ -00070530: 6d69 616c 7328 632c 6429 0a20 202a 2049 mials(c,d). * I │ │ │ │ -00070540: 6e70 7574 733a 0a20 2020 2020 202a 2063 nputs:. * c │ │ │ │ -00070550: 2c20 616e 202a 6e6f 7465 2069 6e74 6567 , an *note integ │ │ │ │ -00070560: 6572 3a20 284d 6163 6175 6c61 7932 446f er: (Macaulay2Do │ │ │ │ -00070570: 6329 5a5a 2c2c 2063 6f64 696d 656e 7369 c)ZZ,, codimensi │ │ │ │ -00070580: 6f6e 2069 6e20 7768 6963 6820 746f 2077 on in which to w │ │ │ │ -00070590: 6f72 6b0a 2020 2020 2020 2a20 642c 2061 ork. * d, a │ │ │ │ -000705a0: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ -000705b0: 2028 4d61 6361 756c 6179 3244 6f63 295a (Macaulay2Doc)Z │ │ │ │ -000705c0: 5a2c 2c20 6465 6772 6565 206f 6620 7468 Z,, degree of th │ │ │ │ -000705d0: 6520 6d6f 6e6f 6d69 616c 7320 746f 2074 e monomials to t │ │ │ │ -000705e0: 616b 650a 2020 2a20 2a6e 6f74 6520 4f70 ake. * *note Op │ │ │ │ -000705f0: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ -00070600: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ -00070610: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ -00070620: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ -00070630: 732c 3a0a 2020 2020 2020 2a20 4f70 7469 s,:. * Opti │ │ │ │ -00070640: 6d69 736d 203d 3e20 2e2e 2e2c 2064 6566 mism => ..., def │ │ │ │ -00070650: 6175 6c74 2076 616c 7565 2030 0a20 202a ault value 0. * │ │ │ │ -00070660: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00070670: 2a20 542c 2061 202a 6e6f 7465 2074 616c * T, a *note tal │ │ │ │ -00070680: 6c79 3a20 284d 6163 6175 6c61 7932 446f ly: (Macaulay2Do │ │ │ │ -00070690: 6329 5461 6c6c 792c 2c20 0a0a 4465 7363 c)Tally,, ..Desc │ │ │ │ -000706a0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -000706b0: 3d3d 3d0a 0a74 616c 6c69 6573 2074 6865 ===..tallies the │ │ │ │ -000706c0: 2073 6571 7565 6e63 6573 206f 6620 422d sequences of B- │ │ │ │ -000706d0: 7261 6e6b 7320 7468 6174 206f 6363 7572 ranks that occur │ │ │ │ -000706e0: 2066 6f72 2069 6465 616c 7320 6765 6e65 for ideals gene │ │ │ │ -000706f0: 7261 7465 6420 6279 2070 6169 7273 206f rated by pairs o │ │ │ │ -00070700: 660a 6d6f 6e6f 6d69 616c 7320 696e 2052 f.monomials in R │ │ │ │ -00070710: 203d 2053 2f28 642d 7468 2070 6f77 6572 = S/(d-th power │ │ │ │ -00070720: 7320 6f66 2074 6865 2076 6172 6961 626c s of the variabl │ │ │ │ -00070730: 6573 292c 2077 6974 6820 6675 6c6c 2063 es), with full c │ │ │ │ -00070740: 6f6d 706c 6578 6974 7920 283d 6329 3b0a omplexity (=c);. │ │ │ │ -00070750: 7468 6174 2069 732c 2066 6f72 2061 6e20 that is, for an │ │ │ │ -00070760: 6170 7072 6f70 7269 6174 6520 7379 7a79 appropriate syzy │ │ │ │ -00070770: 6779 204d 206f 6620 4d30 203d 2052 2f28 gy M of M0 = R/( │ │ │ │ -00070780: 6d31 2c20 6d32 2920 7768 6572 6520 6d31 m1, m2) where m1 │ │ │ │ -00070790: 2061 6e64 206d 3220 6172 650a 6d6f 6e6f and m2 are.mono │ │ │ │ -000707a0: 6d69 616c 7320 6f66 2074 6865 2073 616d mials of the sam │ │ │ │ -000707b0: 6520 6465 6772 6565 2e0a 0a2b 2d2d 2d2d e degree...+---- │ │ │ │ +000704f0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +00070500: 2020 2020 2020 2054 203d 2054 776f 4d6f T = TwoMo │ │ │ │ +00070510: 6e6f 6d69 616c 7328 632c 6429 0a20 202a nomials(c,d). * │ │ │ │ +00070520: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +00070530: 2063 2c20 616e 202a 6e6f 7465 2069 6e74 c, an *note int │ │ │ │ +00070540: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ +00070550: 446f 6329 5a5a 2c2c 2063 6f64 696d 656e Doc)ZZ,, codimen │ │ │ │ +00070560: 7369 6f6e 2069 6e20 7768 6963 6820 746f sion in which to │ │ │ │ +00070570: 2077 6f72 6b0a 2020 2020 2020 2a20 642c work. * d, │ │ │ │ +00070580: 2061 6e20 2a6e 6f74 6520 696e 7465 6765 an *note intege │ │ │ │ +00070590: 723a 2028 4d61 6361 756c 6179 3244 6f63 r: (Macaulay2Doc │ │ │ │ +000705a0: 295a 5a2c 2c20 6465 6772 6565 206f 6620 )ZZ,, degree of │ │ │ │ +000705b0: 7468 6520 6d6f 6e6f 6d69 616c 7320 746f the monomials to │ │ │ │ +000705c0: 2074 616b 650a 2020 2a20 2a6e 6f74 6520 take. * *note │ │ │ │ +000705d0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a Optional inputs: │ │ │ │ +000705e0: 2028 4d61 6361 756c 6179 3244 6f63 2975 (Macaulay2Doc)u │ │ │ │ +000705f0: 7369 6e67 2066 756e 6374 696f 6e73 2077 sing functions w │ │ │ │ +00070600: 6974 6820 6f70 7469 6f6e 616c 2069 6e70 ith optional inp │ │ │ │ +00070610: 7574 732c 3a0a 2020 2020 2020 2a20 4f70 uts,:. * Op │ │ │ │ +00070620: 7469 6d69 736d 203d 3e20 2e2e 2e2c 2064 timism => ..., d │ │ │ │ +00070630: 6566 6175 6c74 2076 616c 7565 2030 0a20 efault value 0. │ │ │ │ +00070640: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +00070650: 2020 2a20 542c 2061 202a 6e6f 7465 2074 * T, a *note t │ │ │ │ +00070660: 616c 6c79 3a20 284d 6163 6175 6c61 7932 ally: (Macaulay2 │ │ │ │ +00070670: 446f 6329 5461 6c6c 792c 2c20 0a0a 4465 Doc)Tally,, ..De │ │ │ │ +00070680: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00070690: 3d3d 3d3d 3d0a 0a74 616c 6c69 6573 2074 =====..tallies t │ │ │ │ +000706a0: 6865 2073 6571 7565 6e63 6573 206f 6620 he sequences of │ │ │ │ +000706b0: 422d 7261 6e6b 7320 7468 6174 206f 6363 B-ranks that occ │ │ │ │ +000706c0: 7572 2066 6f72 2069 6465 616c 7320 6765 ur for ideals ge │ │ │ │ +000706d0: 6e65 7261 7465 6420 6279 2070 6169 7273 nerated by pairs │ │ │ │ +000706e0: 206f 660a 6d6f 6e6f 6d69 616c 7320 696e of.monomials in │ │ │ │ +000706f0: 2052 203d 2053 2f28 642d 7468 2070 6f77 R = S/(d-th pow │ │ │ │ +00070700: 6572 7320 6f66 2074 6865 2076 6172 6961 ers of the varia │ │ │ │ +00070710: 626c 6573 292c 2077 6974 6820 6675 6c6c bles), with full │ │ │ │ +00070720: 2063 6f6d 706c 6578 6974 7920 283d 6329 complexity (=c) │ │ │ │ +00070730: 3b0a 7468 6174 2069 732c 2066 6f72 2061 ;.that is, for a │ │ │ │ +00070740: 6e20 6170 7072 6f70 7269 6174 6520 7379 n appropriate sy │ │ │ │ +00070750: 7a79 6779 204d 206f 6620 4d30 203d 2052 zygy M of M0 = R │ │ │ │ +00070760: 2f28 6d31 2c20 6d32 2920 7768 6572 6520 /(m1, m2) where │ │ │ │ +00070770: 6d31 2061 6e64 206d 3220 6172 650a 6d6f m1 and m2 are.mo │ │ │ │ +00070780: 6e6f 6d69 616c 7320 6f66 2074 6865 2073 nomials of the s │ │ │ │ +00070790: 616d 6520 6465 6772 6565 2e0a 0a2b 2d2d ame degree...+-- │ │ │ │ +000707a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000707b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000707c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000707d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000707e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000707f0: 2d2b 0a7c 6931 203a 2073 6574 5261 6e64 -+.|i1 : setRand │ │ │ │ -00070800: 6f6d 5365 6564 2030 2020 2020 2020 2020 omSeed 0 │ │ │ │ -00070810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070820: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00070830: 7365 7474 696e 6720 7261 6e64 6f6d 2073 setting random s │ │ │ │ -00070840: 6565 6420 746f 2030 2020 2020 2020 2020 eed to 0 │ │ │ │ +000707d0: 2d2d 2d2b 0a7c 6931 203a 2073 6574 5261 ---+.|i1 : setRa │ │ │ │ +000707e0: 6e64 6f6d 5365 6564 2030 2020 2020 2020 ndomSeed 0 │ │ │ │ +000707f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070800: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ +00070810: 2d20 7365 7474 696e 6720 7261 6e64 6f6d - setting random │ │ │ │ +00070820: 2073 6565 6420 746f 2030 2020 2020 2020 seed to 0 │ │ │ │ +00070830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070840: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00070850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070860: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00070870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070890: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ -000708a0: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -000708b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000708c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000708d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -000708e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000708f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070900: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -00070910: 2074 776f 4d6f 6e6f 6d69 616c 7328 322c twoMonomials(2, │ │ │ │ -00070920: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ -00070930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070940: 207c 0a7c 202d 2d20 7573 6564 2030 2e37 |.| -- used 0.7 │ │ │ │ -00070950: 3931 3031 3773 2028 6370 7529 3b20 302e 91017s (cpu); 0. │ │ │ │ -00070960: 3539 3936 3436 7320 2874 6872 6561 6429 599646s (thread) │ │ │ │ -00070970: 3b20 3073 2028 6763 297c 0a7c 3220 2020 ; 0s (gc)|.|2 │ │ │ │ +00070860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070870: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00070880: 203d 2030 2020 2020 2020 2020 2020 2020 = 0 │ │ │ │ +00070890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000708a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000708b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +000708c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000708d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000708e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +000708f0: 203a 2074 776f 4d6f 6e6f 6d69 616c 7328 : twoMonomials( │ │ │ │ +00070900: 322c 3329 2020 2020 2020 2020 2020 2020 2,3) │ │ │ │ +00070910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070920: 2020 207c 0a7c 202d 2d20 7573 6564 2031 |.| -- used 1 │ │ │ │ +00070930: 2e33 3333 3873 2028 6370 7529 3b20 302e .3338s (cpu); 0. │ │ │ │ +00070940: 3736 3839 3038 7320 2874 6872 6561 6429 768908s (thread) │ │ │ │ +00070950: 3b20 3073 2028 6763 2920 207c 0a7c 3220 ; 0s (gc) |.|2 │ │ │ │ +00070960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000709a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000709b0: 207c 0a7c 5461 6c6c 797b 7b7b 312c 2031 |.|Tally{{{1, 1 │ │ │ │ -000709c0: 7d7d 203d 3e20 3220 2020 2020 2020 207d }} => 2 } │ │ │ │ -000709d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000709e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000709f0: 2020 7b7b 322c 2032 7d2c 207b 312c 2032 {{2, 2}, {1, 2 │ │ │ │ -00070a00: 7d7d 203d 3e20 3420 2020 2020 2020 2020 }} => 4 │ │ │ │ +00070990: 2020 207c 0a7c 5461 6c6c 797b 7b7b 312c |.|Tally{{{1, │ │ │ │ +000709a0: 2031 7d7d 203d 3e20 3220 2020 2020 2020 1}} => 2 │ │ │ │ +000709b0: 207d 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +000709c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000709d0: 2020 2020 7b7b 322c 2032 7d2c 207b 312c {{2, 2}, {1, │ │ │ │ +000709e0: 2032 7d7d 203d 3e20 3420 2020 2020 2020 2}} => 4 │ │ │ │ +000709f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070a00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00070a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070a20: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00070a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070a50: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00070a60: 7573 6564 2030 2e34 3133 3435 3873 2028 used 0.413458s ( │ │ │ │ -00070a70: 6370 7529 3b20 302e 3336 3633 3934 7320 cpu); 0.366394s │ │ │ │ -00070a80: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00070a90: 297c 0a7c 3320 2020 2020 2020 2020 2020 )|.|3 │ │ │ │ -00070aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070ac0: 2020 2020 2020 2020 207c 0a7c 5461 6c6c |.|Tall │ │ │ │ -00070ad0: 797b 7b7b 322c 2032 7d2c 207b 312c 2032 y{{{2, 2}, {1, 2 │ │ │ │ -00070ae0: 7d7d 203d 3e20 327d 2020 2020 2020 2020 }} => 2} │ │ │ │ -00070af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070b00: 207c 0a7c 2020 2020 2020 7b7b 332c 2033 |.| {{3, 3 │ │ │ │ -00070b10: 7d2c 207b 322c 2033 7d7d 203d 3e20 3120 }, {2, 3}} => 1 │ │ │ │ +00070a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070a30: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ +00070a40: 2d20 7573 6564 2030 2e36 3837 3435 3573 - used 0.687455s │ │ │ │ +00070a50: 2028 6370 7529 3b20 302e 3434 3634 3933 (cpu); 0.446493 │ │ │ │ +00070a60: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00070a70: 6763 297c 0a7c 3320 2020 2020 2020 2020 gc)|.|3 │ │ │ │ +00070a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070aa0: 2020 2020 2020 2020 2020 207c 0a7c 5461 |.|Ta │ │ │ │ +00070ab0: 6c6c 797b 7b7b 322c 2032 7d2c 207b 312c lly{{{2, 2}, {1, │ │ │ │ +00070ac0: 2032 7d7d 203d 3e20 327d 2020 2020 2020 2}} => 2} │ │ │ │ +00070ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070ae0: 2020 207c 0a7c 2020 2020 2020 7b7b 332c |.| {{3, │ │ │ │ +00070af0: 2033 7d2c 207b 322c 2033 7d7d 203d 3e20 3}, {2, 3}} => │ │ │ │ +00070b00: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00070b10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00070b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070b30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00070b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070b70: 207c 0a7c 202d 2d20 7573 6564 2030 2e31 |.| -- used 0.1 │ │ │ │ -00070b80: 3930 3035 7320 2863 7075 293b 2030 2e31 9005s (cpu); 0.1 │ │ │ │ -00070b90: 3339 3937 3473 2028 7468 7265 6164 293b 39974s (thread); │ │ │ │ -00070ba0: 2030 7320 2867 6329 207c 0a7c 3420 2020 0s (gc) |.|4 │ │ │ │ +00070b50: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ +00070b60: 2e33 3330 3532 3373 2028 6370 7529 3b20 .330523s (cpu); │ │ │ │ +00070b70: 302e 3137 3230 3933 7320 2874 6872 6561 0.172093s (threa │ │ │ │ +00070b80: 6429 3b20 3073 2028 6763 297c 0a7c 3420 d); 0s (gc)|.|4 │ │ │ │ +00070b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070be0: 207c 0a7c 5461 6c6c 797b 7b7b 322c 2032 |.|Tally{{{2, 2 │ │ │ │ -00070bf0: 7d2c 207b 312c 2032 7d7d 203d 3e20 317d }, {1, 2}} => 1} │ │ │ │ -00070c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070c10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00070bc0: 2020 207c 0a7c 5461 6c6c 797b 7b7b 322c |.|Tally{{{2, │ │ │ │ +00070bd0: 2032 7d2c 207b 312c 2032 7d7d 203d 3e20 2}, {1, 2}} => │ │ │ │ +00070be0: 317d 2020 2020 2020 2020 2020 2020 2020 1} │ │ │ │ +00070bf0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00070c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070c50: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ -00070c60: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -00070c70: 2074 776f 4d6f 6e6f 6d69 616c 733a 2074 twoMonomials: t │ │ │ │ -00070c80: 776f 4d6f 6e6f 6d69 616c 732c 202d 2d20 woMonomials, -- │ │ │ │ -00070c90: 7461 6c6c 7920 7468 6520 7365 7175 656e tally the sequen │ │ │ │ -00070ca0: 6365 7320 6f66 2042 5261 6e6b 7320 666f ces of BRanks fo │ │ │ │ -00070cb0: 720a 2020 2020 6365 7274 6169 6e20 6578 r. certain ex │ │ │ │ -00070cc0: 616d 706c 6573 0a0a 5761 7973 2074 6f20 amples..Ways to │ │ │ │ -00070cd0: 7573 6520 7477 6f4d 6f6e 6f6d 6961 6c73 use twoMonomials │ │ │ │ -00070ce0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -00070cf0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00070d00: 2022 7477 6f4d 6f6e 6f6d 6961 6c73 285a "twoMonomials(Z │ │ │ │ -00070d10: 5a2c 5a5a 2922 0a0a 466f 7220 7468 6520 Z,ZZ)"..For the │ │ │ │ -00070d20: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00070d30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00070d40: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00070d50: 7477 6f4d 6f6e 6f6d 6961 6c73 3a20 7477 twoMonomials: tw │ │ │ │ -00070d60: 6f4d 6f6e 6f6d 6961 6c73 2c20 6973 2061 oMonomials, is a │ │ │ │ -00070d70: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ -00070d80: 6e63 7469 6f6e 2077 6974 680a 6f70 7469 nction with.opti │ │ │ │ -00070d90: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ -00070da0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00070db0: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +00070c30: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ +00070c40: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +00070c50: 7465 2074 776f 4d6f 6e6f 6d69 616c 733a te twoMonomials: │ │ │ │ +00070c60: 2074 776f 4d6f 6e6f 6d69 616c 732c 202d twoMonomials, - │ │ │ │ +00070c70: 2d20 7461 6c6c 7920 7468 6520 7365 7175 - tally the sequ │ │ │ │ +00070c80: 656e 6365 7320 6f66 2042 5261 6e6b 7320 ences of BRanks │ │ │ │ +00070c90: 666f 720a 2020 2020 6365 7274 6169 6e20 for. certain │ │ │ │ +00070ca0: 6578 616d 706c 6573 0a0a 5761 7973 2074 examples..Ways t │ │ │ │ +00070cb0: 6f20 7573 6520 7477 6f4d 6f6e 6f6d 6961 o use twoMonomia │ │ │ │ +00070cc0: 6c73 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ls:.============ │ │ │ │ +00070cd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00070ce0: 202a 2022 7477 6f4d 6f6e 6f6d 6961 6c73 * "twoMonomials │ │ │ │ +00070cf0: 285a 5a2c 5a5a 2922 0a0a 466f 7220 7468 (ZZ,ZZ)"..For th │ │ │ │ +00070d00: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +00070d10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00070d20: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +00070d30: 6520 7477 6f4d 6f6e 6f6d 6961 6c73 3a20 e twoMonomials: │ │ │ │ +00070d40: 7477 6f4d 6f6e 6f6d 6961 6c73 2c20 6973 twoMonomials, is │ │ │ │ +00070d50: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00070d60: 6675 6e63 7469 6f6e 2077 6974 680a 6f70 function with.op │ │ │ │ +00070d70: 7469 6f6e 733a 2028 4d61 6361 756c 6179 tions: (Macaulay │ │ │ │ +00070d80: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +00070d90: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ +00070da0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +00070db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -00070e10: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00070e20: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00070e30: 696e 0a2f 6275 696c 642f 7265 7072 6f64 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250371.Node: mak │ │ │ │ +000712a0: 6548 6f6d 6f74 6f70 6965 737f 3235 3934 eHomotopies.2594 │ │ │ │ +000712b0: 3539 0a4e 6f64 653a 206d 616b 6548 6f6d 59.Node: makeHom │ │ │ │ +000712c0: 6f74 6f70 6965 7331 7f33 3437 3232 300a otopies1.347220. │ │ │ │ +000712d0: 4e6f 6465 3a20 6d61 6b65 486f 6d6f 746f Node: makeHomoto │ │ │ │ +000712e0: 7069 6573 4f6e 486f 6d6f 6c6f 6779 7f33 piesOnHomology.3 │ │ │ │ +000712f0: 3438 3835 340a 4e6f 6465 3a20 6d61 6b65 48854.Node: make │ │ │ │ +00071300: 4d6f 6475 6c65 7f33 3530 3439 390a 4e6f Module.350499.No │ │ │ │ +00071310: 6465 3a20 6d61 6b65 547f 3336 3131 3937 de: makeT.361197 │ │ │ │ +00071320: 0a4e 6f64 653a 206d 6174 7269 7846 6163 .Node: matrixFac │ │ │ │ +00071330: 746f 7269 7a61 7469 6f6e 7f33 3635 3339 torization.36539 │ │ │ │ +00071340: 360a 4e6f 6465 3a20 6d66 426f 756e 647f 6.Node: mfBound. │ │ │ │ +00071350: 3337 3432 3438 0a4e 6f64 653a 206d 6f64 374248.Node: mod │ │ │ │ +00071360: 756c 6541 7345 7874 7f33 3736 3134 300a uleAsExt.376140. │ │ │ │ +00071370: 4e6f 6465 3a20 6e65 7745 7874 7f33 3832 Node: newExt.382 │ │ │ │ +00071380: 3338 380a 4e6f 6465 3a20 6f64 6445 7874 388.Node: oddExt │ │ │ │ +00071390: 4d6f 6475 6c65 7f34 3130 3937 340a 4e6f Module.410974.No │ │ │ │ +000713a0: 6465 3a20 4f70 7469 6d69 736d 7f34 3137 de: Optimism.417 │ │ │ │ +000713b0: 3236 340a 4e6f 6465 3a20 4f75 7452 696e 264.Node: OutRin │ │ │ │ +000713c0: 677f 3431 3837 3930 0a4e 6f64 653a 2070 g.418790.Node: p │ │ │ │ +000713d0: 7369 4d61 7073 7f34 3230 3133 370a 4e6f siMaps.420137.No │ │ │ │ +000713e0: 6465 3a20 7265 6775 6c61 7269 7479 5365 de: regularitySe │ │ │ │ +000713f0: 7175 656e 6365 7f34 3231 3637 370a 4e6f quence.421677.No │ │ │ │ +00071400: 6465 3a20 5332 7f34 3235 3233 320a 4e6f de: S2.425232.No │ │ │ │ +00071410: 6465 3a20 5368 616d 6173 687f 3433 3633 de: Shamash.4363 │ │ │ │ +00071420: 3633 0a4e 6f64 653a 2073 706c 6974 7469 63.Node: splitti │ │ │ │ +00071430: 6e67 737f 3434 3431 3032 0a4e 6f64 653a ngs.444102.Node: │ │ │ │ +00071440: 2073 7461 626c 6548 6f6d 7f34 3439 3332 stableHom.44932 │ │ │ │ +00071450: 310a 4e6f 6465 3a20 7375 6d54 776f 4d6f 1.Node: sumTwoMo │ │ │ │ +00071460: 6e6f 6d69 616c 737f 3435 3035 3538 0a4e nomials.450558.N │ │ │ │ +00071470: 6f64 653a 2054 6174 6552 6573 6f6c 7574 ode: TateResolut │ │ │ │ +00071480: 696f 6e7f 3435 3330 3438 0a4e 6f64 653a ion.453048.Node: │ │ │ │ +00071490: 2074 656e 736f 7257 6974 6843 6f6d 706f tensorWithCompo │ │ │ │ +000714a0: 6e65 6e74 737f 3435 3734 3235 0a4e 6f64 nents.457425.Nod │ │ │ │ +000714b0: 653a 2074 6f41 7272 6179 7f34 3538 3837 e: toArray.45887 │ │ │ │ +000714c0: 320a 4e6f 6465 3a20 7477 6f4d 6f6e 6f6d 2.Node: twoMonom │ │ │ │ +000714d0: 6961 6c73 7f34 3539 3739 340a 1f0a 456e ials.459794...En │ │ │ │ +000714e0: 6420 5461 6720 5461 626c 650a d Tag Table. │ │ ├── ./usr/share/info/ConnectionMatrices.info.gz │ │ │ ├── ConnectionMatrices.info │ │ │ │ @@ -2415,31 +2415,31 @@ │ │ │ │ 000096e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000096f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00009700: 7c69 3920 3a20 656c 6170 7365 6454 696d |i9 : elapsedTim │ │ │ │ 00009710: 6520 4120 3d20 636f 6e6e 6563 7469 6f6e e A = connection │ │ │ │ 00009720: 4d61 7472 6963 6573 2049 3b20 2020 2020 Matrices I; │ │ │ │ 00009730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009740: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00009750: 7c20 2d2d 2032 2e38 3139 3873 2065 6c61 | -- 2.8198s ela │ │ │ │ -00009760: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ +00009750: 7c20 2d2d 2032 2e34 3834 3939 7320 656c | -- 2.48499s el │ │ │ │ +00009760: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00009770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009790: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000097a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000097b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000097c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000097d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000097e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000097f0: 7c69 3130 203a 2065 6c61 7073 6564 5469 |i10 : elapsedTi │ │ │ │ 00009800: 6d65 2061 7373 6572 7420 6973 496e 7465 me assert isInte │ │ │ │ 00009810: 6772 6162 6c65 2041 2020 2020 2020 2020 grable A │ │ │ │ 00009820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009830: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00009840: 7c20 2d2d 2035 2e39 3938 7320 656c 6170 | -- 5.998s elap │ │ │ │ -00009850: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ +00009840: 7c20 2d2d 2034 2e33 3334 3037 7320 656c | -- 4.33407s el │ │ │ │ +00009850: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00009860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009880: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00009890: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000098a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000098b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000098c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -4559,15 +4559,15 @@ │ │ │ │ 00011ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011d00: 2b0a 7c69 3134 203a 2065 6c61 7073 6564 +.|i14 : elapsed │ │ │ │ 00011d10: 5469 6d65 2067 203d 2067 6175 6765 4d61 Time g = gaugeMa │ │ │ │ 00011d20: 7472 6978 2849 2c20 4229 3b20 2020 2020 trix(I, B); │ │ │ │ 00011d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011d50: 7c0a 7c20 2d2d 202e 3639 3531 3373 2065 |.| -- .69513s e │ │ │ │ +00011d50: 7c0a 7c20 2d2d 202e 3531 3834 3373 2065 |.| -- .51843s e │ │ │ │ 00011d60: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00011d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011da0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4589,30 +4589,30 @@ │ │ │ │ 00011ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011ee0: 2b0a 7c69 3135 203a 2065 6c61 7073 6564 +.|i15 : elapsed │ │ │ │ 00011ef0: 5469 6d65 2041 3120 3d20 6761 7567 6554 Time A1 = gaugeT │ │ │ │ 00011f00: 7261 6e73 666f 726d 2867 2c20 4129 3b20 ransform(g, A); │ │ │ │ 00011f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011f30: 7c0a 7c20 2d2d 2031 2e34 3731 3937 7320 |.| -- 1.47197s │ │ │ │ +00011f30: 7c0a 7c20 2d2d 2031 2e31 3737 3137 7320 |.| -- 1.17717s │ │ │ │ 00011f40: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00011f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00011f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011fd0: 2b0a 7c69 3136 203a 2065 6c61 7073 6564 +.|i16 : elapsed │ │ │ │ 00011fe0: 5469 6d65 2061 7373 6572 7420 6973 496e Time assert isIn │ │ │ │ 00011ff0: 7465 6772 6162 6c65 2041 3120 2020 2020 tegrable A1 │ │ │ │ 00012000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012020: 7c0a 7c20 2d2d 202e 3937 3436 3539 7320 |.| -- .974659s │ │ │ │ +00012020: 7c0a 7c20 2d2d 202e 3936 3833 3332 7320 |.| -- .968332s │ │ │ │ 00012030: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00012040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012070: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00012080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5030,31 +5030,31 @@ │ │ │ │ 00013a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013a70: 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 656c -----+.|i19 : el │ │ │ │ 00013a80: 6170 7365 6454 696d 6520 4132 203d 2067 apsedTime A2 = g │ │ │ │ 00013a90: 6175 6765 5472 616e 7366 6f72 6d28 6368 augeTransform(ch │ │ │ │ 00013aa0: 616e 6765 4570 732c 2041 3129 3b20 2020 angeEps, A1); │ │ │ │ 00013ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013ac0: 2020 2020 207c 0a7c 202d 2d20 2e34 3630 |.| -- .460 │ │ │ │ -00013ad0: 3535 3273 2065 6c61 7073 6564 2020 2020 552s elapsed │ │ │ │ +00013ac0: 2020 2020 207c 0a7c 202d 2d20 2e33 3438 |.| -- .348 │ │ │ │ +00013ad0: 3438 3773 2065 6c61 7073 6564 2020 2020 487s elapsed │ │ │ │ 00013ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013b10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00013b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b60: 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20 656c -----+.|i20 : el │ │ │ │ 00013b70: 6170 7365 6454 696d 6520 6173 7365 7274 apsedTime assert │ │ │ │ 00013b80: 2069 7349 6e74 6567 7261 626c 6520 4132 isIntegrable A2 │ │ │ │ 00013b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013bb0: 2020 2020 207c 0a7c 202d 2d20 2e37 3938 |.| -- .798 │ │ │ │ -00013bc0: 3331 7320 656c 6170 7365 6420 2020 2020 31s elapsed │ │ │ │ +00013bb0: 2020 2020 207c 0a7c 202d 2d20 2e37 3430 |.| -- .740 │ │ │ │ +00013bc0: 3034 3273 2065 6c61 7073 6564 2020 2020 042s elapsed │ │ │ │ 00013bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013c00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00013c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5440,30 +5440,30 @@ │ │ │ │ 000153f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00015410: 6937 203a 2065 6c61 7073 6564 5469 6d65 i7 : elapsedTime │ │ │ │ 00015420: 2041 203d 2063 6f6e 6e65 6374 696f 6e4d A = connectionM │ │ │ │ 00015430: 6174 7269 6365 7320 493b 2020 2020 2020 atrices I; │ │ │ │ 00015440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015450: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00015460: 202d 2d20 2e31 3839 3933 3773 2065 6c61 -- .189937s ela │ │ │ │ +00015460: 202d 2d20 2e32 3037 3231 3973 2065 6c61 -- .207219s ela │ │ │ │ 00015470: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00015480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000154a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 000154b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000154c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000154d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000154e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000154f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00015500: 6938 203a 2065 6c61 7073 6564 5469 6d65 i8 : elapsedTime │ │ │ │ 00015510: 2061 7373 6572 7420 6973 496e 7465 6772 assert isIntegr │ │ │ │ 00015520: 6162 6c65 2041 2020 2020 2020 2020 2020 able A │ │ │ │ 00015530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00015550: 202d 2d20 2e31 3532 3332 3673 2065 6c61 -- .152326s ela │ │ │ │ +00015550: 202d 2d20 2e31 3832 3930 3373 2065 6c61 -- .182903s ela │ │ │ │ 00015560: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00015570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015590: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 000155a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000155b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000155c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/Cremona.info.gz │ │ │ ├── Cremona.info │ │ │ │ @@ -147,16 +147,16 @@ │ │ │ │ 00000920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000930: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2074 -------+.|i2 : t │ │ │ │ 00000940: 696d 6520 7068 6920 3d20 746f 4d61 7020 ime phi = toMap │ │ │ │ 00000950: 6d69 6e6f 7273 2833 2c6d 6174 7269 787b minors(3,matrix{ │ │ │ │ 00000960: 7b74 5f30 2e2e 745f 347d 2c7b 745f 312e {t_0..t_4},{t_1. │ │ │ │ 00000970: 2e74 5f35 7d2c 7b74 5f32 2e2e 745f 367d .t_5},{t_2..t_6} │ │ │ │ 00000980: 7d29 2020 2020 207c 0a7c 202d 2d20 7573 }) |.| -- us │ │ │ │ -00000990: 6564 2030 2e30 3034 3230 3331 3373 2028 ed 0.00420313s ( │ │ │ │ -000009a0: 6370 7529 3b20 302e 3030 3431 3939 3537 cpu); 0.00419957 │ │ │ │ +00000990: 6564 2030 2e30 3035 3739 3333 3373 2028 ed 0.00579333s ( │ │ │ │ +000009a0: 6370 7529 3b20 302e 3030 3537 3932 3037 cpu); 0.00579207 │ │ │ │ 000009b0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 000009c0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 000009d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000009e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000009f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -322,17 +322,17 @@ │ │ │ │ 00001410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001420: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 -------+.|i3 : t │ │ │ │ 00001430: 696d 6520 4a20 3d20 6b65 726e 656c 2870 ime J = kernel(p │ │ │ │ 00001440: 6869 2c32 2920 2020 2020 2020 2020 2020 hi,2) │ │ │ │ 00001450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001470: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001480: 6564 2030 2e31 3232 3738 3173 2028 6370 ed 0.122781s (cp │ │ │ │ -00001490: 7529 3b20 302e 3036 3536 3732 7320 2874 u); 0.065672s (t │ │ │ │ -000014a0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00001480: 6564 2030 2e31 3833 3939 3473 2028 6370 ed 0.183994s (cp │ │ │ │ +00001490: 7529 3b20 302e 3038 3635 3531 3373 2028 u); 0.0865513s ( │ │ │ │ +000014a0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 000014b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000014c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000014d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000014e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000014f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001510: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ @@ -387,18 +387,18 @@ │ │ │ │ 00001820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001830: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 -------+.|i4 : t │ │ │ │ 00001840: 696d 6520 6465 6772 6565 4d61 7020 7068 ime degreeMap ph │ │ │ │ 00001850: 6920 2020 2020 2020 2020 2020 2020 2020 i │ │ │ │ 00001860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001880: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001890: 6564 2030 2e30 3239 3239 3139 7320 2863 ed 0.0292919s (c │ │ │ │ -000018a0: 7075 293b 2030 2e30 3239 3239 3673 2028 pu); 0.029296s ( │ │ │ │ -000018b0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -000018c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00001890: 6564 2030 2e30 3336 3139 3234 7320 2863 ed 0.0361924s (c │ │ │ │ +000018a0: 7075 293b 2030 2e30 3336 3230 3036 7320 pu); 0.0362006s │ │ │ │ +000018b0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +000018c0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 000018d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000018e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000018f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001920: 2020 2020 2020 207c 0a7c 6f34 203d 2031 |.|o4 = 1 │ │ │ │ 00001930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -412,16 +412,16 @@ │ │ │ │ 000019b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000019c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 -------+.|i5 : t │ │ │ │ 000019d0: 696d 6520 7072 6f6a 6563 7469 7665 4465 ime projectiveDe │ │ │ │ 000019e0: 6772 6565 7320 7068 6920 2020 2020 2020 grees phi │ │ │ │ 000019f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a10: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001a20: 6564 2030 2e36 3137 3930 3773 2028 6370 ed 0.617907s (cp │ │ │ │ -00001a30: 7529 3b20 302e 3433 3637 3434 7320 2874 u); 0.436744s (t │ │ │ │ +00001a20: 6564 2030 2e37 3432 3933 3973 2028 6370 ed 0.742939s (cp │ │ │ │ +00001a30: 7529 3b20 302e 3532 3631 3435 7320 2874 u); 0.526145s (t │ │ │ │ 00001a40: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00001a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -447,18 +447,18 @@ │ │ │ │ 00001be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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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 3032 3239 3836 3673 2028 ed 0.00229866s ( │ │ │ │ +00001e80: 6564 2030 2e30 3032 3937 3430 3473 2028 ed 0.00297404s ( │ │ │ │ 00001e90: 6370 7520 2020 2020 2020 2020 2020 2020 cpu │ │ │ │ 00001ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -567,15 +567,15 @@ │ │ │ │ 00002360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002370: 2d2d 2d2d 2d2d 2d7c 0a7c 2c44 6f6d 696e -------|.|,Domin │ │ │ │ 00002380: 616e 743d 3e4a 2920 2020 2020 2020 2020 ant=>J) │ │ │ │ 00002390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023c0: 2020 2020 2020 207c 0a7c 293b 2030 2e30 |.|); 0.0 │ │ │ │ -000023d0: 3032 3239 3934 3573 2028 7468 7265 6164 0229945s (thread │ │ │ │ +000023d0: 3032 3938 3036 3973 2028 7468 7265 6164 0298069s (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 │ │ │ │ @@ -832,17 +832,17 @@ │ │ │ │ 000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003400: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074 -------+.|i8 : t │ │ │ │ 00003410: 696d 6520 7073 6920 3d20 696e 7665 7273 ime psi = invers │ │ │ │ 00003420: 654d 6170 2070 6869 2020 2020 2020 2020 eMap phi │ │ │ │ 00003430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003450: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00003460: 6564 2030 2e34 3631 3037 3773 2028 6370 ed 0.461077s (cp │ │ │ │ -00003470: 7529 3b20 302e 3339 3234 3432 7320 2874 u); 0.392442s (t │ │ │ │ -00003480: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00003460: 6564 2030 2e34 3238 3273 2028 6370 7529 ed 0.4282s (cpu) │ │ │ │ +00003470: 3b20 302e 3432 3832 3131 7320 2874 6872 ; 0.428211s (thr │ │ │ │ +00003480: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 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 │ │ │ │ 000034f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ @@ -1117,18 +1117,18 @@ │ │ │ │ 000045c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000045d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 -------+.|i9 : t │ │ │ │ 000045e0: 696d 6520 6973 496e 7665 7273 654d 6170 ime isInverseMap │ │ │ │ 000045f0: 2870 6869 2c70 7369 2920 2020 2020 2020 (phi,psi) │ │ │ │ 00004600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004620: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00004630: 6564 2030 2e30 3039 3936 3636 3473 2028 ed 0.00996664s ( │ │ │ │ -00004640: 6370 7529 3b20 302e 3030 3939 3731 3231 cpu); 0.00997121 │ │ │ │ -00004650: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -00004660: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +00004630: 6564 2030 2e30 3132 3439 3773 2028 6370 ed 0.012497s (cp │ │ │ │ +00004640: 7529 3b20 302e 3031 3235 3031 3373 2028 u); 0.0125013s ( │ │ │ │ +00004650: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00004660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004670: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00004680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000046a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000046b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000046c0: 2020 2020 2020 207c 0a7c 6f39 203d 2074 |.|o9 = t │ │ │ │ 000046d0: 7275 6520 2020 2020 2020 2020 2020 2020 rue │ │ │ │ @@ -1142,16 +1142,16 @@ │ │ │ │ 00004750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004760: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ 00004770: 7469 6d65 2064 6567 7265 654d 6170 2070 time degreeMap p │ │ │ │ 00004780: 7369 2020 2020 2020 2020 2020 2020 2020 si │ │ │ │ 00004790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000047a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000047b0: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -000047c0: 6564 2030 2e33 3237 3939 3973 2028 6370 ed 0.327999s (cp │ │ │ │ -000047d0: 7529 3b20 302e 3230 3334 3834 7320 2874 u); 0.203484s (t │ │ │ │ +000047c0: 6564 2030 2e36 3138 3439 3273 2028 6370 ed 0.618492s (cp │ │ │ │ +000047d0: 7529 3b20 302e 3330 3735 3939 7320 2874 u); 0.307599s (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 │ │ │ │ @@ -1167,17 +1167,17 @@ │ │ │ │ 000048e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000048f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ 00004900: 7469 6d65 2070 726f 6a65 6374 6976 6544 time projectiveD │ │ │ │ 00004910: 6567 7265 6573 2070 7369 2020 2020 2020 egrees psi │ │ │ │ 00004920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004940: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00004950: 6564 2035 2e31 3334 3634 7320 2863 7075 ed 5.13464s (cpu │ │ │ │ -00004960: 293b 2034 2e34 3433 3036 7320 2874 6872 ); 4.44306s (thr │ │ │ │ -00004970: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00004950: 6564 2036 2e36 3232 3773 2028 6370 7529 ed 6.6227s (cpu) │ │ │ │ +00004960: 3b20 362e 3039 3135 3573 2028 7468 7265 ; 6.09155s (thre │ │ │ │ +00004970: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00004980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004990: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000049a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049e0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20 |.|o11 = │ │ │ │ @@ -1214,17 +1214,17 @@ │ │ │ │ 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: 3235 3837 7320 2863 7075 293b 2030 2e30 2587s (cpu); 0.0 │ │ │ │ -00004c50: 3032 3235 3938 7320 2874 6872 6561 6429 022598s (thread) │ │ │ │ -00004c60: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +00004c40: 3731 3434 3373 2028 6370 7529 3b20 302e 71443s (cpu); 0. │ │ │ │ +00004c50: 3030 3237 3139 3139 7320 2874 6872 6561 00271919s (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 | │ │ │ │ 00004cd0: 0a7c 6f31 3220 3d20 2d2d 2072 6174 696f .|o12 = -- ratio │ │ │ │ @@ -1493,18 +1493,18 @@ │ │ │ │ 00005d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00005d60: 0a7c 6931 3320 3a20 7469 6d65 2070 6869 .|i13 : time phi │ │ │ │ 00005d70: 203d 2072 6174 696f 6e61 6c4d 6170 2870 = rationalMap(p │ │ │ │ 00005d80: 6869 2c44 6f6d 696e 616e 743d 3e32 2920 hi,Dominant=>2) │ │ │ │ 00005d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005da0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00005db0: 0a7c 202d 2d20 7573 6564 2030 2e31 3533 .| -- used 0.153 │ │ │ │ -00005dc0: 3137 3473 2028 6370 7529 3b20 302e 3038 174s (cpu); 0.08 │ │ │ │ -00005dd0: 3635 3639 3373 2028 7468 7265 6164 293b 65693s (thread); │ │ │ │ -00005de0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e32 3132 .| -- used 0.212 │ │ │ │ +00005dc0: 3135 3973 2028 6370 7529 3b20 302e 3130 159s (cpu); 0.10 │ │ │ │ +00005dd0: 3038 3638 7320 2874 6872 6561 6429 3b20 0868s (thread); │ │ │ │ +00005de0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00005df0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00005e00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00005e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00005e50: 0a7c 6f31 3320 3d20 2d2d 2072 6174 696f .|o13 = -- ratio │ │ │ │ @@ -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 3033 .| -- used 0.503 │ │ │ │ -00008700: 3132 3773 2028 6370 7529 3b20 302e 3432 127s (cpu); 0.42 │ │ │ │ -00008710: 3238 3831 7320 2874 6872 6561 6429 3b20 2881s (thread); │ │ │ │ +000086f0: 0a7c 202d 2d20 7573 6564 2030 2e34 3930 .| -- used 0.490 │ │ │ │ +00008700: 3834 3573 2028 6370 7529 3b20 302e 3439 845s (cpu); 0.49 │ │ │ │ +00008710: 3038 3531 7320 2874 6872 6561 6429 3b20 0851s (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 | │ │ │ │ @@ -2708,18 +2708,18 @@ │ │ │ │ 0000a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000a950: 0a7c 6931 3520 3a20 7469 6d65 2064 6567 .|i15 : time deg │ │ │ │ 0000a960: 7265 6573 2070 6869 5e28 2d31 2920 2020 rees phi^(-1) │ │ │ │ 0000a970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a990: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e33 3133 .| -- used 0.313 │ │ │ │ -0000a9b0: 3139 3873 2028 6370 7529 3b20 302e 3235 198s (cpu); 0.25 │ │ │ │ -0000a9c0: 3937 3138 7320 2874 6872 6561 6429 3b20 9718s (thread); │ │ │ │ -0000a9d0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e34 3830 .| -- used 0.480 │ │ │ │ +0000a9b0: 3438 3473 2028 6370 7529 3b20 302e 3336 484s (cpu); 0.36 │ │ │ │ +0000a9c0: 3730 3673 2028 7468 7265 6164 293b 2030 706s (thread); 0 │ │ │ │ +0000a9d0: 7320 2867 6329 2020 2020 2020 2020 2020 s (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 | │ │ │ │ 0000aa40: 0a7c 6f31 3520 3d20 7b35 2c20 3135 2c20 .|o15 = {5, 15, │ │ │ │ @@ -2743,17 +2743,17 @@ │ │ │ │ 0000ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000ab80: 0a7c 6931 3620 3a20 7469 6d65 2064 6567 .|i16 : time deg │ │ │ │ 0000ab90: 7265 6573 2070 6869 2020 2020 2020 2020 rees phi │ │ │ │ 0000aba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000abb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000abc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3138 .| -- used 0.018 │ │ │ │ -0000abe0: 3231 3134 7320 2863 7075 293b 2030 2e30 2114s (cpu); 0.0 │ │ │ │ -0000abf0: 3137 3732 3139 7320 2874 6872 6561 6429 177219s (thread) │ │ │ │ +0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3332 .| -- used 0.032 │ │ │ │ +0000abe0: 3330 3231 7320 2863 7075 293b 2030 2e30 3021s (cpu); 0.0 │ │ │ │ +0000abf0: 3230 3231 3138 7320 2874 6872 6561 6429 202118s (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 | │ │ │ │ @@ -2779,16 +2779,16 @@ │ │ │ │ 0000ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000adb0: 0a7c 6931 3720 3a20 7469 6d65 2064 6573 .|i17 : time des │ │ │ │ 0000adc0: 6372 6962 6520 7068 6920 2020 2020 2020 cribe phi │ │ │ │ 0000add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000adf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3033 .| -- used 0.003 │ │ │ │ -0000ae10: 3135 3530 3473 2028 6370 7529 3b20 302e 15504s (cpu); 0. │ │ │ │ -0000ae20: 3030 3331 3535 3735 7320 2874 6872 6561 00315575s (threa │ │ │ │ +0000ae10: 3939 3336 3673 2028 6370 7529 3b20 302e 99366s (cpu); 0. │ │ │ │ +0000ae20: 3030 3339 3632 3634 7320 2874 6872 6561 00396264s (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 3130 .| -- used 0.010 │ │ │ │ -0000b220: 3039 3373 2028 6370 7529 3b20 302e 3031 093s (cpu); 0.01 │ │ │ │ -0000b230: 3030 3933 3773 2028 7468 7265 6164 293b 00937s (thread); │ │ │ │ -0000b240: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3132 .| -- used 0.012 │ │ │ │ +0000b220: 3137 3039 7320 2863 7075 293b 2030 2e30 1709s (cpu); 0.0 │ │ │ │ +0000b230: 3132 3133 3037 7320 2874 6872 6561 6429 121307s (thread) │ │ │ │ +0000b240: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0000b250: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b260: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b2a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b2b0: 0a7c 6f31 3820 3d20 7261 7469 6f6e 616c .|o18 = rational │ │ │ │ @@ -2923,18 +2923,18 @@ │ │ │ │ 0000b6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000b6c0: 0a7c 6931 3920 3a20 7469 6d65 2028 662c .|i19 : time (f, │ │ │ │ 0000b6d0: 6729 203d 2067 7261 7068 2070 6869 5e2d g) = graph phi^- │ │ │ │ 0000b6e0: 313b 2066 3b20 2020 2020 2020 2020 2020 1; f; │ │ │ │ 0000b6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b700: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3039 .| -- used 0.009 │ │ │ │ -0000b720: 3739 3130 3773 2028 6370 7529 3b20 302e 79107s (cpu); 0. │ │ │ │ -0000b730: 3030 3937 3931 3739 7320 2874 6872 6561 00979179s (threa │ │ │ │ -0000b740: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3131 .| -- used 0.011 │ │ │ │ +0000b720: 3137 3038 7320 2863 7075 293b 2030 2e30 1708s (cpu); 0.0 │ │ │ │ +0000b730: 3131 3138 3138 7320 2874 6872 6561 6429 111818s (thread) │ │ │ │ +0000b740: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0000b750: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b760: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b7a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b7b0: 0a7c 6f32 3020 3a20 4d75 6c74 6968 6f6d .|o20 : Multihom │ │ │ │ @@ -2958,18 +2958,18 @@ │ │ │ │ 0000b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000b8f0: 0a7c 6932 3120 3a20 7469 6d65 2064 6567 .|i21 : time deg │ │ │ │ 0000b900: 7265 6573 2066 2020 2020 2020 2020 2020 rees f │ │ │ │ 0000b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b930: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000b940: 0a7c 202d 2d20 7573 6564 2031 2e32 3836 .| -- used 1.286 │ │ │ │ -0000b950: 3535 7320 2863 7075 293b 2030 2e39 3233 55s (cpu); 0.923 │ │ │ │ -0000b960: 3232 3473 2028 7468 7265 6164 293b 2030 224s (thread); 0 │ │ │ │ -0000b970: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ +0000b940: 0a7c 202d 2d20 7573 6564 2031 2e32 3833 .| -- used 1.283 │ │ │ │ +0000b950: 3836 7320 2863 7075 293b 2031 2e30 3434 86s (cpu); 1.044 │ │ │ │ +0000b960: 3533 7320 2874 6872 6561 6429 3b20 3073 53s (thread); 0s │ │ │ │ +0000b970: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0000b980: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b990: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b9e0: 0a7c 6f32 3120 3d20 7b39 3034 2c20 3530 .|o21 = {904, 50 │ │ │ │ @@ -2993,18 +2993,18 @@ │ │ │ │ 0000bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000bb20: 0a7c 6932 3220 3a20 7469 6d65 2064 6567 .|i22 : time deg │ │ │ │ 0000bb30: 7265 6520 6620 2020 2020 2020 2020 2020 ree f │ │ │ │ 0000bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000bb70: 0a7c 202d 2d20 7573 6564 2031 2e36 3733 .| -- used 1.673 │ │ │ │ -0000bb80: 3265 2d30 3573 2028 6370 7529 3b20 312e 2e-05s (cpu); 1. │ │ │ │ -0000bb90: 3633 3331 652d 3035 7320 2874 6872 6561 6331e-05s (threa │ │ │ │ -0000bba0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0000bb70: 0a7c 202d 2d20 7573 6564 2032 2e38 3731 .| -- used 2.871 │ │ │ │ +0000bb80: 3565 2d30 3573 2028 6370 7529 3b20 322e 5e-05s (cpu); 2. │ │ │ │ +0000bb90: 3435 3965 2d30 3573 2028 7468 7265 6164 459e-05s (thread │ │ │ │ +0000bba0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0000bbb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bc10: 0a7c 6f32 3220 3d20 3120 2020 2020 2020 .|o22 = 1 │ │ │ │ @@ -3019,16 +3019,16 @@ │ │ │ │ 0000bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000bcb0: 0a7c 6932 3320 3a20 7469 6d65 2064 6573 .|i23 : time des │ │ │ │ 0000bcc0: 6372 6962 6520 6620 2020 2020 2020 2020 cribe f │ │ │ │ 0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bcf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bd00: 0a7c 202d 2d20 7573 6564 2030 2e30 3031 .| -- used 0.001 │ │ │ │ -0000bd10: 3731 3937 3873 2028 6370 7529 3b20 302e 71978s (cpu); 0. │ │ │ │ -0000bd20: 3030 3137 3330 3839 7320 2874 6872 6561 00173089s (threa │ │ │ │ +0000bd10: 3733 3733 3273 2028 6370 7529 3b20 302e 73732s (cpu); 0. │ │ │ │ +0000bd20: 3030 3137 3433 3237 7320 2874 6872 6561 00174327s (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 | │ │ │ │ @@ -4676,17 +4676,17 @@ │ │ │ │ 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 3336 3031 3773 2028 6370 7529 3b20 0436017s (cpu); │ │ │ │ -000124b0: 302e 3030 3034 3239 3830 3673 2028 7468 0.000429806s (th │ │ │ │ -000124c0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +000124a0: 3034 3634 3873 2028 6370 7529 3b20 302e 04648s (cpu); 0. │ │ │ │ +000124b0: 3030 3034 3539 3130 3773 2028 7468 7265 000459107s (thre │ │ │ │ +000124c0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 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 │ │ │ │ 00012530: 7c0a 7c6f 3420 3d20 2d2d 2072 6174 696f |.|o4 = -- ratio │ │ │ │ @@ -4746,16 +4746,16 @@ │ │ │ │ 00012890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000128a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000128b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000128c0: 6935 203a 2074 696d 6520 7072 6f6a 6563 i5 : time projec │ │ │ │ 000128d0: 7469 7665 4465 6772 6565 7328 7073 692c tiveDegrees(psi, │ │ │ │ 000128e0: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ 000128f0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00012900: 2d20 7573 6564 2030 2e32 3832 3832 3973 - used 0.282829s │ │ │ │ -00012910: 2028 6370 7529 3b20 302e 3137 3130 3234 (cpu); 0.171024 │ │ │ │ +00012900: 2d20 7573 6564 2030 2e34 3230 3639 3673 - used 0.420696s │ │ │ │ +00012910: 2028 6370 7529 3b20 302e 3232 3131 3831 (cpu); 0.221181 │ │ │ │ 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 │ │ │ │ @@ -4766,17 +4766,17 @@ │ │ │ │ 000129d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000129e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000129f0: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520 ---+.|i6 : time │ │ │ │ 00012a00: 7261 7469 6f6e 616c 4d61 7020 7073 6920 rationalMap psi │ │ │ │ 00012a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e35 |.| -- used 0.5 │ │ │ │ -00012a40: 3832 3334 7320 2863 7075 293b 2030 2e34 8234s (cpu); 0.4 │ │ │ │ -00012a50: 3532 3631 3473 2028 7468 7265 6164 293b 52614s (thread); │ │ │ │ -00012a60: 2030 7320 2867 6329 2020 2020 2020 207c 0s (gc) | │ │ │ │ +00012a40: 3032 3734 3973 2028 6370 7529 3b20 302e 02749s (cpu); 0. │ │ │ │ +00012a50: 3339 3936 3137 7320 2874 6872 6561 6429 399617s (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 -- │ │ │ │ 00012ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5189,17 +5189,17 @@ │ │ │ │ 00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a --------+.|i14 : │ │ │ │ 00014460: 2074 696d 6520 5420 3d20 6162 7374 7261 time T = abstra │ │ │ │ 00014470: 6374 5261 7469 6f6e 616c 4d61 7028 492c ctRationalMap(I, │ │ │ │ 00014480: 224f 4144 5022 2920 2020 2020 2020 2020 "OADP") │ │ │ │ 00014490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144a0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -000144b0: 6420 302e 3135 3333 3039 7320 2863 7075 d 0.153309s (cpu │ │ │ │ -000144c0: 293b 2030 2e30 3734 3637 3131 7320 2874 ); 0.0746711s (t │ │ │ │ -000144d0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +000144b0: 6420 302e 3230 3930 3732 7320 2863 7075 d 0.209072s (cpu │ │ │ │ +000144c0: 293b 2030 2e31 3131 3336 3973 2028 7468 ); 0.111369s (th │ │ │ │ +000144d0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 000144e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00014500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014540: 2020 7c0a 7c6f 3134 203d 202d 2d20 7261 |.|o14 = -- ra │ │ │ │ @@ -5265,16 +5265,16 @@ │ │ │ │ 00014900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014930: 2d2d 2b0a 7c69 3135 203a 2074 696d 6520 --+.|i15 : time │ │ │ │ 00014940: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ 00014950: 7328 542c 3229 2020 2020 2020 2020 2020 s(T,2) │ │ │ │ 00014960: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00014970: 7365 6420 332e 3634 3838 3173 2028 6370 sed 3.64881s (cp │ │ │ │ -00014980: 7529 3b20 312e 3936 3237 3373 2028 7468 u); 1.96273s (th │ │ │ │ +00014970: 7365 6420 352e 3236 3636 3173 2028 6370 sed 5.26661s (cp │ │ │ │ +00014980: 7529 3b20 322e 3436 3630 3673 2028 7468 u); 2.46606s (th │ │ │ │ 00014990: 7265 6164 293b 2030 7320 2867 6329 7c0a read); 0s (gc)|. │ │ │ │ 000149a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000149b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149d0: 2020 2020 7c0a 7c6f 3135 203d 2033 2020 |.|o15 = 3 │ │ │ │ 000149e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5291,16 +5291,16 @@ │ │ │ │ 00014aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 00014ad0: 3620 3a20 7469 6d65 2054 3220 3d20 5420 6 : time T2 = T │ │ │ │ 00014ae0: 2a20 5420 2020 2020 2020 2020 2020 2020 * T │ │ │ │ 00014af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b00: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00014b10: 7365 6420 342e 3233 3339 652d 3035 7320 sed 4.2339e-05s │ │ │ │ -00014b20: 2863 7075 293b 2034 2e32 3138 3965 2d30 (cpu); 4.2189e-0 │ │ │ │ +00014b10: 7365 6420 332e 3831 3538 652d 3035 7320 sed 3.8158e-05s │ │ │ │ +00014b20: 2863 7075 293b 2033 2e36 3639 3465 2d30 (cpu); 3.6694e-0 │ │ │ │ 00014b30: 3573 2028 7468 7265 6164 293b 2030 7320 5s (thread); 0s │ │ │ │ 00014b40: 2867 6329 207c 0a7c 2020 2020 2020 2020 (gc) |.| │ │ │ │ 00014b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b80: 2020 7c0a 7c6f 3136 203d 202d 2d20 7261 |.|o16 = -- ra │ │ │ │ 00014b90: 7469 6f6e 616c 206d 6170 202d 2d20 2020 tional map -- │ │ │ │ @@ -5344,18 +5344,18 @@ │ │ │ │ 00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014e20: 2d2b 0a7c 6931 3720 3a20 7469 6d65 2070 -+.|i17 : time p │ │ │ │ 00014e30: 726f 6a65 6374 6976 6544 6567 7265 6573 rojectiveDegrees │ │ │ │ 00014e40: 2854 322c 3229 2020 2020 2020 2020 2020 (T2,2) │ │ │ │ 00014e50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014e60: 7c20 2d2d 2075 7365 6420 352e 3831 3035 | -- used 5.8105 │ │ │ │ -00014e70: 3973 2028 6370 7529 3b20 332e 3039 3431 9s (cpu); 3.0941 │ │ │ │ -00014e80: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ -00014e90: 2867 6329 2020 2020 2020 207c 0a7c 2020 (gc) |.| │ │ │ │ +00014e60: 7c20 2d2d 2075 7365 6420 382e 3334 3330 | -- used 8.3430 │ │ │ │ +00014e70: 3173 2028 6370 7529 3b20 332e 3937 3033 1s (cpu); 3.9703 │ │ │ │ +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 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 2020 │ │ │ │ @@ -5430,17 +5430,17 @@ │ │ │ │ 00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 00015380: 3120 3a20 7469 6d65 2066 203d 2072 6174 1 : time f = rat │ │ │ │ 00015390: 696f 6e61 6c4d 6170 2054 2020 2020 2020 ionalMap T │ │ │ │ 000153a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000153c0: 0a7c 202d 2d20 7573 6564 2034 2e35 3639 .| -- used 4.569 │ │ │ │ -000153d0: 3836 7320 2863 7075 293b 2032 2e35 3731 86s (cpu); 2.571 │ │ │ │ -000153e0: 3235 7320 2874 6872 6561 6429 3b20 3073 25s (thread); 0s │ │ │ │ +000153c0: 0a7c 202d 2d20 7573 6564 2036 2e39 3632 .| -- used 6.962 │ │ │ │ +000153d0: 3632 7320 2863 7075 293b 2033 2e34 3336 62s (cpu); 3.436 │ │ │ │ +000153e0: 3431 7320 2874 6872 6561 6429 3b20 3073 41s (thread); 0s │ │ │ │ 000153f0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00015400: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00015410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015440: 2020 2020 2020 207c 0a7c 6f32 3120 3d20 |.|o21 = │ │ │ │ 00015450: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020 -- rational map │ │ │ │ @@ -6678,18 +6678,18 @@ │ │ │ │ 0001a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a160: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 0001a170: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ 0001a180: 6572 7365 4d61 703a 2073 7465 7020 3130 erseMap: step 10 │ │ │ │ 0001a190: 206f 6620 3130 2020 2020 2020 2020 2020 of 10 │ │ │ │ 0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a1b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001a1c0: 2d2d 2075 7365 6420 302e 3233 3836 3833 -- used 0.238683 │ │ │ │ -0001a1d0: 7320 2863 7075 293b 2030 2e31 3934 3436 s (cpu); 0.19446 │ │ │ │ -0001a1e0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -0001a1f0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +0001a1c0: 2d2d 2075 7365 6420 302e 3335 3036 3137 -- used 0.350617 │ │ │ │ +0001a1d0: 7320 2863 7075 293b 2030 2e32 3533 3735 s (cpu); 0.25375 │ │ │ │ +0001a1e0: 3873 2028 7468 7265 6164 293b 2030 7320 8s (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 │ │ │ │ 0001a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a250: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0001a260: 3320 3d20 2d2d 2072 6174 696f 6e61 6c20 3 = -- rational │ │ │ │ @@ -8043,17 +8043,17 @@ │ │ │ │ 0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 0001f6c0: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ 0001f6d0: 6572 7365 4d61 703a 2073 7465 7020 3320 erseMap: step 3 │ │ │ │ 0001f6e0: 6f66 2033 2020 2020 2020 2020 2020 2020 of 3 │ │ │ │ 0001f6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f700: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001f710: 2d2d 2075 7365 6420 302e 3138 3538 3137 -- used 0.185817 │ │ │ │ -0001f720: 7320 2863 7075 293b 2030 2e31 3530 3034 s (cpu); 0.15004 │ │ │ │ -0001f730: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ +0001f710: 2d2d 2075 7365 6420 302e 3238 3638 3031 -- used 0.286801 │ │ │ │ +0001f720: 7320 2863 7075 293b 2030 2e31 3834 3439 s (cpu); 0.18449 │ │ │ │ +0001f730: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ 0001f740: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0001f750: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -10405,16 +10405,16 @@ │ │ │ │ 00028a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a50: 2020 2020 207c 0a7c 2d2d 2061 7070 726f |.|-- appro │ │ │ │ 00028a60: 7869 6d61 7465 496e 7665 7273 654d 6170 ximateInverseMap │ │ │ │ 00028a70: 3a20 7374 6570 2033 206f 6620 3320 2020 : step 3 of 3 │ │ │ │ 00028a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028aa0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00028ab0: 2032 2e32 3331 3939 7320 2863 7075 293b 2.23199s (cpu); │ │ │ │ -00028ac0: 2031 2e37 3233 3232 7320 2874 6872 6561 1.72322s (threa │ │ │ │ +00028ab0: 2032 2e32 3132 3433 7320 2863 7075 293b 2.21243s (cpu); │ │ │ │ +00028ac0: 2031 2e38 3235 3931 7320 2874 6872 6561 1.82591s (threa │ │ │ │ 00028ad0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028af0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00028b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -11710,16 +11710,16 @@ │ │ │ │ 0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbe0: 2020 2020 207c 0a7c 4365 7274 6966 793a |.|Certify: │ │ │ │ 0002dbf0: 206f 7574 7075 7420 6365 7274 6966 6965 output certifie │ │ │ │ 0002dc00: 6421 2020 2020 2020 2020 2020 2020 2020 d! │ │ │ │ 0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc30: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0002dc40: 2033 2e31 3238 3533 7320 2863 7075 293b 3.12853s (cpu); │ │ │ │ -0002dc50: 2032 2e35 3237 3338 7320 2874 6872 6561 2.52738s (threa │ │ │ │ +0002dc40: 2033 2e32 3738 3031 7320 2863 7075 293b 3.27801s (cpu); │ │ │ │ +0002dc50: 2032 2e38 3032 3335 7320 2874 6872 6561 2.80235s (threa │ │ │ │ 0002dc60: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0002dc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13366,17 +13366,17 @@ │ │ │ │ 00034350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034380: 2b0a 7c69 3320 3a20 7469 6d65 2043 6865 +.|i3 : time Che │ │ │ │ 00034390: 726e 5363 6877 6172 747a 4d61 6350 6865 rnSchwartzMacPhe │ │ │ │ 000343a0: 7273 6f6e 2043 2020 2020 2020 2020 2020 rson C │ │ │ │ 000343b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000343c0: 7573 6564 2032 2e31 3333 3137 7320 2863 used 2.13317s (c │ │ │ │ -000343d0: 7075 293b 2031 2e31 3537 3173 2028 7468 pu); 1.1571s (th │ │ │ │ -000343e0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +000343c0: 7573 6564 2032 2e39 3535 3335 7320 2863 used 2.95535s (c │ │ │ │ +000343d0: 7075 293b 2031 2e34 3233 3135 7320 2874 pu); 1.42315s (t │ │ │ │ +000343e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 000343f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00034400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00034430: 2020 2020 2034 2020 2020 2033 2020 2020 4 3 │ │ │ │ 00034440: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00034450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13409,17 +13409,17 @@ │ │ │ │ 00034600: 4368 6572 6e53 6368 7761 7274 7a4d 6163 ChernSchwartzMac │ │ │ │ 00034610: 5068 6572 736f 6e28 432c 4365 7274 6966 Pherson(C,Certif │ │ │ │ 00034620: 793d 3e74 7275 6529 2020 2020 7c0a 7c43 y=>true) |.|C │ │ │ │ 00034630: 6572 7469 6679 3a20 6f75 7470 7574 2063 ertify: output c │ │ │ │ 00034640: 6572 7469 6669 6564 2120 2020 2020 2020 ertified! │ │ │ │ 00034650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034660: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00034670: 2031 2e32 3838 3834 7320 2863 7075 293b 1.28884s (cpu); │ │ │ │ -00034680: 2030 2e39 3038 3838 3273 2028 7468 7265 0.908882s (thre │ │ │ │ -00034690: 6164 293b 2030 7320 2867 6329 2020 7c0a ad); 0s (gc) |. │ │ │ │ +00034670: 2031 2e37 3833 3234 7320 2863 7075 293b 1.78324s (cpu); │ │ │ │ +00034680: 2031 2e31 3437 3335 7320 2874 6872 6561 1.14735s (threa │ │ │ │ +00034690: 6429 3b20 3073 2028 6763 2920 2020 7c0a d); 0s (gc) |. │ │ │ │ 000346a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000346b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000346c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000346d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000346e0: 2034 2020 2020 2033 2020 2020 2032 2020 4 3 2 │ │ │ │ 000346f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13619,16 +13619,16 @@ │ │ │ │ 00035320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00035330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 -----------+.|i9 │ │ │ │ 00035340: 203a 2074 696d 6520 4368 6572 6e43 6c61 : time ChernCla │ │ │ │ 00035350: 7373 2047 2020 2020 2020 2020 2020 2020 ss G │ │ │ │ 00035360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035380: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00035390: 2d20 7573 6564 2030 2e33 3338 3037 3473 - used 0.338074s │ │ │ │ -000353a0: 2028 6370 7529 3b20 302e 3139 3237 3538 (cpu); 0.192758 │ │ │ │ +00035390: 2d20 7573 6564 2030 2e34 3432 3339 3973 - used 0.442399s │ │ │ │ +000353a0: 2028 6370 7529 3b20 302e 3231 3534 3531 (cpu); 0.215451 │ │ │ │ 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 │ │ │ │ @@ -13679,17 +13679,17 @@ │ │ │ │ 000356e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000356f0: 2020 2020 2020 2020 2020 207c 0a7c 4365 |.|Ce │ │ │ │ 00035700: 7274 6966 793a 206f 7574 7075 7420 6365 rtify: output ce │ │ │ │ 00035710: 7274 6966 6965 6421 2020 2020 2020 2020 rtified! │ │ │ │ 00035720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035740: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00035750: 2d20 7573 6564 2030 2e31 3136 3832 3773 - used 0.116827s │ │ │ │ -00035760: 2028 6370 7529 3b20 302e 3034 3430 3036 (cpu); 0.044006 │ │ │ │ -00035770: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ +00035750: 2d20 7573 6564 2030 2e32 3331 3032 3673 - used 0.231026s │ │ │ │ +00035760: 2028 6370 7529 3b20 302e 3035 3838 3933 (cpu); 0.058893 │ │ │ │ +00035770: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ 00035780: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00035790: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000357a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ @@ -16336,16 +16336,16 @@ │ │ │ │ 0003fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003fd00: 2d2b 0a7c 6935 203a 2074 696d 6520 6465 -+.|i5 : time de │ │ │ │ 0003fd10: 6772 6565 4d61 7020 7068 6920 2020 2020 greeMap phi │ │ │ │ 0003fd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fd50: 207c 0a7c 202d 2d20 7573 6564 2030 2e30 |.| -- used 0.0 │ │ │ │ -0003fd60: 3434 3939 3338 7320 2863 7075 293b 2030 449938s (cpu); 0 │ │ │ │ -0003fd70: 2e30 3434 3936 3837 7320 2874 6872 6561 .0449687s (threa │ │ │ │ +0003fd60: 3535 3937 3434 7320 2863 7075 293b 2030 559744s (cpu); 0 │ │ │ │ +0003fd70: 2e30 3535 3937 3339 7320 2874 6872 6561 .0559739s (threa │ │ │ │ 0003fd80: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0003fd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fda0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003fdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -17510,18 +17510,18 @@ │ │ │ │ 00044650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044670: 2d2b 0a7c 6937 203a 2074 696d 6520 6465 -+.|i7 : time de │ │ │ │ 00044680: 6772 6565 4d61 7020 7068 6927 2020 2020 greeMap phi' │ │ │ │ 00044690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000446a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000446b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000446c0: 207c 0a7c 202d 2d20 7573 6564 2031 2e31 |.| -- used 1.1 │ │ │ │ -000446d0: 3238 3273 2028 6370 7529 3b20 302e 3638 282s (cpu); 0.68 │ │ │ │ -000446e0: 3632 3435 7320 2874 6872 6561 6429 3b20 6245s (thread); │ │ │ │ -000446f0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +000446c0: 207c 0a7c 202d 2d20 7573 6564 2031 2e35 |.| -- used 1.5 │ │ │ │ +000446d0: 3536 3833 7320 2863 7075 293b 2030 2e38 5683s (cpu); 0.8 │ │ │ │ +000446e0: 3832 3131 3773 2028 7468 7265 6164 293b 82117s (thread); │ │ │ │ +000446f0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00044700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044710: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00044720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044760: 207c 0a7c 6f37 203d 2031 3420 2020 2020 |.|o7 = 14 │ │ │ │ @@ -18325,16 +18325,16 @@ │ │ │ │ 00047940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00047950: 2d2d 2b0a 7c69 3220 3a20 7469 6d65 2045 --+.|i2 : time E │ │ │ │ 00047960: 756c 6572 4368 6172 6163 7465 7269 7374 ulerCharacterist │ │ │ │ 00047970: 6963 2049 2020 2020 2020 2020 2020 2020 ic I │ │ │ │ 00047980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000479a0: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -000479b0: 3239 3139 3673 2028 6370 7529 3b20 302e 29196s (cpu); 0. │ │ │ │ -000479c0: 3136 3333 3638 7320 2874 6872 6561 6429 163368s (thread) │ │ │ │ +000479b0: 3430 3635 3033 7320 2863 7075 293b 2030 406503s (cpu); 0 │ │ │ │ +000479c0: 2e32 3137 3533 7320 2874 6872 6561 6429 .21753s (thread) │ │ │ │ 000479d0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 000479e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000479f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00047a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -18355,16 +18355,16 @@ │ │ │ │ 00047b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b30: 2020 7c0a 7c43 6572 7469 6679 3a20 6f75 |.|Certify: ou │ │ │ │ 00047b40: 7470 7574 2063 6572 7469 6669 6564 2120 tput certified! │ │ │ │ 00047b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b80: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00047b90: 3031 3230 3232 3773 2028 6370 7529 3b20 0120227s (cpu); │ │ │ │ -00047ba0: 302e 3031 3135 3331 3773 2028 7468 7265 0.0115317s (thre │ │ │ │ +00047b90: 3033 3634 3932 3173 2028 6370 7529 3b20 0364921s (cpu); │ │ │ │ +00047ba0: 302e 3031 3632 3631 3773 2028 7468 7265 0.0162617s (thre │ │ │ │ 00047bb0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00047bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047bd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00047be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -19033,18 +19033,18 @@ │ │ │ │ 0004a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0004a5a0: 3420 3a20 7469 6d65 2066 6f72 6365 496d 4 : time forceIm │ │ │ │ 0004a5b0: 6167 6528 5068 692c 6964 6561 6c20 305f age(Phi,ideal 0_ │ │ │ │ 0004a5c0: 2874 6172 6765 7420 5068 6929 2920 2020 (target Phi)) │ │ │ │ 0004a5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a5e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0004a5f0: 2d2d 2075 7365 6420 302e 3030 3036 3837 -- used 0.000687 │ │ │ │ -0004a600: 3736 7320 2863 7075 293b 2030 2e30 3030 76s (cpu); 0.000 │ │ │ │ -0004a610: 3637 3935 3134 7320 2874 6872 6561 6429 679514s (thread) │ │ │ │ -0004a620: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0004a5f0: 2d2d 2075 7365 6420 302e 3030 3038 3834 -- used 0.000884 │ │ │ │ +0004a600: 3033 3573 2028 6370 7529 3b20 302e 3030 035s (cpu); 0.00 │ │ │ │ +0004a610: 3038 3736 3133 3673 2028 7468 7265 6164 0876136s (thread │ │ │ │ +0004a620: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0004a630: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0004a640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0004a690: 3520 3a20 5068 693b 2020 2020 2020 2020 5 : Phi; │ │ │ │ @@ -19645,16 +19645,16 @@ │ │ │ │ 0004cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004cbd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 -------+.|i3 : t │ │ │ │ 0004cbe0: 696d 6520 2870 312c 7032 2920 3d20 6772 ime (p1,p2) = gr │ │ │ │ 0004cbf0: 6170 6820 7068 693b 2020 2020 2020 2020 aph phi; │ │ │ │ 0004cc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004cc20: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -0004cc30: 6564 2030 2e30 3136 3238 3032 7320 2863 ed 0.0162802s (c │ │ │ │ -0004cc40: 7075 293b 2030 2e30 3135 3939 3934 7320 pu); 0.0159994s │ │ │ │ +0004cc30: 6564 2030 2e30 3430 3135 3437 7320 2863 ed 0.0401547s (c │ │ │ │ +0004cc40: 7075 293b 2030 2e30 3230 3636 3131 7320 pu); 0.0206611s │ │ │ │ 0004cc50: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 0004cc60: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0004cc70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0004cc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004cc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004cca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004ccb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -20942,16 +20942,16 @@ │ │ │ │ 00051cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051ce0: 2d2d 2d2d 2b0a 7c69 3920 3a20 7469 6d65 ----+.|i9 : time │ │ │ │ 00051cf0: 2067 203d 2067 7261 7068 2070 323b 2020 g = graph p2; │ │ │ │ 00051d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d30: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00051d40: 302e 3033 3233 3639 3773 2028 6370 7529 0.0323697s (cpu) │ │ │ │ -00051d50: 3b20 302e 3033 3139 3138 3673 2028 7468 ; 0.0319186s (th │ │ │ │ +00051d40: 302e 3036 3539 3137 3973 2028 6370 7529 0.0659179s (cpu) │ │ │ │ +00051d50: 3b20 302e 3034 3337 3033 3273 2028 7468 ; 0.0437032s (th │ │ │ │ 00051d60: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00051d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 00051d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -21662,17 +21662,17 @@ │ │ │ │ 000549d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000549e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ 000549f0: 7469 6d65 2069 6465 616c 2070 6869 2020 time ideal phi │ │ │ │ 00054a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a30: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00054a40: 7365 6420 302e 3030 3338 3135 3138 7320 sed 0.00381518s │ │ │ │ -00054a50: 2863 7075 293b 2030 2e30 3033 3831 3431 (cpu); 0.0038141 │ │ │ │ -00054a60: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ +00054a40: 7365 6420 302e 3030 3432 3934 3235 7320 sed 0.00429425s │ │ │ │ +00054a50: 2863 7075 293b 2030 2e30 3034 3239 3131 (cpu); 0.0042911 │ │ │ │ +00054a60: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ 00054a70: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00054a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00054a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ @@ -22297,18 +22297,18 @@ │ │ │ │ 00057180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00057190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ 000571a0: 7469 6d65 2069 6465 616c 2070 6869 2720 time ideal phi' │ │ │ │ 000571b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571e0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000571f0: 7365 6420 302e 3039 3330 3239 3173 2028 sed 0.0930291s ( │ │ │ │ -00057200: 6370 7529 3b20 302e 3039 3330 3334 3773 cpu); 0.0930347s │ │ │ │ -00057210: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -00057220: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +000571f0: 7365 6420 302e 3131 3136 3135 7320 2863 sed 0.111615s (c │ │ │ │ +00057200: 7075 293b 2030 2e31 3131 3631 3473 2028 pu); 0.111614s ( │ │ │ │ +00057210: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00057220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057230: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00057240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057280: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ 00057290: 6964 6561 6c20 3120 2020 2020 2020 2020 ideal 1 │ │ │ │ @@ -24856,16 +24856,16 @@ │ │ │ │ 00061170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061180: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 696d -----+.|i3 : tim │ │ │ │ 00061190: 6520 696e 7665 7273 6520 7068 6920 2020 e inverse phi │ │ │ │ 000611a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000611b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000611c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000611d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000611e0: 2030 2e30 3538 3030 3933 7320 2863 7075 0.0580093s (cpu │ │ │ │ -000611f0: 293b 2030 2e30 3538 3031 3032 7320 2874 ); 0.0580102s (t │ │ │ │ +000611e0: 2030 2e30 3635 3933 3638 7320 2863 7075 0.0659368s (cpu │ │ │ │ +000611f0: 293b 2030 2e30 3635 3839 3639 7320 2874 ); 0.0658969s (t │ │ │ │ 00061200: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00061210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061220: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00061230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -27855,16 +27855,16 @@ │ │ │ │ 0006cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ccf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ 0006cd00: 2074 696d 6520 7073 6920 3d20 696e 7665 time psi = inve │ │ │ │ 0006cd10: 7273 654d 6170 2070 6869 2020 2020 2020 rseMap phi │ │ │ │ 0006cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd40: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -0006cd50: 7573 6564 2030 2e32 3035 3839 3873 2028 used 0.205898s ( │ │ │ │ -0006cd60: 6370 7529 3b20 302e 3132 3830 3436 7320 cpu); 0.128046s │ │ │ │ +0006cd50: 7573 6564 2030 2e32 3430 3735 3873 2028 used 0.240758s ( │ │ │ │ +0006cd60: 6370 7529 3b20 302e 3133 3438 3132 7320 cpu); 0.134812s │ │ │ │ 0006cd70: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 0006cd80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0006cd90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0006cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -28540,16 +28540,16 @@ │ │ │ │ 0006f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006f7c0: 2d2b 0a7c 6935 203a 2074 696d 6520 7073 -+.|i5 : time ps │ │ │ │ 0006f7d0: 6920 3d20 696e 7665 7273 654d 6170 2070 i = inverseMap p │ │ │ │ 0006f7e0: 6869 2020 2020 2020 2020 2020 2020 2020 hi │ │ │ │ 0006f7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f810: 207c 0a7c 202d 2d20 7573 6564 2030 2e34 |.| -- used 0.4 │ │ │ │ -0006f820: 3731 3138 3873 2028 6370 7529 3b20 302e 71188s (cpu); 0. │ │ │ │ -0006f830: 3331 3332 3035 7320 2874 6872 6561 6429 313205s (thread) │ │ │ │ +0006f820: 3632 3036 3473 2028 6370 7529 3b20 302e 62064s (cpu); 0. │ │ │ │ +0006f830: 3234 3637 3538 7320 2874 6872 6561 6429 246758s (thread) │ │ │ │ 0006f840: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0006f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f860: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -29536,18 +29536,18 @@ │ │ │ │ 000735f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00073600: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ 00073610: 7469 6d65 2069 7342 6972 6174 696f 6e61 time isBirationa │ │ │ │ 00073620: 6c20 7068 6920 2020 2020 2020 2020 2020 l phi │ │ │ │ 00073630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073650: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00073660: 7365 6420 302e 3031 3736 3736 7320 2863 sed 0.017676s (c │ │ │ │ -00073670: 7075 293b 2030 2e30 3137 3637 3537 7320 pu); 0.0176757s │ │ │ │ -00073680: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00073690: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00073660: 7365 6420 302e 3032 3237 3538 7320 2863 sed 0.022758s (c │ │ │ │ +00073670: 7075 293b 2030 2e30 3232 3735 3773 2028 pu); 0.022757s ( │ │ │ │ +00073680: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00073690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000736a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000736b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000736c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000736d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000736e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000736f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ 00073700: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ @@ -29566,18 +29566,18 @@ │ │ │ │ 000737d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000737e0: 2020 2020 2020 2020 7c0a 7c43 6572 7469 |.|Certi │ │ │ │ 000737f0: 6679 3a20 6f75 7470 7574 2063 6572 7469 fy: output certi │ │ │ │ 00073800: 6669 6564 2120 2020 2020 2020 2020 2020 fied! │ │ │ │ 00073810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073830: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00073840: 7365 6420 302e 3031 3434 3739 3373 2028 sed 0.0144793s ( │ │ │ │ -00073850: 6370 7529 3b20 302e 3031 3430 3839 7320 cpu); 0.014089s │ │ │ │ -00073860: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00073870: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00073840: 7365 6420 302e 3032 3935 3037 3373 2028 sed 0.0295073s ( │ │ │ │ +00073850: 6370 7529 3b20 302e 3031 3730 3039 3973 cpu); 0.0170099s │ │ │ │ +00073860: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +00073870: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00073880: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00073890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738d0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ 000738e0: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ @@ -29739,18 +29739,18 @@ │ │ │ │ 000742a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000742b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 000742c0: 0a7c 4365 7274 6966 793a 206f 7574 7075 .|Certify: outpu │ │ │ │ 000742d0: 7420 6365 7274 6966 6965 6421 2020 2020 t certified! │ │ │ │ 000742e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000742f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00074300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00074310: 0a7c 202d 2d20 7573 6564 2032 2e34 3237 .| -- used 2.427 │ │ │ │ -00074320: 3833 7320 2863 7075 293b 2031 2e39 3937 83s (cpu); 1.997 │ │ │ │ -00074330: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ -00074340: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00074310: 0a7c 202d 2d20 7573 6564 2032 2e36 3236 .| -- used 2.626 │ │ │ │ +00074320: 3235 7320 2863 7075 293b 2032 2e32 3339 25s (cpu); 2.239 │ │ │ │ +00074330: 3634 7320 2874 6872 6561 6429 3b20 3073 64s (thread); 0s │ │ │ │ +00074340: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00074350: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00074360: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00074370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00074380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00074390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000743a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 000743b0: 0a7c 6f33 203d 2074 7275 6520 2020 2020 .|o3 = true │ │ │ │ @@ -30174,18 +30174,18 @@ │ │ │ │ 00075dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075de0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00075df0: 0a7c 4365 7274 6966 793a 206f 7574 7075 .|Certify: outpu │ │ │ │ 00075e00: 7420 6365 7274 6966 6965 6421 2020 2020 t certified! │ │ │ │ 00075e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075e30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00075e40: 0a7c 202d 2d20 7573 6564 2033 2e36 3130 .| -- used 3.610 │ │ │ │ -00075e50: 3536 7320 2863 7075 293b 2032 2e34 3039 56s (cpu); 2.409 │ │ │ │ -00075e60: 3933 7320 2874 6872 6561 6429 3b20 3073 93s (thread); 0s │ │ │ │ -00075e70: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00075e40: 0a7c 202d 2d20 7573 6564 2034 2e32 3337 .| -- used 4.237 │ │ │ │ +00075e50: 3773 2028 6370 7529 3b20 322e 3931 3731 7s (cpu); 2.9171 │ │ │ │ +00075e60: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00075e70: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00075e80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00075e90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00075ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075ed0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00075ee0: 0a7c 6f37 203d 2066 616c 7365 2020 2020 .|o7 = false │ │ │ │ @@ -31483,17 +31483,17 @@ │ │ │ │ 0007afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0007afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0007afc0: 7c69 3220 3a20 7469 6d65 206b 6572 6e65 |i2 : time kerne │ │ │ │ 0007afd0: 6c28 7068 692c 3129 2020 2020 2020 2020 l(phi,1) │ │ │ │ 0007afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b000: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0007b010: 7c20 2d2d 2075 7365 6420 302e 3031 3739 | -- used 0.0179 │ │ │ │ -0007b020: 3333 3673 2028 6370 7529 3b20 302e 3031 336s (cpu); 0.01 │ │ │ │ -0007b030: 3739 3239 3173 2028 7468 7265 6164 293b 79291s (thread); │ │ │ │ +0007b010: 7c20 2d2d 2075 7365 6420 302e 3032 3037 | -- used 0.0207 │ │ │ │ +0007b020: 3530 3173 2028 6370 7529 3b20 302e 3032 501s (cpu); 0.02 │ │ │ │ +0007b030: 3037 3531 3173 2028 7468 7265 6164 293b 07511s (thread); │ │ │ │ 0007b040: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0007b050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0007b060: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0007b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b0a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -31523,18 +31523,18 @@ │ │ │ │ 0007b220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0007b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0007b240: 7c69 3320 3a20 7469 6d65 206b 6572 6e65 |i3 : time kerne │ │ │ │ 0007b250: 6c28 7068 692c 3229 2020 2020 2020 2020 l(phi,2) │ │ │ │ 0007b260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b280: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0007b290: 7c20 2d2d 2075 7365 6420 302e 3834 3136 | -- used 0.8416 │ │ │ │ -0007b2a0: 3937 7320 2863 7075 293b 2030 2e34 3433 97s (cpu); 0.443 │ │ │ │ -0007b2b0: 3930 3473 2028 7468 7265 6164 293b 2030 904s (thread); 0 │ │ │ │ -0007b2c0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ +0007b290: 7c20 2d2d 2075 7365 6420 312e 3235 3131 | -- used 1.2511 │ │ │ │ +0007b2a0: 3473 2028 6370 7529 3b20 302e 3536 3432 4s (cpu); 0.5642 │ │ │ │ +0007b2b0: 3335 7320 2874 6872 6561 6429 3b20 3073 35s (thread); 0s │ │ │ │ +0007b2c0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0007b2d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0007b2e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0007b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b320: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0007b330: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ @@ -32424,17 +32424,17 @@ │ │ │ │ 0007ea70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0007ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0007ea90: 7c69 3320 3a20 7469 6d65 2070 6172 616d |i3 : time param │ │ │ │ 0007eaa0: 6574 7269 7a65 204c 2020 2020 2020 2020 etrize L │ │ │ │ 0007eab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007eac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007ead0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0007eae0: 7c20 2d2d 2075 7365 6420 302e 3030 3532 | -- used 0.0052 │ │ │ │ -0007eaf0: 3933 3635 7320 2863 7075 293b 2030 2e30 9365s (cpu); 0.0 │ │ │ │ -0007eb00: 3035 3238 3734 3273 2028 7468 7265 6164 0528742s (thread │ │ │ │ +0007eae0: 7c20 2d2d 2075 7365 6420 302e 3030 3637 | -- used 0.0067 │ │ │ │ +0007eaf0: 3135 3831 7320 2863 7075 293b 2030 2e30 1581s (cpu); 0.0 │ │ │ │ +0007eb00: 3036 3731 3035 3273 2028 7468 7265 6164 0671052s (thread │ │ │ │ 0007eb10: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0007eb20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0007eb30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0007eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007eb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007eb70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -32934,18 +32934,18 @@ │ │ │ │ 00080a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00080a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00080a70: 7c69 3520 3a20 7469 6d65 2070 6172 616d |i5 : time param │ │ │ │ 00080a80: 6574 7269 7a65 2051 2020 2020 2020 2020 etrize Q │ │ │ │ 00080a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00080ac0: 7c20 2d2d 2075 7365 6420 302e 3538 3137 | -- used 0.5817 │ │ │ │ -00080ad0: 3234 7320 2863 7075 293b 2030 2e33 3939 24s (cpu); 0.399 │ │ │ │ -00080ae0: 3936 7320 2874 6872 6561 6429 3b20 3073 96s (thread); 0s │ │ │ │ -00080af0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00080ac0: 7c20 2d2d 2075 7365 6420 302e 3630 3838 | -- used 0.6088 │ │ │ │ +00080ad0: 3237 7320 2863 7075 293b 2030 2e34 3534 27s (cpu); 0.454 │ │ │ │ +00080ae0: 3037 3773 2028 7468 7265 6164 293b 2030 077s (thread); 0 │ │ │ │ +00080af0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00080b00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00080b10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00080b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080b50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00080b60: 7c6f 3520 3d20 2d2d 2072 6174 696f 6e61 |o5 = -- rationa │ │ │ │ @@ -34395,18 +34395,18 @@ │ │ │ │ 000865a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000865b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000865c0: 6932 203a 2074 696d 6520 7020 3d20 706f i2 : time p = po │ │ │ │ 000865d0: 696e 7420 736f 7572 6365 2066 2020 2020 int source f │ │ │ │ 000865e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000865f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086600: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00086610: 202d 2d20 7573 6564 2030 2e34 3031 3434 -- used 0.40144 │ │ │ │ -00086620: 3673 2028 6370 7529 3b20 302e 3139 3632 6s (cpu); 0.1962 │ │ │ │ -00086630: 3631 7320 2874 6872 6561 6429 3b20 3073 61s (thread); 0s │ │ │ │ -00086640: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00086610: 202d 2d20 7573 6564 2030 2e35 3432 3631 -- used 0.54261 │ │ │ │ +00086620: 3873 2028 6370 7529 3b20 302e 3234 3134 8s (cpu); 0.2414 │ │ │ │ +00086630: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ +00086640: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00086650: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00086660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000866a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000866b0: 6f32 203d 2069 6465 616c 2028 7920 2020 o2 = ideal (y │ │ │ │ @@ -34510,17 +34510,17 @@ │ │ │ │ 00086cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00086ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00086cf0: 6933 203a 2074 696d 6520 7020 3d3d 2066 i3 : time p == f │ │ │ │ 00086d00: 5e2a 2066 2070 2020 2020 2020 2020 2020 ^* f p │ │ │ │ 00086d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086d30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00086d40: 202d 2d20 7573 6564 2030 2e32 3036 3836 -- used 0.20686 │ │ │ │ -00086d50: 3173 2028 6370 7529 3b20 302e 3133 3131 1s (cpu); 0.1311 │ │ │ │ -00086d60: 3133 7320 2874 6872 6561 6429 3b20 3073 13s (thread); 0s │ │ │ │ +00086d40: 202d 2d20 7573 6564 2030 2e32 3533 3334 -- used 0.25334 │ │ │ │ +00086d50: 3673 2028 6370 7529 3b20 302e 3134 3531 6s (cpu); 0.1451 │ │ │ │ +00086d60: 3438 7320 2874 6872 6561 6429 3b20 3073 48s (thread); 0s │ │ │ │ 00086d70: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00086d80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00086d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -34842,17 +34842,17 @@ │ │ │ │ 00088190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000881a0: 2020 207c 0a7c 4365 7274 6966 793a 206f |.|Certify: o │ │ │ │ 000881b0: 7574 7075 7420 6365 7274 6966 6965 6421 utput certified! │ │ │ │ 000881c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000881d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000881e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000881f0: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -00088200: 2e30 3136 3033 3734 7320 2863 7075 293b .0160374s (cpu); │ │ │ │ -00088210: 2030 2e30 3135 3538 3136 7320 2874 6872 0.0155816s (thr │ │ │ │ -00088220: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00088200: 2e30 3333 3637 3573 2028 6370 7529 3b20 .033675s (cpu); │ │ │ │ +00088210: 302e 3031 3832 3539 3773 2028 7468 7265 0.0182597s (thre │ │ │ │ +00088220: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00088230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088240: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00088250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088290: 2020 207c 0a7c 6f33 203d 207b 312c 2032 |.|o3 = {1, 2 │ │ │ │ @@ -34972,16 +34972,16 @@ │ │ │ │ 000889b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000889c0: 2020 207c 0a7c 4365 7274 6966 793a 206f |.|Certify: o │ │ │ │ 000889d0: 7574 7075 7420 6365 7274 6966 6965 6421 utput certified! │ │ │ │ 000889e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000889f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a10: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -00088a20: 2e30 3131 3839 3336 7320 2863 7075 293b .0118936s (cpu); │ │ │ │ -00088a30: 2030 2e30 3131 3531 3534 7320 2874 6872 0.0115154s (thr │ │ │ │ +00088a20: 2e30 3236 3831 3639 7320 2863 7075 293b .0268169s (cpu); │ │ │ │ +00088a30: 2030 2e30 3134 3835 3735 7320 2874 6872 0.0148575s (thr │ │ │ │ 00088a40: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00088a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a60: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00088a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -35296,18 +35296,18 @@ │ │ │ │ 00089df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00089e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00089e10: 2d2d 2d2b 0a7c 6937 203a 2074 696d 6520 ---+.|i7 : time │ │ │ │ 00089e20: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ 00089e30: 7320 7068 6920 2020 2020 2020 2020 2020 s phi │ │ │ │ 00089e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00089e60: 2020 207c 0a7c 202d 2d20 7573 6564 2034 |.| -- used 4 │ │ │ │ -00089e70: 2e37 3939 652d 3035 7320 2863 7075 293b .799e-05s (cpu); │ │ │ │ -00089e80: 2034 2e31 3835 3865 2d30 3573 2028 7468 4.1858e-05s (th │ │ │ │ -00089e90: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +00089e60: 2020 207c 0a7c 202d 2d20 7573 6564 2037 |.| -- used 7 │ │ │ │ +00089e70: 2e35 3665 2d30 3573 2028 6370 7529 3b20 .56e-05s (cpu); │ │ │ │ +00089e80: 362e 3336 3539 652d 3035 7320 2874 6872 6.3659e-05s (thr │ │ │ │ +00089e90: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00089ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089eb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00089ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089f00: 2020 207c 0a7c 6f37 203d 207b 312c 2032 |.|o7 = {1, 2 │ │ │ │ @@ -35332,17 +35332,17 @@ │ │ │ │ 0008a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0008a040: 2d2d 2d2b 0a7c 6938 203a 2074 696d 6520 ---+.|i8 : time │ │ │ │ 0008a050: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ 0008a060: 7328 7068 692c 4e75 6d44 6567 7265 6573 s(phi,NumDegrees │ │ │ │ 0008a070: 3d3e 3129 2020 2020 2020 2020 2020 2020 =>1) │ │ │ │ 0008a080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a090: 2020 207c 0a7c 202d 2d20 7573 6564 2032 |.| -- used 2 │ │ │ │ -0008a0a0: 2e37 3136 3165 2d30 3573 2028 6370 7529 .7161e-05s (cpu) │ │ │ │ -0008a0b0: 3b20 322e 3639 3365 2d30 3573 2028 7468 ; 2.693e-05s (th │ │ │ │ -0008a0c0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +0008a0a0: 2e31 3538 3465 2d30 3573 2028 6370 7529 .1584e-05s (cpu) │ │ │ │ +0008a0b0: 3b20 322e 3138 3231 652d 3035 7320 2874 ; 2.1821e-05s (t │ │ │ │ +0008a0c0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0008a0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a0e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0008a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a130: 2020 207c 0a7c 6f38 203d 207b 342c 2031 |.|o8 = {4, 1 │ │ │ │ @@ -37824,18 +37824,18 @@ │ │ │ │ 00093bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00093c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00093c10: 2b0a 7c69 3420 3a20 7469 6d65 2070 6869 +.|i4 : time phi │ │ │ │ 00093c20: 2120 3b20 2020 2020 2020 2020 2020 2020 ! ; │ │ │ │ 00093c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00093c60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3035 |.| -- used 0.05 │ │ │ │ -00093c70: 3237 3939 7320 2863 7075 293b 2030 2e30 2799s (cpu); 0.0 │ │ │ │ -00093c80: 3532 3339 3534 7320 2874 6872 6561 6429 523954s (thread) │ │ │ │ -00093c90: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +00093c60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3130 |.| -- used 0.10 │ │ │ │ +00093c70: 3530 3473 2028 6370 7529 3b20 302e 3036 504s (cpu); 0.06 │ │ │ │ +00093c80: 3934 3933 3473 2028 7468 7265 6164 293b 94934s (thread); │ │ │ │ +00093c90: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00093ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093cb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00093cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093d00: 7c0a 7c6f 3420 3a20 5261 7469 6f6e 616c |.|o4 : Rational │ │ │ │ @@ -37999,17 +37999,17 @@ │ │ │ │ 000946e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000946f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00094700: 2b0a 7c69 3920 3a20 7469 6d65 2070 6869 +.|i9 : time phi │ │ │ │ 00094710: 2120 3b20 2020 2020 2020 2020 2020 2020 ! ; │ │ │ │ 00094720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00094730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00094740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00094750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3033 |.| -- used 0.03 │ │ │ │ -00094760: 3337 3430 3673 2028 6370 7529 3b20 302e 37406s (cpu); 0. │ │ │ │ -00094770: 3033 3333 3938 3273 2028 7468 7265 6164 0333982s (thread │ │ │ │ +00094750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3035 |.| -- used 0.05 │ │ │ │ +00094760: 3533 3034 3773 2028 6370 7529 3b20 302e 53047s (cpu); 0. │ │ │ │ +00094770: 3034 3333 3330 3373 2028 7468 7265 6164 0433303s (thread │ │ │ │ 00094780: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00094790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000947b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -40043,17 +40043,17 @@ │ │ │ │ 0009c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0009c6c0: 7c69 3620 3a20 7469 6d65 2070 6869 5e2a |i6 : time phi^* │ │ │ │ 0009c6d0: 2a20 7120 2020 2020 2020 2020 2020 2020 * q │ │ │ │ 0009c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c700: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0009c710: 7c20 2d2d 2075 7365 6420 302e 3135 3831 | -- used 0.1581 │ │ │ │ -0009c720: 3836 7320 2863 7075 293b 2030 2e31 3538 86s (cpu); 0.158 │ │ │ │ -0009c730: 3134 3873 2028 7468 7265 6164 293b 2030 148s (thread); 0 │ │ │ │ +0009c710: 7c20 2d2d 2075 7365 6420 302e 3138 3233 | -- used 0.1823 │ │ │ │ +0009c720: 3633 7320 2863 7075 293b 2030 2e31 3832 63s (cpu); 0.182 │ │ │ │ +0009c730: 3336 3373 2028 7468 7265 6164 293b 2030 363s (thread); 0 │ │ │ │ 0009c740: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0009c750: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0009c760: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0009c770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c7a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -42164,17 +42164,17 @@ │ │ │ │ 000a4b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000a4b40: 2d2b 0a7c 6933 203a 2074 696d 6520 7068 -+.|i3 : time ph │ │ │ │ 000a4b50: 6920 3d20 7261 7469 6f6e 616c 4d61 7028 i = rationalMap( │ │ │ │ 000a4b60: 562c 332c 3229 2020 2020 2020 2020 2020 V,3,2) │ │ │ │ 000a4b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4b90: 207c 0a7c 202d 2d20 7573 6564 2030 2e31 |.| -- used 0.1 │ │ │ │ -000a4ba0: 3032 3634 3173 2028 6370 7529 3b20 302e 02641s (cpu); 0. │ │ │ │ -000a4bb0: 3130 3236 3173 2028 7468 7265 6164 293b 10261s (thread); │ │ │ │ -000a4bc0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +000a4ba0: 3138 3134 3673 2028 6370 7529 3b20 302e 18146s (cpu); 0. │ │ │ │ +000a4bb0: 3131 3831 3438 7320 2874 6872 6561 6429 118148s (thread) │ │ │ │ +000a4bc0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 000a4bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4be0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000a4bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4c30: 207c 0a7c 6f33 203d 202d 2d20 7261 7469 |.|o3 = -- rati │ │ │ │ @@ -43696,16 +43696,16 @@ │ │ │ │ 000aaaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000aab00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ 000aab10: 7469 6d65 2070 6869 203d 2072 6174 696f time phi = ratio │ │ │ │ 000aab20: 6e61 6c4d 6170 2044 2020 2020 2020 2020 nalMap D │ │ │ │ 000aab30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aab40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aab50: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000aab60: 7365 6420 302e 3033 3133 3338 3573 2028 sed 0.0313385s ( │ │ │ │ -000aab70: 6370 7529 3b20 302e 3033 3133 3430 3273 cpu); 0.0313402s │ │ │ │ +000aab60: 7365 6420 302e 3033 3637 3735 3973 2028 sed 0.0367759s ( │ │ │ │ +000aab70: 6370 7529 3b20 302e 3033 3636 3635 3873 cpu); 0.0366658s │ │ │ │ 000aab80: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 000aab90: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000aaba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000aabb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aabc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aabd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aabe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -44706,16 +44706,16 @@ │ │ │ │ 000aea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000aea20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 --------+.|i7 : │ │ │ │ 000aea30: 7469 6d65 203f 2069 6d61 6765 2870 6869 time ? image(phi │ │ │ │ 000aea40: 2c22 4634 2229 2020 2020 2020 2020 2020 ,"F4") │ │ │ │ 000aea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aea60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aea70: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000aea80: 7365 6420 312e 3039 3132 3973 2028 6370 sed 1.09129s (cp │ │ │ │ -000aea90: 7529 3b20 302e 3731 3732 3038 7320 2874 u); 0.717208s (t │ │ │ │ +000aea80: 7365 6420 312e 3734 3230 3473 2028 6370 sed 1.74204s (cp │ │ │ │ +000aea90: 7529 3b20 302e 3731 3836 3637 7320 2874 u); 0.718667s (t │ │ │ │ 000aeaa0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 000aeab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000aead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46244,16 +46244,16 @@ │ │ │ │ 000b4a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b4a40: 2d2d 2d2d 2b0a 7c69 3420 3a20 7469 6d65 ----+.|i4 : time │ │ │ │ 000b4a50: 2053 6567 7265 436c 6173 7320 5820 2020 SegreClass X │ │ │ │ 000b4a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4a90: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000b4aa0: 302e 3832 3235 3235 7320 2863 7075 293b 0.822525s (cpu); │ │ │ │ -000b4ab0: 2030 2e35 3237 3537 3173 2028 7468 7265 0.527571s (thre │ │ │ │ +000b4aa0: 302e 3934 3631 3836 7320 2863 7075 293b 0.946186s (cpu); │ │ │ │ +000b4ab0: 2030 2e36 3038 3331 3473 2028 7468 7265 0.608314s (thre │ │ │ │ 000b4ac0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000b4ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000b4af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46299,16 +46299,16 @@ │ │ │ │ 000b4da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b4db0: 2d2d 2d2d 2b0a 7c69 3520 3a20 7469 6d65 ----+.|i5 : time │ │ │ │ 000b4dc0: 2053 6567 7265 436c 6173 7320 6c69 6674 SegreClass lift │ │ │ │ 000b4dd0: 2858 2c50 3729 2020 2020 2020 2020 2020 (X,P7) │ │ │ │ 000b4de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e00: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000b4e10: 302e 3437 3637 3832 7320 2863 7075 293b 0.476782s (cpu); │ │ │ │ -000b4e20: 2030 2e33 3035 3535 3473 2028 7468 7265 0.305554s (thre │ │ │ │ +000b4e10: 302e 3735 3434 3834 7320 2863 7075 293b 0.754484s (cpu); │ │ │ │ +000b4e20: 2030 2e34 3233 3135 3173 2028 7468 7265 0.423151s (thre │ │ │ │ 000b4e30: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000b4e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000b4e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46359,16 +46359,16 @@ │ │ │ │ 000b5160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5170: 2020 2020 7c0a 7c43 6572 7469 6679 3a20 |.|Certify: │ │ │ │ 000b5180: 6f75 7470 7574 2063 6572 7469 6669 6564 output certified │ │ │ │ 000b5190: 2120 2020 2020 2020 2020 2020 2020 2020 ! │ │ │ │ 000b51a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b51b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b51c0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000b51d0: 302e 3032 3134 3434 3573 2028 6370 7529 0.0214445s (cpu) │ │ │ │ -000b51e0: 3b20 302e 3032 3038 3832 3173 2028 7468 ; 0.0208821s (th │ │ │ │ +000b51d0: 302e 3034 3138 3632 3973 2028 6370 7529 0.0418629s (cpu) │ │ │ │ +000b51e0: 3b20 302e 3032 3730 3334 3773 2028 7468 ; 0.0270347s (th │ │ │ │ 000b51f0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 000b5200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000b5220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46419,17 +46419,17 @@ │ │ │ │ 000b5520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5530: 2020 2020 7c0a 7c43 6572 7469 6679 3a20 |.|Certify: │ │ │ │ 000b5540: 6f75 7470 7574 2063 6572 7469 6669 6564 output certified │ │ │ │ 000b5550: 2120 2020 2020 2020 2020 2020 2020 2020 ! │ │ │ │ 000b5560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5580: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000b5590: 302e 3039 3733 3739 3473 2028 6370 7529 0.0973794s (cpu) │ │ │ │ -000b55a0: 3b20 302e 3039 3639 3234 3173 2028 7468 ; 0.0969241s (th │ │ │ │ -000b55b0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +000b5590: 302e 3133 3673 2028 6370 7529 3b20 302e 0.136s (cpu); 0. │ │ │ │ +000b55a0: 3132 3332 3873 2028 7468 7265 6164 293b 12328s (thread); │ │ │ │ +000b55b0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 000b55c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b55d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000b55e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b55f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5620: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ @@ -46535,26 +46535,26 @@ │ │ │ │ 000b5c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5c70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b5c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5cc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b5cd0: 6f39 2020 2020 205a 5a20 2020 2020 2020 o9 ZZ │ │ │ │ +000b5cd0: 2020 2020 2020 205a 5a20 2020 2020 2020 ZZ │ │ │ │ 000b5ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b5d20: 203d 202d 2d2d 2d2d 2d5b 7820 2e2e 7820 = ------[x ..x │ │ │ │ -000b5d30: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +000b5d20: 6f39 203d 202d 2d2d 2d2d 2d5b 7820 2e2e o9 = ------[x .. │ │ │ │ +000b5d30: 7820 5d20 2020 2020 2020 2020 2020 2020 x ] │ │ │ │ 000b5d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b5d70: 2020 2031 3030 3030 3320 2030 2020 2036 100003 0 6 │ │ │ │ -000b5d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000b5d70: 2020 2020 2031 3030 3030 3320 2030 2020 100003 0 │ │ │ │ +000b5d80: 2036 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ 000b5d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5db0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b5dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46570,18 +46570,18 @@ │ │ │ │ 000b5e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b5ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000b5eb0: 6931 3020 3a20 7469 6d65 2070 6869 203d i10 : time phi = │ │ │ │ 000b5ec0: 2069 6e76 6572 7365 4d61 7020 746f 4d61 inverseMap toMa │ │ │ │ 000b5ed0: 7028 6d69 6e6f 7273 2832 2c6d 6174 7269 p(minors(2,matri │ │ │ │ 000b5ee0: 787b 7b78 5f30 2c78 5f31 2c78 5f33 2c78 x{{x_0,x_1,x_3,x │ │ │ │ 000b5ef0: 5f34 2c78 5f35 7d2c 7b78 5f31 2c7c 0a7c _4,x_5},{x_1,|.| │ │ │ │ -000b5f00: 202d 2d20 7573 6564 2030 2e32 3031 3031 -- used 0.20101 │ │ │ │ -000b5f10: 7320 2863 7075 293b 2030 2e31 3032 3230 s (cpu); 0.10220 │ │ │ │ -000b5f20: 3573 2028 7468 7265 6164 293b 2030 7320 5s (thread); 0s │ │ │ │ -000b5f30: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +000b5f00: 202d 2d20 7573 6564 2030 2e30 3730 3432 -- used 0.07042 │ │ │ │ +000b5f10: 3238 7320 2863 7075 293b 2030 2e30 3730 28s (cpu); 0.070 │ │ │ │ +000b5f20: 3432 3735 7320 2874 6872 6561 6429 3b20 4275s (thread); │ │ │ │ +000b5f30: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 000b5f40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b5f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5f90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b5fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46775,17 +46775,17 @@ │ │ │ │ 000b6b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b6b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000b6b80: 6931 3120 3a20 7469 6d65 2053 6567 7265 i11 : time Segre │ │ │ │ 000b6b90: 436c 6173 7320 7068 6920 2020 2020 2020 Class phi │ │ │ │ 000b6ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b6bd0: 202d 2d20 7573 6564 2030 2e31 3634 3832 -- used 0.16482 │ │ │ │ -000b6be0: 3273 2028 6370 7529 3b20 302e 3136 3438 2s (cpu); 0.1648 │ │ │ │ -000b6bf0: 3237 7320 2874 6872 6561 6429 3b20 3073 27s (thread); 0s │ │ │ │ +000b6bd0: 202d 2d20 7573 6564 2030 2e34 3332 3335 -- used 0.43235 │ │ │ │ +000b6be0: 3373 2028 6370 7529 3b20 302e 3238 3237 3s (cpu); 0.2827 │ │ │ │ +000b6bf0: 3138 7320 2874 6872 6561 6429 3b20 3073 18s (thread); 0s │ │ │ │ 000b6c00: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000b6c10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b6c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6c60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -46935,17 +46935,17 @@ │ │ │ │ 000b7560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7570: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b7580: 2020 2020 2020 7469 6d65 2053 6567 7265 time Segre │ │ │ │ 000b7590: 436c 6173 7320 4220 2020 2020 2020 2020 Class B │ │ │ │ 000b75a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b75b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b75c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b75d0: 202d 2d20 7573 6564 2030 2e35 3231 3136 -- used 0.52116 │ │ │ │ -000b75e0: 3373 2028 6370 7529 3b20 302e 3333 3038 3s (cpu); 0.3308 │ │ │ │ -000b75f0: 3633 7320 2874 6872 6561 6429 3b20 3073 63s (thread); 0s │ │ │ │ +000b75d0: 202d 2d20 7573 6564 2030 2e35 3138 3430 -- used 0.51840 │ │ │ │ +000b75e0: 3473 2028 6370 7529 3b20 302e 3335 3634 4s (cpu); 0.3564 │ │ │ │ +000b75f0: 3531 7320 2874 6872 6561 6429 3b20 3073 51s (thread); 0s │ │ │ │ 000b7600: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000b7610: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b7620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7660: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -46995,18 +46995,18 @@ │ │ │ │ 000b7920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7930: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b7940: 2020 2020 2020 7469 6d65 2053 6567 7265 time Segre │ │ │ │ 000b7950: 436c 6173 7320 6c69 6674 2842 2c61 6d62 Class lift(B,amb │ │ │ │ 000b7960: 6965 6e74 2072 696e 6720 4229 2020 2020 ient ring B) │ │ │ │ 000b7970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000b7990: 202d 2d20 7573 6564 2031 2e36 3433 3933 -- used 1.64393 │ │ │ │ -000b79a0: 7320 2863 7075 293b 2031 2e30 3931 3538 s (cpu); 1.09158 │ │ │ │ -000b79b0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -000b79c0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +000b7990: 202d 2d20 7573 6564 2031 2e38 3830 3535 -- used 1.88055 │ │ │ │ +000b79a0: 7320 2863 7075 293b 2031 2e30 3630 3873 s (cpu); 1.0608s │ │ │ │ +000b79b0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +000b79c0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000b79d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b79e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b79f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7a20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000b7a30: 2020 2020 2020 2020 2020 2039 2020 2020 9 │ │ │ │ @@ -47245,17 +47245,17 @@ │ │ │ │ 000b88c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b88d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000b88e0: 7c69 3120 3a20 7469 6d65 2061 7070 6c79 |i1 : time apply │ │ │ │ 000b88f0: 2831 2e2e 3132 2c69 202d 3e20 6465 7363 (1..12,i -> desc │ │ │ │ 000b8900: 7269 6265 2073 7065 6369 616c 4372 656d ribe specialCrem │ │ │ │ 000b8910: 6f6e 6154 7261 6e73 666f 726d 6174 696f onaTransformatio │ │ │ │ 000b8920: 6e28 692c 5a5a 2f33 3333 3129 2920 7c0a n(i,ZZ/3331)) |. │ │ │ │ -000b8930: 7c20 2d2d 2075 7365 6420 312e 3436 3831 | -- used 1.4681 │ │ │ │ -000b8940: 3573 2028 6370 7529 3b20 312e 3132 3531 5s (cpu); 1.1251 │ │ │ │ -000b8950: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ +000b8930: 7c20 2d2d 2075 7365 6420 312e 3733 3638 | -- used 1.7368 │ │ │ │ +000b8940: 3973 2028 6370 7529 3b20 312e 3237 3039 9s (cpu); 1.2709 │ │ │ │ +000b8950: 3173 2028 7468 7265 6164 293b 2030 7320 1s (thread); 0s │ │ │ │ 000b8960: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000b8970: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000b8980: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000b8990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b89a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b89b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b89c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -48068,16 +48068,16 @@ │ │ │ │ 000bbc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000bbc40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ 000bbc50: 2074 696d 6520 7370 6563 6961 6c43 7562 time specialCub │ │ │ │ 000bbc60: 6963 5472 616e 7366 6f72 6d61 7469 6f6e icTransformation │ │ │ │ 000bbc70: 2039 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ 000bbc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbc90: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000bbca0: 7573 6564 2030 2e30 3939 3136 3736 7320 used 0.0991676s │ │ │ │ -000bbcb0: 2863 7075 293b 2030 2e30 3939 3136 3834 (cpu); 0.0991684 │ │ │ │ +000bbca0: 7573 6564 2030 2e30 3935 3939 3536 7320 used 0.0959956s │ │ │ │ +000bbcb0: 2863 7075 293b 2030 2e30 3935 3939 3634 (cpu); 0.0959964 │ │ │ │ 000bbcc0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 000bbcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbce0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000bbcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -50518,16 +50518,16 @@ │ │ │ │ 000c5550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000c5560: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ 000c5570: 2074 696d 6520 6465 7363 7269 6265 206f time describe o │ │ │ │ 000c5580: 6f20 2020 2020 2020 2020 2020 2020 2020 o │ │ │ │ 000c5590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c55a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c55b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000c55c0: 7573 6564 2030 2e30 3139 3039 3134 7320 used 0.0190914s │ │ │ │ -000c55d0: 2863 7075 293b 2030 2e30 3139 3039 3273 (cpu); 0.019092s │ │ │ │ +000c55c0: 7573 6564 2030 2e30 3139 3539 3537 7320 used 0.0195957s │ │ │ │ +000c55d0: 2863 7075 293b 2030 2e30 3139 3539 3773 (cpu); 0.019597s │ │ │ │ 000c55e0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 000c55f0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000c5600: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000c5610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c5620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c5630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c5640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -50732,17 +50732,17 @@ │ │ │ │ 000c62b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000c62c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 000c62d0: 203a 2074 696d 6520 7370 6563 6961 6c51 : time specialQ │ │ │ │ 000c62e0: 7561 6472 6174 6963 5472 616e 7366 6f72 uadraticTransfor │ │ │ │ 000c62f0: 6d61 7469 6f6e 2034 2020 2020 2020 2020 mation 4 │ │ │ │ 000c6300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c6310: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -000c6320: 2d20 7573 6564 2030 2e30 3735 3439 3538 - used 0.0754958 │ │ │ │ -000c6330: 7320 2863 7075 293b 2030 2e30 3735 3439 s (cpu); 0.07549 │ │ │ │ -000c6340: 3738 7320 2874 6872 6561 6429 3b20 3073 78s (thread); 0s │ │ │ │ +000c6320: 2d20 7573 6564 2030 2e30 3835 3238 3839 - used 0.0852889 │ │ │ │ +000c6330: 7320 2863 7075 293b 2030 2e30 3835 3238 s (cpu); 0.08528 │ │ │ │ +000c6340: 3933 7320 2874 6872 6561 6429 3b20 3073 93s (thread); 0s │ │ │ │ 000c6350: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000c6360: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000c6370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c6380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c6390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c63a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c63b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ @@ -51287,18 +51287,18 @@ │ │ │ │ 000c8560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000c8570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 000c8580: 203a 2074 696d 6520 6465 7363 7269 6265 : time describe │ │ │ │ 000c8590: 206f 6f20 2020 2020 2020 2020 2020 2020 oo │ │ │ │ 000c85a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c85b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c85c0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -000c85d0: 2d20 7573 6564 2030 2e31 3036 3238 3373 - used 0.106283s │ │ │ │ -000c85e0: 2028 6370 7529 3b20 302e 3033 3531 3132 (cpu); 0.035112 │ │ │ │ -000c85f0: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ -000c8600: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +000c85d0: 2d20 7573 6564 2030 2e31 3531 3736 3273 - used 0.151762s │ │ │ │ +000c85e0: 2028 6370 7529 3b20 302e 3033 3333 3638 (cpu); 0.033368 │ │ │ │ +000c85f0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +000c8600: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 000c8610: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000c8620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c8630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c8640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c8650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c8660: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ 000c8670: 203d 2072 6174 696f 6e61 6c20 6d61 7020 = rational map │ │ │ │ @@ -52398,17 +52398,17 @@ │ │ │ │ 000ccad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000ccae0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ 000ccaf0: 7469 6d65 2070 6869 2720 3d20 7661 6c75 time phi' = valu │ │ │ │ 000ccb00: 6520 7374 723b 2020 2020 2020 2020 2020 e str; │ │ │ │ 000ccb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccb30: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000ccb40: 302e 3032 3435 3337 7320 2863 7075 293b 0.024537s (cpu); │ │ │ │ -000ccb50: 2030 2e30 3234 3533 3635 7320 2874 6872 0.0245365s (thr │ │ │ │ -000ccb60: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +000ccb40: 302e 3032 3536 3934 3373 2028 6370 7529 0.0256943s (cpu) │ │ │ │ +000ccb50: 3b20 302e 3032 3536 3934 3273 2028 7468 ; 0.0256942s (th │ │ │ │ +000ccb60: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 000ccb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccb80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000ccb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 000ccbd0: 3420 3a20 5261 7469 6f6e 616c 4d61 7020 4 : RationalMap │ │ │ │ @@ -52422,16 +52422,16 @@ │ │ │ │ 000ccc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000ccc60: 2d2d 2d2d 2b0a 7c69 3520 3a20 7469 6d65 ----+.|i5 : time │ │ │ │ 000ccc70: 2064 6573 6372 6962 6520 7068 6927 2020 describe phi' │ │ │ │ 000ccc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cccb0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3030 |.| -- used 0.00 │ │ │ │ -000cccc0: 3534 3832 3538 7320 2863 7075 293b 2030 548258s (cpu); 0 │ │ │ │ -000cccd0: 2e30 3035 3438 3333 3773 2028 7468 7265 .00548337s (thre │ │ │ │ +000cccc0: 3636 3932 3338 7320 2863 7075 293b 2030 669238s (cpu); 0 │ │ │ │ +000cccd0: 2e30 3036 3730 3138 3173 2028 7468 7265 .00670181s (thre │ │ │ │ 000ccce0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000cccf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000ccd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccd40: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ @@ -52488,17 +52488,17 @@ │ │ │ │ 000cd070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000cd080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 000cd090: 3620 3a20 7469 6d65 2064 6573 6372 6962 6 : time describ │ │ │ │ 000cd0a0: 6520 696e 7665 7273 6520 7068 6927 2020 e inverse phi' │ │ │ │ 000cd0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd0d0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000cd0e0: 7365 6420 302e 3030 3435 3432 3137 7320 sed 0.00454217s │ │ │ │ -000cd0f0: 2863 7075 293b 2030 2e30 3034 3534 3736 (cpu); 0.0045476 │ │ │ │ -000cd100: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ +000cd0e0: 7365 6420 302e 3030 3536 3037 3032 7320 sed 0.00560702s │ │ │ │ +000cd0f0: 2863 7075 293b 2030 2e30 3035 3631 3736 (cpu); 0.0056176 │ │ │ │ +000cd100: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ 000cd110: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000cd120: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000cd130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd170: 7c0a 7c6f 3620 3d20 7261 7469 6f6e 616c |.|o6 = rational │ │ ├── ./usr/share/info/DGAlgebras.info.gz │ │ │ ├── DGAlgebras.info │ │ │ │ @@ -1231,17 +1231,17 @@ │ │ │ │ 00004ce0: 3a20 4842 203d 2048 4820 4220 2020 2020 : HB = HH B │ │ │ │ 00004cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004d20: 2020 2020 2020 2020 2020 7c0a 7c46 696e |.|Fin │ │ │ │ 00004d30: 6469 6e67 2065 6173 7920 7265 6c61 7469 ding easy relati │ │ │ │ 00004d40: 6f6e 7320 2020 2020 2020 2020 2020 3a20 ons : │ │ │ │ -00004d50: 202d 2d20 7573 6564 2030 2e30 3230 3831 -- used 0.02081 │ │ │ │ -00004d60: 3932 7320 2863 7075 293b 2030 2e30 3139 92s (cpu); 0.019 │ │ │ │ -00004d70: 3734 3134 7320 2020 2020 7c0a 7c20 2020 7414s |.| │ │ │ │ +00004d50: 202d 2d20 7573 6564 2030 2e30 3631 3037 -- used 0.06107 │ │ │ │ +00004d60: 3637 7320 2863 7075 293b 2030 2e30 3237 67s (cpu); 0.027 │ │ │ │ +00004d70: 3936 3273 2020 2020 2020 7c0a 7c20 2020 962s |.| │ │ │ │ 00004d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004dc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ 00004dd0: 3d20 4842 2020 2020 2020 2020 2020 2020 = HB │ │ │ │ 00004de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1495,16 +1495,16 @@ │ │ │ │ 00005d60: 6779 416c 6765 6272 6128 432c 4765 6e44 gyAlgebra(C,GenD │ │ │ │ 00005d70: 6567 7265 654c 696d 6974 3d3e 342c 5265 egreeLimit=>4,Re │ │ │ │ 00005d80: 6c44 6567 7265 654c 696d 6974 3d3e 3429 lDegreeLimit=>4) │ │ │ │ 00005d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005da0: 207c 0a7c 4669 6e64 696e 6720 6561 7379 |.|Finding easy │ │ │ │ 00005db0: 2072 656c 6174 696f 6e73 2020 2020 2020 relations │ │ │ │ 00005dc0: 2020 2020 203a 2020 2d2d 2075 7365 6420 : -- used │ │ │ │ -00005dd0: 302e 3031 3938 3733 3173 2028 6370 7529 0.0198731s (cpu) │ │ │ │ -00005de0: 3b20 302e 3031 3835 3939 3973 2020 2020 ; 0.0185999s │ │ │ │ +00005dd0: 302e 3033 3437 3531 3673 2028 6370 7529 0.0347516s (cpu) │ │ │ │ +00005de0: 3b20 302e 3032 3230 3731 3873 2020 2020 ; 0.0220718s │ │ │ │ 00005df0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00005e00: 2020 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: 207c 0a7c 2020 2020 2020 205a 5a20 2020 |.| ZZ │ │ │ │ 00005e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2723,17 +2723,17 @@ │ │ │ │ 0000aa20: 3720 3a20 484b 5220 3d20 4848 204b 5220 7 : HKR = HH KR │ │ │ │ 0000aa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa60: 2020 2020 2020 2020 2020 2020 7c0a 7c46 |.|F │ │ │ │ 0000aa70: 696e 6469 6e67 2065 6173 7920 7265 6c61 inding easy rela │ │ │ │ 0000aa80: 7469 6f6e 7320 2020 2020 2020 2020 2020 tions │ │ │ │ -0000aa90: 3a20 202d 2d20 7573 6564 2030 2e31 3137 : -- used 0.117 │ │ │ │ -0000aaa0: 3837 3773 2028 6370 7529 3b20 302e 3037 877s (cpu); 0.07 │ │ │ │ -0000aab0: 3737 3334 3173 2020 2020 2020 7c0a 7c20 77341s |.| │ │ │ │ +0000aa90: 3a20 202d 2d20 7573 6564 2030 2e32 3931 : -- used 0.291 │ │ │ │ +0000aaa0: 3136 3373 2028 6370 7529 3b20 302e 3037 163s (cpu); 0.07 │ │ │ │ +0000aab0: 3532 3339 3973 2020 2020 2020 7c0a 7c20 52399s |.| │ │ │ │ 0000aac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0000ab10: 3720 3d20 484b 5220 2020 2020 2020 2020 7 = HKR │ │ │ │ 0000ab20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2838,17 +2838,17 @@ │ │ │ │ 0000b150: 3130 203a 2048 4b52 2720 3d20 4848 206b 10 : HKR' = HH k │ │ │ │ 0000b160: 6f73 7a75 6c43 6f6d 706c 6578 4447 4120 oszulComplexDGA │ │ │ │ 0000b170: 5227 2020 2020 2020 2020 2020 2020 2020 R' │ │ │ │ 0000b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b190: 2020 2020 2020 2020 2020 2020 7c0a 7c46 |.|F │ │ │ │ 0000b1a0: 696e 6469 6e67 2065 6173 7920 7265 6c61 inding easy rela │ │ │ │ 0000b1b0: 7469 6f6e 7320 2020 2020 2020 2020 2020 tions │ │ │ │ -0000b1c0: 3a20 202d 2d20 7573 6564 2030 2e35 3136 : -- used 0.516 │ │ │ │ -0000b1d0: 3236 3573 2028 6370 7529 3b20 302e 3437 265s (cpu); 0.47 │ │ │ │ -0000b1e0: 3030 3573 2020 2020 2020 2020 7c0a 7c20 005s |.| │ │ │ │ +0000b1c0: 3a20 202d 2d20 7573 6564 2030 2e36 3834 : -- used 0.684 │ │ │ │ +0000b1d0: 3933 3873 2028 6370 7529 3b20 302e 3636 938s (cpu); 0.66 │ │ │ │ +0000b1e0: 3937 3135 7320 2020 2020 2020 7c0a 7c20 9715s |.| │ │ │ │ 0000b1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b230: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0000b240: 3130 203d 2048 4b52 2720 2020 2020 2020 10 = HKR' │ │ │ │ 0000b250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4569,16 +4569,16 @@ │ │ │ │ 00011d80: 2048 4820 6720 2020 2020 2020 2020 2020 HH g │ │ │ │ 00011d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011dc0: 2020 7c0a 7c46 696e 6469 6e67 2065 6173 |.|Finding eas │ │ │ │ 00011dd0: 7920 7265 6c61 7469 6f6e 7320 2020 2020 y relations │ │ │ │ 00011de0: 2020 2020 2020 3a20 202d 2d20 7573 6564 : -- used │ │ │ │ -00011df0: 2030 2e30 3136 3734 3231 7320 2863 7075 0.0167421s (cpu │ │ │ │ -00011e00: 293b 2030 2e30 3135 3733 3233 7320 2020 ); 0.0157323s │ │ │ │ +00011df0: 2030 2e30 3430 3136 3834 7320 2863 7075 0.0401684s (cpu │ │ │ │ +00011e00: 293b 2030 2e30 3139 3439 3935 7320 2020 ); 0.0194995s │ │ │ │ 00011e10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011e70: 2020 2020 2020 2020 2020 2020 2020 205a Z │ │ │ │ @@ -6296,16 +6296,16 @@ │ │ │ │ 00018970: 6c6f 6779 416c 6765 6272 6128 4129 2020 logyAlgebra(A) │ │ │ │ 00018980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000189a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 000189b0: 0a7c 4669 6e64 696e 6720 6561 7379 2072 .|Finding easy r │ │ │ │ 000189c0: 656c 6174 696f 6e73 2020 2020 2020 2020 elations │ │ │ │ 000189d0: 2020 203a 2020 2d2d 2075 7365 6420 302e : -- used 0. │ │ │ │ -000189e0: 3032 3033 3937 3873 2028 6370 7529 3b20 0203978s (cpu); │ │ │ │ -000189f0: 302e 3031 3830 3635 3973 2020 2020 207c 0.0180659s | │ │ │ │ +000189e0: 3033 3638 3634 3473 2028 6370 7529 3b20 0368644s (cpu); │ │ │ │ +000189f0: 302e 3032 3339 3432 3173 2020 2020 207c 0.0239421s | │ │ │ │ 00018a00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00018a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 2020 207c | │ │ │ │ 00018a50: 0a7c 6f34 203d 2048 4120 2020 2020 2020 .|o4 = HA │ │ │ │ 00018a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -15306,17 +15306,17 @@ │ │ │ │ 0003bc90: 6937 203a 2048 4867 203d 2048 4820 6720 i7 : HHg = HH g │ │ │ │ 0003bca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bcd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0003bce0: 4669 6e64 696e 6720 6561 7379 2072 656c Finding easy rel │ │ │ │ 0003bcf0: 6174 696f 6e73 2020 2020 2020 2020 2020 ations │ │ │ │ -0003bd00: 203a 2020 2d2d 2075 7365 6420 302e 3032 : -- used 0.02 │ │ │ │ -0003bd10: 3434 3735 3273 2028 6370 7529 3b20 302e 44752s (cpu); 0. │ │ │ │ -0003bd20: 3032 3239 3734 7320 2020 2020 207c 0a7c 022974s |.| │ │ │ │ +0003bd00: 203a 2020 2d2d 2075 7365 6420 302e 3034 : -- used 0.04 │ │ │ │ +0003bd10: 3636 3931 3673 2028 6370 7529 3b20 302e 66916s (cpu); 0. │ │ │ │ +0003bd20: 3031 3935 3634 3773 2020 2020 207c 0a7c 0195647s |.| │ │ │ │ 0003bd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bd70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0003bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bd90: 2020 2020 2020 2020 205a 5a20 2020 2020 ZZ │ │ │ │ @@ -15749,17 +15749,17 @@ │ │ │ │ 0003d840: 3a20 4841 203d 2068 6f6d 6f6c 6f67 7941 : HA = homologyA │ │ │ │ 0003d850: 6c67 6562 7261 2841 2920 2020 2020 2020 lgebra(A) │ │ │ │ 0003d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d880: 2020 2020 2020 2020 2020 7c0a 7c46 696e |.|Fin │ │ │ │ 0003d890: 6469 6e67 2065 6173 7920 7265 6c61 7469 ding easy relati │ │ │ │ 0003d8a0: 6f6e 7320 2020 2020 2020 2020 2020 3a20 ons : │ │ │ │ -0003d8b0: 202d 2d20 7573 6564 2030 2e30 3138 3436 -- used 0.01846 │ │ │ │ -0003d8c0: 3273 2028 6370 7529 3b20 302e 3031 3736 2s (cpu); 0.0176 │ │ │ │ -0003d8d0: 3631 3973 2020 2020 2020 7c0a 7c20 2020 619s |.| │ │ │ │ +0003d8b0: 202d 2d20 7573 6564 2030 2e30 3735 3332 -- used 0.07532 │ │ │ │ +0003d8c0: 3635 7320 2863 7075 293b 2030 2e30 3330 65s (cpu); 0.030 │ │ │ │ +0003d8d0: 3234 3932 7320 2020 2020 7c0a 7c20 2020 2492s |.| │ │ │ │ 0003d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d920: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ 0003d930: 3d20 4841 2020 2020 2020 2020 2020 2020 = HA │ │ │ │ 0003d940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -15912,16 +15912,16 @@ │ │ │ │ 0003e270: 6f6c 6f67 7941 6c67 6562 7261 2841 2920 ologyAlgebra(A) │ │ │ │ 0003e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e2b0: 7c0a 7c46 696e 6469 6e67 2065 6173 7920 |.|Finding easy │ │ │ │ 0003e2c0: 7265 6c61 7469 6f6e 7320 2020 2020 2020 relations │ │ │ │ 0003e2d0: 2020 2020 3a20 202d 2d20 7573 6564 2030 : -- used 0 │ │ │ │ -0003e2e0: 2e30 3834 3031 3236 7320 2863 7075 293b .0840126s (cpu); │ │ │ │ -0003e2f0: 2030 2e30 3831 3033 3939 7320 2020 2020 0.0810399s │ │ │ │ +0003e2e0: 2e34 3531 3231 3873 2028 6370 7529 3b20 .451218s (cpu); │ │ │ │ +0003e2f0: 302e 3135 3133 3435 7320 2020 2020 2020 0.151345s │ │ │ │ 0003e300: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e320: 2020 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: 7c0a 7c6f 3820 3d20 4841 2020 2020 2020 |.|o8 = HA │ │ │ │ 0003e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -16277,16 +16277,16 @@ │ │ │ │ 0003f940: 6d6f 6c6f 6779 416c 6765 6272 6128 4129 mologyAlgebra(A) │ │ │ │ 0003f950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003f980: 7c0a 7c46 696e 6469 6e67 2065 6173 7920 |.|Finding easy │ │ │ │ 0003f990: 7265 6c61 7469 6f6e 7320 2020 2020 2020 relations │ │ │ │ 0003f9a0: 2020 2020 3a20 202d 2d20 7573 6564 2030 : -- used 0 │ │ │ │ -0003f9b0: 2e30 3532 3435 3234 7320 2863 7075 293b .0524524s (cpu); │ │ │ │ -0003f9c0: 2030 2e30 3530 3633 3838 7320 2020 2020 0.0506388s │ │ │ │ +0003f9b0: 2e31 3534 3933 7320 2863 7075 293b 2030 .15493s (cpu); 0 │ │ │ │ +0003f9c0: 2e30 3831 3831 3737 7320 2020 2020 2020 .0818177s │ │ │ │ 0003f9d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003f9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003f9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fa20: 7c0a 7c6f 3136 203d 2048 4120 2020 2020 |.|o16 = HA │ │ │ │ 0003fa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -16488,17 +16488,17 @@ │ │ │ │ 00040670: 3231 203a 2048 4220 3d20 686f 6d6f 6c6f 21 : HB = homolo │ │ │ │ 00040680: 6779 416c 6765 6272 6128 422c 4765 6e44 gyAlgebra(B,GenD │ │ │ │ 00040690: 6567 7265 654c 696d 6974 3d3e 372c 5265 egreeLimit=>7,Re │ │ │ │ 000406a0: 6c44 6567 7265 654c 696d 6974 3d3e 3134 lDegreeLimit=>14 │ │ │ │ 000406b0: 2920 2020 2020 2020 2020 2020 7c0a 7c46 ) |.|F │ │ │ │ 000406c0: 696e 6469 6e67 2065 6173 7920 7265 6c61 inding easy rela │ │ │ │ 000406d0: 7469 6f6e 7320 2020 2020 2020 2020 2020 tions │ │ │ │ -000406e0: 3a20 202d 2d20 7573 6564 2030 2e30 3139 : -- used 0.019 │ │ │ │ -000406f0: 3735 3636 7320 2863 7075 293b 2030 2e30 7566s (cpu); 0.0 │ │ │ │ -00040700: 3138 3535 3473 2020 2020 2020 7c0a 7c20 18554s |.| │ │ │ │ +000406e0: 3a20 202d 2d20 7573 6564 2030 2e30 3539 : -- used 0.059 │ │ │ │ +000406f0: 3938 7320 2863 7075 293b 2030 2e30 3237 98s (cpu); 0.027 │ │ │ │ +00040700: 3037 3132 7320 2020 2020 2020 7c0a 7c20 0712s |.| │ │ │ │ 00040710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00040720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00040730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00040740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00040750: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 00040760: 3231 203d 2048 4220 2020 2020 2020 2020 21 = HB │ │ │ │ 00040770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -17245,17 +17245,17 @@ │ │ │ │ 000435c0: 203d 2048 4828 4b52 2920 2020 2020 2020 = HH(KR) │ │ │ │ 000435d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000435e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000435f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043600: 2020 2020 2020 207c 0a7c 4669 6e64 696e |.|Findin │ │ │ │ 00043610: 6720 6561 7379 2072 656c 6174 696f 6e73 g easy relations │ │ │ │ 00043620: 2020 2020 2020 2020 2020 203a 2020 2d2d : -- │ │ │ │ -00043630: 2075 7365 6420 302e 3031 3630 3734 3273 used 0.0160742s │ │ │ │ -00043640: 2028 6370 7529 3b20 302e 3031 3436 3938 (cpu); 0.014698 │ │ │ │ -00043650: 3873 2020 2020 207c 0a7c 2020 2020 2020 8s |.| │ │ │ │ +00043630: 2075 7365 6420 302e 3036 3237 3137 3573 used 0.0627175s │ │ │ │ +00043640: 2028 6370 7529 3b20 302e 3032 3139 3930 (cpu); 0.021990 │ │ │ │ +00043650: 3473 2020 2020 207c 0a7c 2020 2020 2020 4s |.| │ │ │ │ 00043660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000436a0: 2020 2020 2020 207c 0a7c 6f37 203d 2048 |.|o7 = H │ │ │ │ 000436b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000436c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -17611,16 +17611,16 @@ │ │ │ │ 00044ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044cc0: 2b0a 7c69 3620 3a20 484b 5220 3d20 4848 +.|i6 : HKR = HH │ │ │ │ 00044cd0: 284b 5229 2020 2020 2020 2020 2020 2020 (KR) │ │ │ │ 00044ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044d00: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00044d10: 2d20 7573 6564 2030 2e31 3734 3235 3973 - used 0.174259s │ │ │ │ -00044d20: 2028 6370 7529 3b20 302e 3133 3034 3834 (cpu); 0.130484 │ │ │ │ +00044d10: 2d20 7573 6564 2030 2e33 3235 3139 3773 - used 0.325197s │ │ │ │ +00044d20: 2028 6370 7529 3b20 302e 3231 3736 3933 (cpu); 0.217693 │ │ │ │ 00044d30: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00044d40: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00044d50: 2020 2020 2020 7c0a 7c46 696e 6469 6e67 |.|Finding │ │ │ │ 00044d60: 2065 6173 7920 7265 6c61 7469 6f6e 7320 easy relations │ │ │ │ 00044d70: 2020 2020 2020 2020 2020 3a20 2020 2020 : │ │ │ │ 00044d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -22288,16 +22288,16 @@ │ │ │ │ 000570f0: 6d61 7373 6579 5472 6970 6c65 5072 6f64 masseyTripleProd │ │ │ │ 00057100: 7563 7428 4b52 2c7a 312c 7a32 2c7a 3329 uct(KR,z1,z2,z3) │ │ │ │ 00057110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057130: 7c0a 7c46 696e 6469 6e67 2065 6173 7920 |.|Finding easy │ │ │ │ 00057140: 7265 6c61 7469 6f6e 7320 2020 2020 2020 relations │ │ │ │ 00057150: 2020 2020 3a20 202d 2d20 7573 6564 2030 : -- used 0 │ │ │ │ -00057160: 2e35 3332 3137 3773 2028 6370 7529 3b20 .532177s (cpu); │ │ │ │ -00057170: 302e 3438 3132 3639 7320 2020 2020 2020 0.481269s │ │ │ │ +00057160: 2e38 3037 3235 3873 2028 6370 7529 3b20 .807258s (cpu); │ │ │ │ +00057170: 302e 3636 3636 3136 7320 2020 2020 2020 0.666616s │ │ │ │ 00057180: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00057190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000571e0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -22791,17 +22791,17 @@ │ │ │ │ 00059060: 7c69 3520 3a20 4820 3d20 4848 284b 5229 |i5 : H = HH(KR) │ │ │ │ 00059070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000590a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000590b0: 7c46 696e 6469 6e67 2065 6173 7920 7265 |Finding easy re │ │ │ │ 000590c0: 6c61 7469 6f6e 7320 2020 2020 2020 2020 lations │ │ │ │ -000590d0: 2020 3a20 202d 2d20 7573 6564 2030 2e31 : -- used 0.1 │ │ │ │ -000590e0: 3434 3936 3373 2028 6370 7529 3b20 302e 44963s (cpu); 0. │ │ │ │ -000590f0: 3134 3130 3538 7320 2020 2020 2020 7c0a 141058s |. │ │ │ │ +000590d0: 2020 3a20 202d 2d20 7573 6564 2030 2e32 : -- used 0.2 │ │ │ │ +000590e0: 3336 3532 3273 2028 6370 7529 3b20 302e 36522s (cpu); 0. │ │ │ │ +000590f0: 3137 3838 3833 7320 2020 2020 2020 7c0a 178883s |. │ │ │ │ 00059100: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00059110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059140: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00059150: 7c6f 3520 3d20 4820 2020 2020 2020 2020 |o5 = H │ │ │ │ 00059160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -26223,17 +26223,17 @@ │ │ │ │ 000666e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000666f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066700: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2048 -------+.|i4 : H │ │ │ │ 00066710: 4220 3d20 746f 7241 6c67 6562 7261 2852 B = torAlgebra(R │ │ │ │ 00066720: 2c53 2c47 656e 4465 6772 6565 4c69 6d69 ,S,GenDegreeLimi │ │ │ │ 00066730: 743d 3e34 2c52 656c 4465 6772 6565 4c69 t=>4,RelDegreeLi │ │ │ │ 00066740: 6d69 743d 3e38 2920 2020 2020 2020 2020 mit=>8) │ │ │ │ -00066750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3436 |.| -- used 0.46 │ │ │ │ -00066760: 3538 3235 7320 2863 7075 293b 2030 2e34 5825s (cpu); 0.4 │ │ │ │ -00066770: 3034 3931 3173 2028 7468 7265 6164 293b 04911s (thread); │ │ │ │ +00066750: 7c0a 7c20 2d2d 2075 7365 6420 302e 3731 |.| -- used 0.71 │ │ │ │ +00066760: 3336 3838 7320 2863 7075 293b 2030 2e35 3688s (cpu); 0.5 │ │ │ │ +00066770: 3533 3637 3273 2028 7468 7265 6164 293b 53672s (thread); │ │ │ │ 00066780: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00066790: 2020 2020 2020 2020 207c 0a7c 4669 6e64 |.|Find │ │ │ │ 000667a0: 696e 6720 6561 7379 2072 656c 6174 696f ing easy relatio │ │ │ │ 000667b0: 6e73 2020 2020 2020 2020 2020 203a 2020 ns : │ │ │ │ 000667c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000667d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000667e0: 2020 7c0a 7c6f 3420 3d20 4842 2020 2020 |.|o4 = HB │ │ ├── ./usr/share/info/EdgeIdeals.info.gz │ │ │ ├── EdgeIdeals.info │ │ │ │ @@ -7842,16 +7842,16 @@ │ │ │ │ 0001ea10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001ea30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea60: 2020 7c0a 7c6f 3420 3d20 4879 7065 7247 |.|o4 = HyperG │ │ │ │ 0001ea70: 7261 7068 7b22 6564 6765 7322 203d 3e20 raph{"edges" => │ │ │ │ -0001ea80: 7b7b 622c 2063 7d2c 207b 612c 2065 7d2c {{b, c}, {a, e}, │ │ │ │ -0001ea90: 207b 632c 2064 7d7d 7d20 2020 2020 2020 {c, d}}} │ │ │ │ +0001ea80: 7b7b 632c 2064 7d2c 207b 622c 2063 7d2c {{c, d}, {b, c}, │ │ │ │ +0001ea90: 207b 612c 2065 7d7d 7d20 2020 2020 2020 {a, e}}} │ │ │ │ 0001eaa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0001eab0: 2020 2020 2020 2022 7269 6e67 2220 3d3e "ring" => │ │ │ │ 0001eac0: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 0001ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eae0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0001eaf0: 2020 2020 2020 2020 2022 7665 7274 6963 "vertic │ │ │ │ 0001eb00: 6573 2220 3d3e 207b 612c 2062 2c20 632c es" => {a, b, c, │ │ │ │ @@ -21417,16 +21417,16 @@ │ │ │ │ 00053a80: 2020 207c 0a7c 6f33 203d 2048 7970 6572 |.|o3 = Hyper │ │ │ │ 00053a90: 4772 6170 687b 2265 6467 6573 2220 3d3e Graph{"edges" => │ │ │ │ 00053aa0: 207b 7b78 202c 2078 202c 2078 207d 2c20 {{x , x , x }, │ │ │ │ 00053ab0: 7b78 202c 2078 207d 2c20 7b78 202c 2078 {x , x }, {x , x │ │ │ │ 00053ac0: 202c 2078 202c 2078 207d 7d7d 2020 2020 , x , x }}} │ │ │ │ 00053ad0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053af0: 2020 2020 3220 2020 3420 2020 3520 2020 2 4 5 │ │ │ │ -00053b00: 2020 3220 2020 3320 2020 2020 3120 2020 2 3 1 │ │ │ │ +00053af0: 2020 2020 3220 2020 3320 2020 3420 2020 2 3 4 │ │ │ │ +00053b00: 2020 3220 2020 3520 2020 2020 3120 2020 2 5 1 │ │ │ │ 00053b10: 3320 2020 3420 2020 3520 2020 2020 2020 3 4 5 │ │ │ │ 00053b20: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053b30: 2020 2020 2020 2272 696e 6722 203d 3e20 "ring" => │ │ │ │ 00053b40: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00053b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053b70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ @@ -21467,17 +21467,17 @@ │ │ │ │ 00053da0: 2020 207c 0a7c 6f34 203d 2048 7970 6572 |.|o4 = Hyper │ │ │ │ 00053db0: 4772 6170 687b 2265 6467 6573 2220 3d3e Graph{"edges" => │ │ │ │ 00053dc0: 207b 7b78 202c 2078 202c 2078 207d 2c20 {{x , x , x }, │ │ │ │ 00053dd0: 7b78 202c 2078 207d 2c20 7b78 202c 2078 {x , x }, {x , x │ │ │ │ 00053de0: 202c 2078 202c 2078 207d 7d7d 2020 2020 , x , x }}} │ │ │ │ 00053df0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053e10: 2020 2020 3320 2020 3420 2020 3520 2020 3 4 5 │ │ │ │ -00053e20: 2020 3120 2020 3320 2020 2020 3120 2020 1 3 1 │ │ │ │ -00053e30: 3220 2020 3420 2020 3520 2020 2020 2020 2 4 5 │ │ │ │ +00053e10: 2020 2020 3220 2020 3420 2020 3520 2020 2 4 5 │ │ │ │ +00053e20: 2020 3120 2020 3520 2020 2020 3120 2020 1 5 1 │ │ │ │ +00053e30: 3220 2020 3320 2020 3420 2020 2020 2020 2 3 4 │ │ │ │ 00053e40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053e50: 2020 2020 2020 2272 696e 6722 203d 3e20 "ring" => │ │ │ │ 00053e60: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00053e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053e90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053ea0: 2020 2020 2020 2276 6572 7469 6365 7322 "vertices" │ │ ├── ./usr/share/info/EigenSolver.info.gz │ │ │ ├── EigenSolver.info │ │ │ │ @@ -171,15 +171,15 @@ │ │ │ │ 00000aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000ac0: 2b0a 7c69 3320 3a20 656c 6170 7365 6454 +.|i3 : elapsedT │ │ │ │ 00000ad0: 696d 6520 736f 6c73 203d 207a 6572 6f44 ime sols = zeroD │ │ │ │ 00000ae0: 696d 536f 6c76 6520 493b 2020 2020 2020 imSolve I; │ │ │ │ 00000af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00000b10: 7c0a 7c20 2d2d 202e 3234 3332 3432 7320 |.| -- .243242s │ │ │ │ +00000b10: 7c0a 7c20 2d2d 202e 3235 3138 3737 7320 |.| -- .251877s │ │ │ │ 00000b20: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00000b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b60: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00000b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/Elimination.info.gz │ │ │ ├── Elimination.info │ │ │ │ @@ -336,17 +336,17 @@ │ │ │ │ 000014f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00001520: 6934 203a 2074 696d 6520 656c 696d 696e i4 : time elimin │ │ │ │ 00001530: 6174 6528 782c 6964 6561 6c28 662c 6729 ate(x,ideal(f,g) │ │ │ │ 00001540: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00001550: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00001560: 7573 6564 2030 2e30 3032 3639 3937 3473 used 0.00269974s │ │ │ │ -00001570: 2028 6370 7529 3b20 302e 3030 3236 3937 (cpu); 0.002697 │ │ │ │ -00001580: 3034 7320 2874 6872 6561 6429 3b20 3073 04s (thread); 0s │ │ │ │ +00001560: 7573 6564 2030 2e30 3033 3238 3337 3473 used 0.00328374s │ │ │ │ +00001570: 2028 6370 7529 3b20 302e 3030 3332 3831 (cpu); 0.003281 │ │ │ │ +00001580: 3234 7320 2874 6872 6561 6429 3b20 3073 24s (thread); 0s │ │ │ │ 00001590: 2028 6763 297c 0a7c 2020 2020 2020 2020 (gc)|.| │ │ │ │ 000015a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000015b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000015c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000015d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000015e0: 2020 2020 2020 2020 2020 3220 2020 2032 2 2 │ │ │ │ 000015f0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ @@ -366,17 +366,17 @@ │ │ │ │ 000016d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000016e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000016f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00001700: 6935 203a 2074 696d 6520 6964 6561 6c20 i5 : time ideal │ │ │ │ 00001710: 7265 7375 6c74 616e 7428 662c 672c 7829 resultant(f,g,x) │ │ │ │ 00001720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001730: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00001740: 7573 6564 2030 2e30 3031 3636 3836 3373 used 0.00166863s │ │ │ │ -00001750: 2028 6370 7529 3b20 302e 3030 3136 3639 (cpu); 0.001669 │ │ │ │ -00001760: 3032 7320 2874 6872 6561 6429 3b20 3073 02s (thread); 0s │ │ │ │ +00001740: 7573 6564 2030 2e30 3031 3833 3536 3373 used 0.00183563s │ │ │ │ +00001750: 2028 6370 7529 3b20 302e 3030 3138 3337 (cpu); 0.001837 │ │ │ │ +00001760: 3937 7320 2874 6872 6561 6429 3b20 3073 97s (thread); 0s │ │ │ │ 00001770: 2028 6763 297c 0a7c 2020 2020 2020 2020 (gc)|.| │ │ │ │ 00001780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000017a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000017b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000017c0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ 000017d0: 2032 2020 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ @@ -620,17 +620,17 @@ │ │ │ │ 000026b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000026c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000026d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000026e0: 2b0a 7c69 3420 3a20 7469 6d65 2065 6c69 +.|i4 : time eli │ │ │ │ 000026f0: 6d69 6e61 7465 2878 2c69 6465 616c 2866 minate(x,ideal(f │ │ │ │ 00002700: 2c67 2929 2020 2020 2020 2020 2020 2020 ,g)) │ │ │ │ 00002710: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00002720: 2d2d 2075 7365 6420 302e 3030 3238 3333 -- used 0.002833 │ │ │ │ -00002730: 3033 7320 2863 7075 293b 2030 2e30 3032 03s (cpu); 0.002 │ │ │ │ -00002740: 3833 3035 3873 2028 7468 7265 6164 293b 83058s (thread); │ │ │ │ +00002720: 2d2d 2075 7365 6420 302e 3030 3333 3033 -- used 0.003303 │ │ │ │ +00002730: 3739 7320 2863 7075 293b 2030 2e30 3033 79s (cpu); 0.003 │ │ │ │ +00002740: 3239 3733 3273 2028 7468 7265 6164 293b 29732s (thread); │ │ │ │ 00002750: 2030 7320 2867 6329 7c0a 7c20 2020 2020 0s (gc)|.| │ │ │ │ 00002760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002790: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000027a0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ 000027b0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -650,17 +650,17 @@ │ │ │ │ 00002890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000028a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000028b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000028c0: 2b0a 7c69 3520 3a20 7469 6d65 2069 6465 +.|i5 : time ide │ │ │ │ 000028d0: 616c 2072 6573 756c 7461 6e74 2866 2c67 al resultant(f,g │ │ │ │ 000028e0: 2c78 2920 2020 2020 2020 2020 2020 2020 ,x) │ │ │ │ 000028f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00002900: 2d2d 2075 7365 6420 302e 3030 3136 3836 -- used 0.001686 │ │ │ │ -00002910: 3739 7320 2863 7075 293b 2030 2e30 3031 79s (cpu); 0.001 │ │ │ │ -00002920: 3638 3730 3773 2028 7468 7265 6164 293b 68707s (thread); │ │ │ │ +00002900: 2d2d 2075 7365 6420 302e 3030 3138 3534 -- used 0.001854 │ │ │ │ +00002910: 3337 7320 2863 7075 293b 2030 2e30 3031 37s (cpu); 0.001 │ │ │ │ +00002920: 3835 3537 3373 2028 7468 7265 6164 293b 85573s (thread); │ │ │ │ 00002930: 2030 7320 2867 6329 7c0a 7c20 2020 2020 0s (gc)|.| │ │ │ │ 00002940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002970: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00002980: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 00002990: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -995,17 +995,17 @@ │ │ │ │ 00003e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003e30: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2065 --+.|i4 : time e │ │ │ │ 00003e40: 6c69 6d69 6e61 7465 2869 6465 616c 2866 liminate(ideal(f │ │ │ │ 00003e50: 2c67 292c 7829 2020 2020 2020 2020 2020 ,g),x) │ │ │ │ 00003e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003e80: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e |.| -- used 1. │ │ │ │ -00003e90: 3633 3573 2028 6370 7529 3b20 312e 3336 635s (cpu); 1.36 │ │ │ │ -00003ea0: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ -00003eb0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00003e90: 3532 3938 3173 2028 6370 7529 3b20 312e 52981s (cpu); 1. │ │ │ │ +00003ea0: 3333 3935 3973 2028 7468 7265 6164 293b 33959s (thread); │ │ │ │ +00003eb0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00003ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003ed0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00003ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003f20: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ @@ -1275,16 +1275,16 @@ │ │ │ │ 00004fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004fb0: 2d2d 2b0a 7c69 3520 3a20 7469 6d65 2069 --+.|i5 : time i │ │ │ │ 00004fc0: 6465 616c 2072 6573 756c 7461 6e74 2866 deal resultant(f │ │ │ │ 00004fd0: 2c67 2c78 2920 2020 2020 2020 2020 2020 ,g,x) │ │ │ │ 00004fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005000: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00005010: 3031 3630 3336 3473 2028 6370 7529 3b20 0160364s (cpu); │ │ │ │ -00005020: 302e 3031 3630 3338 3473 2028 7468 7265 0.0160384s (thre │ │ │ │ +00005010: 3031 3735 3536 3973 2028 6370 7529 3b20 0175569s (cpu); │ │ │ │ +00005020: 302e 3031 3735 3632 3273 2028 7468 7265 0.0175622s (thre │ │ │ │ 00005030: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00005040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005050: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00005060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1917,17 +1917,17 @@ │ │ │ │ 000077c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000077d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ 000077e0: 3a20 7469 6d65 2065 6c69 6d69 6e61 7465 : time eliminate │ │ │ │ 000077f0: 2869 6465 616c 2866 2c67 292c 7829 2020 (ideal(f,g),x) │ │ │ │ 00007800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007820: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00007830: 2075 7365 6420 312e 3630 3235 3473 2028 used 1.60254s ( │ │ │ │ -00007840: 6370 7529 3b20 312e 3338 3334 3573 2028 cpu); 1.38345s ( │ │ │ │ -00007850: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00007830: 2075 7365 6420 312e 3538 3130 3473 2028 used 1.58104s ( │ │ │ │ +00007840: 6370 7529 3b20 312e 3430 3739 7320 2874 cpu); 1.4079s (t │ │ │ │ +00007850: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00007860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007870: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00007880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000078a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000078b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000078c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ @@ -2197,18 +2197,18 @@ │ │ │ │ 00008940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008950: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ 00008960: 3a20 7469 6d65 2069 6465 616c 2072 6573 : time ideal res │ │ │ │ 00008970: 756c 7461 6e74 2866 2c67 2c78 2920 2020 ultant(f,g,x) │ │ │ │ 00008980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089a0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -000089b0: 2075 7365 6420 302e 3031 3530 3436 7320 used 0.015046s │ │ │ │ -000089c0: 2863 7075 293b 2030 2e30 3135 3034 3731 (cpu); 0.0150471 │ │ │ │ -000089d0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -000089e0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +000089b0: 2075 7365 6420 302e 3031 3534 3334 3673 used 0.0154346s │ │ │ │ +000089c0: 2028 6370 7529 3b20 302e 3031 3534 3336 (cpu); 0.015436 │ │ │ │ +000089d0: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ +000089e0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000089f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00008a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00008a50: 2020 2020 2020 2020 2020 2037 2020 2020 7 │ │ ├── ./usr/share/info/EnumerationCurves.info.gz │ │ │ ├── EnumerationCurves.info │ │ │ │ @@ -256,16 +256,16 @@ │ │ │ │ 00000ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001000: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7469 ------+.|i1 : ti │ │ │ │ 00001010: 6d65 2066 6f72 206e 2066 726f 6d20 3220 me for n from 2 │ │ │ │ 00001020: 746f 2031 3020 6c69 7374 206c 696e 6573 to 10 list lines │ │ │ │ 00001030: 4879 7065 7273 7572 6661 6365 286e 2920 Hypersurface(n) │ │ │ │ 00001040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001050: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00001060: 6420 302e 3032 3737 3030 3873 2028 6370 d 0.0277008s (cp │ │ │ │ -00001070: 7529 3b20 302e 3032 3737 3031 3273 2028 u); 0.0277012s ( │ │ │ │ +00001060: 6420 302e 3033 3236 3235 3573 2028 6370 d 0.0326255s (cp │ │ │ │ +00001070: 7529 3b20 302e 3033 3236 3237 3473 2028 u); 0.0326274s ( │ │ │ │ 00001080: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00001090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000010a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000010b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000010c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000010d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000010e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -649,17 +649,17 @@ │ │ │ │ 00002880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000028a0: 2d2d 2b0a 7c69 3720 3a20 7469 6d65 2066 --+.|i7 : time f │ │ │ │ 000028b0: 6f72 2044 2069 6e20 5420 6c69 7374 2072 or D in T list r │ │ │ │ 000028c0: 6174 696f 6e61 6c43 7572 7665 2832 2c44 ationalCurve(2,D │ │ │ │ 000028d0: 2920 2d20 7261 7469 6f6e 616c 4375 7276 ) - rationalCurv │ │ │ │ 000028e0: 6528 312c 4429 2f38 7c0a 7c20 2d2d 2075 e(1,D)/8|.| -- u │ │ │ │ -000028f0: 7365 6420 302e 3333 3536 3633 7320 2863 sed 0.335663s (c │ │ │ │ -00002900: 7075 293b 2030 2e32 3833 3934 3573 2028 pu); 0.283945s ( │ │ │ │ -00002910: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000028f0: 7365 6420 302e 3338 3035 3037 7320 2863 sed 0.380507s (c │ │ │ │ +00002900: 7075 293b 2030 2e33 3130 3832 7320 2874 pu); 0.31082s (t │ │ │ │ +00002910: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00002920: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00002930: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00002940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002970: 2020 2020 7c0a 7c6f 3720 3d20 7b36 3039 |.|o7 = {609 │ │ │ │ 00002980: 3235 302c 2039 3232 3838 2c20 3532 3831 250, 92288, 5281 │ │ │ │ @@ -685,16 +685,16 @@ │ │ │ │ 00002ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00002af0: 3820 3a20 7469 6d65 2072 6174 696f 6e61 8 : time rationa │ │ │ │ 00002b00: 6c43 7572 7665 2833 2920 2020 2020 2020 lCurve(3) │ │ │ │ 00002b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002b20: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00002b30: 6564 2030 2e32 3133 3134 3373 2028 6370 ed 0.213143s (cp │ │ │ │ -00002b40: 7529 3b20 302e 3136 3032 3233 7320 2874 u); 0.160223s (t │ │ │ │ +00002b30: 6564 2030 2e31 3333 3631 3873 2028 6370 ed 0.133618s (cp │ │ │ │ +00002b40: 7529 3b20 302e 3133 3336 3239 7320 2874 u); 0.133629s (t │ │ │ │ 00002b50: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00002b60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00002b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002b90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00002ba0: 2020 2020 2038 3536 3435 3735 3030 3020 8564575000 │ │ │ │ 00002bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -718,16 +718,16 @@ │ │ │ │ 00002cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00002d00: 0a7c 6939 203a 2074 696d 6520 666f 7220 .|i9 : time for │ │ │ │ 00002d10: 4420 696e 2054 206c 6973 7420 7261 7469 D in T list rati │ │ │ │ 00002d20: 6f6e 616c 4375 7276 6528 332c 4429 2020 onalCurve(3,D) │ │ │ │ 00002d30: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00002d40: 2075 7365 6420 352e 3431 3534 3373 2028 used 5.41543s ( │ │ │ │ -00002d50: 6370 7529 3b20 342e 3731 3033 3973 2028 cpu); 4.71039s ( │ │ │ │ +00002d40: 2075 7365 6420 352e 3333 3936 3173 2028 used 5.33961s ( │ │ │ │ +00002d50: 6370 7529 3b20 342e 3535 3832 3573 2028 cpu); 4.55825s ( │ │ │ │ 00002d60: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00002d70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00002d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002db0: 7c0a 7c20 2020 2020 2038 3536 3435 3735 |.| 8564575 │ │ │ │ 00002dc0: 3030 3020 2034 3232 3639 3038 3136 2020 000 422690816 │ │ │ │ @@ -757,273 +757,275 @@ │ │ │ │ 00002f40: 6e20 6120 6765 6e65 7261 6c20 7175 696e n a general quin │ │ │ │ 00002f50: 7469 6320 7468 7265 6566 6f6c 6420 6361 tic threefold ca │ │ │ │ 00002f60: 6e20 6265 0a63 6f6d 7075 7465 6420 6173 n be.computed as │ │ │ │ 00002f70: 2066 6f6c 6c6f 7773 3a0a 0a0a 0a2b 2d2d follows:....+-- │ │ │ │ 00002f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00002fb0: 2d2b 0a7c 6931 3020 3a20 7469 6d65 2072 -+.|i10 : time r │ │ │ │ -00002fc0: 6174 696f 6e61 6c43 7572 7665 2833 2920 ationalCurve(3) │ │ │ │ -00002fd0: 2d20 7261 7469 6f6e 616c 4375 7276 6528 - rationalCurve( │ │ │ │ -00002fe0: 3129 2f32 3720 207c 0a7c 202d 2d20 7573 1)/27 |.| -- us │ │ │ │ -00002ff0: 6564 2030 2e32 3235 3573 2028 6370 7529 ed 0.2255s (cpu) │ │ │ │ -00003000: 3b20 302e 3137 3035 3039 7320 2874 6872 ; 0.170509s (thr │ │ │ │ -00003010: 6561 6429 3b20 3073 2028 6763 297c 0a7c ead); 0s (gc)|.| │ │ │ │ -00003020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00002fb0: 2d2d 2d2b 0a7c 6931 3020 3a20 7469 6d65 ---+.|i10 : time │ │ │ │ +00002fc0: 2072 6174 696f 6e61 6c43 7572 7665 2833 rationalCurve(3 │ │ │ │ +00002fd0: 2920 2d20 7261 7469 6f6e 616c 4375 7276 ) - rationalCurv │ │ │ │ +00002fe0: 6528 3129 2f32 3720 2020 207c 0a7c 202d e(1)/27 |.| - │ │ │ │ +00002ff0: 2d20 7573 6564 2030 2e31 3337 3338 3373 - used 0.137383s │ │ │ │ +00003000: 2028 6370 7529 3b20 302e 3133 3733 3932 (cpu); 0.137392 │ │ │ │ +00003010: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00003020: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ 00003030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003050: 2020 207c 0a7c 6f31 3020 3d20 3331 3732 |.|o10 = 3172 │ │ │ │ -00003060: 3036 3337 3520 2020 2020 2020 2020 2020 06375 │ │ │ │ +00003050: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00003060: 3020 3d20 3331 3732 3036 3337 3520 2020 0 = 317206375 │ │ │ │ 00003070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003080: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00003090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003090: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000030a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000030b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000030c0: 0a7c 6f31 3020 3a20 5151 2020 2020 2020 .|o10 : QQ │ │ │ │ -000030d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000030b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000030c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +000030d0: 3020 3a20 5151 2020 2020 2020 2020 2020 0 : QQ │ │ │ │ 000030e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000030f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00003100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000030f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003100: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00003110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468 -----------+..Th │ │ │ │ -00003130: 6520 6e75 6d62 6572 7320 6f66 2072 6174 e numbers of rat │ │ │ │ -00003140: 696f 6e61 6c20 6375 7276 6573 206f 6620 ional curves of │ │ │ │ -00003150: 6465 6772 6565 2033 206f 6e20 6765 6e65 degree 3 on gene │ │ │ │ -00003160: 7261 6c20 636f 6d70 6c65 7465 2069 6e74 ral complete int │ │ │ │ -00003170: 6572 7365 6374 696f 6e0a 4361 6c61 6269 ersection.Calabi │ │ │ │ -00003180: 2d59 6175 2074 6872 6565 666f 6c64 7320 -Yau threefolds │ │ │ │ -00003190: 6361 6e20 6265 2063 6f6d 7075 7465 6420 can be computed │ │ │ │ -000031a0: 6173 2066 6f6c 6c6f 7773 3a0a 0a0a 0a2b as follows:....+ │ │ │ │ -000031b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468 -----------+..Th │ │ │ │ +00003140: 6520 6e75 6d62 6572 7320 6f66 2072 6174 e numbers of rat │ │ │ │ +00003150: 696f 6e61 6c20 6375 7276 6573 206f 6620 ional curves of │ │ │ │ +00003160: 6465 6772 6565 2033 206f 6e20 6765 6e65 degree 3 on gene │ │ │ │ +00003170: 7261 6c20 636f 6d70 6c65 7465 2069 6e74 ral complete int │ │ │ │ +00003180: 6572 7365 6374 696f 6e0a 4361 6c61 6269 ersection.Calabi │ │ │ │ +00003190: 2d59 6175 2074 6872 6565 666f 6c64 7320 -Yau threefolds │ │ │ │ +000031a0: 6361 6e20 6265 2063 6f6d 7075 7465 6420 can be computed │ │ │ │ +000031b0: 6173 2066 6f6c 6c6f 7773 3a0a 0a0a 0a2b as follows:....+ │ │ │ │ 000031c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000031d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000031e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000031f0: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 7469 -----+.|i11 : ti │ │ │ │ -00003200: 6d65 2066 6f72 2044 2069 6e20 5420 6c69 me for D in T li │ │ │ │ -00003210: 7374 2072 6174 696f 6e61 6c43 7572 7665 st rationalCurve │ │ │ │ -00003220: 2833 2c44 2920 2d20 7261 7469 6f6e 616c (3,D) - rational │ │ │ │ -00003230: 4375 7276 6528 312c 4429 2f32 377c 0a7c Curve(1,D)/27|.| │ │ │ │ -00003240: 202d 2d20 7573 6564 2035 2e37 3730 3339 -- used 5.77039 │ │ │ │ -00003250: 7320 2863 7075 293b 2035 2e30 3033 3236 s (cpu); 5.00326 │ │ │ │ -00003260: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -00003270: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ -00003280: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00003290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000031f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003200: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 7469 -----+.|i11 : ti │ │ │ │ +00003210: 6d65 2066 6f72 2044 2069 6e20 5420 6c69 me for D in T li │ │ │ │ +00003220: 7374 2072 6174 696f 6e61 6c43 7572 7665 st rationalCurve │ │ │ │ +00003230: 2833 2c44 2920 2d20 7261 7469 6f6e 616c (3,D) - rational │ │ │ │ +00003240: 4375 7276 6528 312c 4429 2f32 377c 0a7c Curve(1,D)/27|.| │ │ │ │ +00003250: 202d 2d20 7573 6564 2035 2e35 3634 3338 -- used 5.56438 │ │ │ │ +00003260: 7320 2863 7075 293b 2034 2e37 3138 3137 s (cpu); 4.71817 │ │ │ │ +00003270: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00003280: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +00003290: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000032a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000032b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000032c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000032d0: 6f31 3120 3d20 7b33 3137 3230 3633 3735 o11 = {317206375 │ │ │ │ -000032e0: 2c20 3135 3635 3531 3638 2c20 3634 3234 , 15655168, 6424 │ │ │ │ -000032f0: 3332 362c 2031 3631 3135 3034 2c20 3431 326, 1611504, 41 │ │ │ │ -00003300: 3632 3536 7d20 2020 2020 2020 2020 2020 6256} │ │ │ │ -00003310: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00003320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000032c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000032d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000032e0: 6f31 3120 3d20 7b33 3137 3230 3633 3735 o11 = {317206375 │ │ │ │ +000032f0: 2c20 3135 3635 3531 3638 2c20 3634 3234 , 15655168, 6424 │ │ │ │ +00003300: 3332 362c 2031 3631 3135 3034 2c20 3431 326, 1611504, 41 │ │ │ │ +00003310: 3632 3536 7d20 2020 2020 2020 2020 2020 6256} │ │ │ │ +00003320: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00003330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003350: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00003360: 6f31 3120 3a20 4c69 7374 2020 2020 2020 o11 : List │ │ │ │ -00003370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003360: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00003370: 6f31 3120 3a20 4c69 7374 2020 2020 2020 o11 : List │ │ │ │ 00003380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000033a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -000033b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000033a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000033b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 000033c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000033d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000033e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -000033f0: 466f 7220 7261 7469 6f6e 616c 2063 7572 For rational cur │ │ │ │ -00003400: 7665 7320 6f66 2064 6567 7265 6520 343a ves of degree 4: │ │ │ │ -00003410: 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ....+----------- │ │ │ │ -00003420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000033e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00003400: 466f 7220 7261 7469 6f6e 616c 2063 7572 For rational cur │ │ │ │ +00003410: 7665 7320 6f66 2064 6567 7265 6520 343a ves of degree 4: │ │ │ │ +00003420: 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ....+----------- │ │ │ │ 00003430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003440: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a --------+.|i12 : │ │ │ │ -00003450: 2074 696d 6520 7261 7469 6f6e 616c 4375 time rationalCu │ │ │ │ -00003460: 7276 6528 3429 2020 2020 2020 2020 2020 rve(4) │ │ │ │ -00003470: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00003480: 7c20 2d2d 2075 7365 6420 312e 3931 3631 | -- used 1.9161 │ │ │ │ -00003490: 3673 2028 6370 7529 3b20 312e 3730 3237 6s (cpu); 1.7027 │ │ │ │ -000034a0: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ -000034b0: 2867 6329 7c0a 7c20 2020 2020 2020 2020 (gc)|.| │ │ │ │ -000034c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a --------+.|i12 : │ │ │ │ +00003460: 2074 696d 6520 7261 7469 6f6e 616c 4375 time rationalCu │ │ │ │ +00003470: 7276 6528 3429 2020 2020 2020 2020 2020 rve(4) │ │ │ │ +00003480: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00003490: 7c20 2d2d 2075 7365 6420 312e 3535 3736 | -- used 1.5576 │ │ │ │ +000034a0: 3573 2028 6370 7529 3b20 312e 3338 3134 5s (cpu); 1.3814 │ │ │ │ +000034b0: 3173 2028 7468 7265 6164 293b 2030 7320 1s (thread); 0s │ │ │ │ +000034c0: 2867 6329 7c0a 7c20 2020 2020 2020 2020 (gc)|.| │ │ │ │ 000034d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000034e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000034f0: 2020 2031 3535 3137 3932 3637 3936 3837 1551792679687 │ │ │ │ -00003500: 3520 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ -00003510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003520: 7c0a 7c6f 3132 203d 202d 2d2d 2d2d 2d2d |.|o12 = ------- │ │ │ │ -00003530: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020 ------- │ │ │ │ -00003540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003550: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00003560: 2020 2020 2036 3420 2020 2020 2020 2020 64 │ │ │ │ -00003570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003580: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00003590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000034e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000034f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00003500: 2020 2031 3535 3137 3932 3637 3936 3837 1551792679687 │ │ │ │ +00003510: 3520 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ +00003520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003530: 7c0a 7c6f 3132 203d 202d 2d2d 2d2d 2d2d |.|o12 = ------- │ │ │ │ +00003540: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020 ------- │ │ │ │ +00003550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003560: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00003570: 2020 2020 2036 3420 2020 2020 2020 2020 64 │ │ │ │ +00003580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003590: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000035a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000035b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000035c0: 2020 7c0a 7c6f 3132 203a 2051 5120 2020 |.|o12 : QQ │ │ │ │ -000035d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000035c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000035d0: 2020 7c0a 7c6f 3132 203a 2051 5120 2020 |.|o12 : QQ │ │ │ │ 000035e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000035f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -00003600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000035f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003600: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00003610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00003630: 7c69 3133 203a 2074 696d 6520 7261 7469 |i13 : time rati │ │ │ │ -00003640: 6f6e 616c 4375 7276 6528 342c 7b34 2c32 onalCurve(4,{4,2 │ │ │ │ -00003650: 7d29 2020 2020 2020 2020 2020 2020 2020 }) │ │ │ │ -00003660: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00003670: 372e 3139 3131 3673 2028 6370 7529 3b20 7.19116s (cpu); │ │ │ │ -00003680: 352e 3730 3130 3973 2028 7468 7265 6164 5.70109s (thread │ │ │ │ -00003690: 293b 2030 7320 2867 6329 7c0a 7c20 2020 ); 0s (gc)|.| │ │ │ │ -000036a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00003640: 7c69 3133 203a 2074 696d 6520 7261 7469 |i13 : time rati │ │ │ │ +00003650: 6f6e 616c 4375 7276 6528 342c 7b34 2c32 onalCurve(4,{4,2 │ │ │ │ +00003660: 7d29 2020 2020 2020 2020 2020 2020 2020 }) │ │ │ │ +00003670: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ +00003680: 372e 3632 3833 3173 2028 6370 7529 3b20 7.62831s (cpu); │ │ │ │ +00003690: 362e 3132 3438 3473 2028 7468 7265 6164 6.12484s (thread │ │ │ │ +000036a0: 293b 2030 7320 2867 6329 7c0a 7c20 2020 ); 0s (gc)|.| │ │ │ │ 000036b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000036c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000036d0: 7c0a 7c6f 3133 203d 2033 3838 3339 3134 |.|o13 = 3883914 │ │ │ │ -000036e0: 3038 3420 2020 2020 2020 2020 2020 2020 084 │ │ │ │ -000036f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003700: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00003710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000036d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000036e0: 7c0a 7c6f 3133 203d 2033 3838 3339 3134 |.|o13 = 3883914 │ │ │ │ +000036f0: 3038 3420 2020 2020 2020 2020 2020 2020 084 │ │ │ │ +00003700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003710: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00003720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003730: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00003740: 3133 203a 2051 5120 2020 2020 2020 2020 13 : QQ │ │ │ │ -00003750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003740: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00003750: 3133 203a 2051 5120 2020 2020 2020 2020 13 : QQ │ │ │ │ 00003760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003770: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00003780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003780: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00003790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000037a0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6865 206e --------+..The n │ │ │ │ -000037b0: 756d 6265 7220 6f66 2072 6174 696f 6e61 umber of rationa │ │ │ │ -000037c0: 6c20 6375 7276 6573 206f 6620 6465 6772 l curves of degr │ │ │ │ -000037d0: 6565 2034 206f 6e20 6120 6765 6e65 7261 ee 4 on a genera │ │ │ │ -000037e0: 6c20 7175 696e 7469 6320 7468 7265 6566 l quintic threef │ │ │ │ -000037f0: 6f6c 6420 6361 6e20 6265 0a63 6f6d 7075 old can be.compu │ │ │ │ -00003800: 7465 6420 6173 2066 6f6c 6c6f 7773 3a0a ted as follows:. │ │ │ │ -00003810: 0a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ -00003820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000037a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000037b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6865 206e --------+..The n │ │ │ │ +000037c0: 756d 6265 7220 6f66 2072 6174 696f 6e61 umber of rationa │ │ │ │ +000037d0: 6c20 6375 7276 6573 206f 6620 6465 6772 l curves of degr │ │ │ │ +000037e0: 6565 2034 206f 6e20 6120 6765 6e65 7261 ee 4 on a genera │ │ │ │ +000037f0: 6c20 7175 696e 7469 6320 7468 7265 6566 l quintic threef │ │ │ │ +00003800: 6f6c 6420 6361 6e20 6265 0a63 6f6d 7075 old can be.compu │ │ │ │ +00003810: 7465 6420 6173 2066 6f6c 6c6f 7773 3a0a ted as follows:. │ │ │ │ +00003820: 0a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 00003830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003840: 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a 2074 ------+.|i14 : t │ │ │ │ -00003850: 696d 6520 7261 7469 6f6e 616c 4375 7276 ime rationalCurv │ │ │ │ -00003860: 6528 3429 202d 2072 6174 696f 6e61 6c43 e(4) - rationalC │ │ │ │ -00003870: 7572 7665 2832 292f 3820 207c 0a7c 202d urve(2)/8 |.| - │ │ │ │ -00003880: 2d20 7573 6564 2031 2e36 3536 3838 7320 - used 1.65688s │ │ │ │ -00003890: 2863 7075 293b 2031 2e34 3437 3573 2028 (cpu); 1.4475s ( │ │ │ │ -000038a0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -000038b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000038c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003850: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 -------+.|i14 : │ │ │ │ +00003860: 7469 6d65 2072 6174 696f 6e61 6c43 7572 time rationalCur │ │ │ │ +00003870: 7665 2834 2920 2d20 7261 7469 6f6e 616c ve(4) - rational │ │ │ │ +00003880: 4375 7276 6528 3229 2f38 2020 207c 0a7c Curve(2)/8 |.| │ │ │ │ +00003890: 202d 2d20 7573 6564 2031 2e37 3239 3832 -- used 1.72982 │ │ │ │ +000038a0: 7320 2863 7075 293b 2031 2e34 3739 3834 s (cpu); 1.47984 │ │ │ │ +000038b0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +000038c0: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ 000038d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000038e0: 2020 2020 207c 0a7c 6f31 3420 3d20 3234 |.|o14 = 24 │ │ │ │ -000038f0: 3234 3637 3533 3030 3030 2020 2020 2020 2467530000 │ │ │ │ -00003900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003910: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00003920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003940: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00003950: 0a7c 6f31 3420 3a20 5151 2020 2020 2020 .|o14 : QQ │ │ │ │ -00003960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000038e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000038f0: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ +00003900: 3d20 3234 3234 3637 3533 3030 3030 2020 = 242467530000 │ │ │ │ +00003910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00003930: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00003940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003960: 2020 2020 207c 0a7c 6f31 3420 3a20 5151 |.|o14 : QQ │ │ │ │ 00003970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003980: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00003990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003990: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 000039a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000039b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5468 6520 ---------+..The │ │ │ │ -000039c0: 6e75 6d62 6572 7320 6f66 2072 6174 696f numbers of ratio │ │ │ │ -000039d0: 6e61 6c20 6375 7276 6573 206f 6620 6465 nal curves of de │ │ │ │ -000039e0: 6772 6565 2034 206f 6e20 6765 6e65 7261 gree 4 on genera │ │ │ │ -000039f0: 6c20 636f 6d70 6c65 7465 2069 6e74 6572 l complete inter │ │ │ │ -00003a00: 7365 6374 696f 6e73 206f 660a 7479 7065 sections of.type │ │ │ │ -00003a10: 7320 2834 2c32 2920 616e 6420 2833 2c33 s (4,2) and (3,3 │ │ │ │ -00003a20: 2920 696e 205c 6d61 7468 6262 2050 5e35 ) in \mathbb P^5 │ │ │ │ -00003a30: 2063 616e 2062 6520 636f 6d70 7574 6564 can be computed │ │ │ │ -00003a40: 2061 7320 666f 6c6c 6f77 733a 0a0a 0a0a as follows:.... │ │ │ │ -00003a50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00003a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000039b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000039c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000039d0: 2d2b 0a0a 5468 6520 6e75 6d62 6572 7320 -+..The numbers │ │ │ │ +000039e0: 6f66 2072 6174 696f 6e61 6c20 6375 7276 of rational curv │ │ │ │ +000039f0: 6573 206f 6620 6465 6772 6565 2034 206f es of degree 4 o │ │ │ │ +00003a00: 6e20 6765 6e65 7261 6c20 636f 6d70 6c65 n general comple │ │ │ │ +00003a10: 7465 2069 6e74 6572 7365 6374 696f 6e73 te intersections │ │ │ │ +00003a20: 206f 660a 7479 7065 7320 2834 2c32 2920 of.types (4,2) │ │ │ │ +00003a30: 616e 6420 2833 2c33 2920 696e 205c 6d61 and (3,3) in \ma │ │ │ │ +00003a40: 7468 6262 2050 5e35 2063 616e 2062 6520 thbb P^5 can be │ │ │ │ +00003a50: 636f 6d70 7574 6564 2061 7320 666f 6c6c computed as foll │ │ │ │ +00003a60: 6f77 733a 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d ows:....+------- │ │ │ │ 00003a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00003a90: 6931 3520 3a20 7469 6d65 2072 6174 696f i15 : time ratio │ │ │ │ -00003aa0: 6e61 6c43 7572 7665 2834 2c7b 342c 327d nalCurve(4,{4,2} │ │ │ │ -00003ab0: 2920 2d20 7261 7469 6f6e 616c 4375 7276 ) - rationalCurv │ │ │ │ -00003ac0: 6528 322c 7b34 2c32 7d29 2f38 7c0a 7c20 e(2,{4,2})/8|.| │ │ │ │ -00003ad0: 2d2d 2075 7365 6420 372e 3236 3432 3473 -- used 7.26424s │ │ │ │ -00003ae0: 2028 6370 7529 3b20 352e 3933 3734 3473 (cpu); 5.93744s │ │ │ │ -00003af0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -00003b00: 6329 2020 2020 2020 2020 207c 0a7c 2020 c) |.| │ │ │ │ -00003b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003aa0: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 7469 -----+.|i15 : ti │ │ │ │ +00003ab0: 6d65 2072 6174 696f 6e61 6c43 7572 7665 me rationalCurve │ │ │ │ +00003ac0: 2834 2c7b 342c 327d 2920 2d20 7261 7469 (4,{4,2}) - rati │ │ │ │ +00003ad0: 6f6e 616c 4375 7276 6528 322c 7b34 2c32 onalCurve(2,{4,2 │ │ │ │ +00003ae0: 7d29 2f38 7c0a 7c20 2d2d 2075 7365 6420 })/8|.| -- used │ │ │ │ +00003af0: 372e 3432 3034 3873 2028 6370 7529 3b20 7.42048s (cpu); │ │ │ │ +00003b00: 352e 3930 3431 7320 2874 6872 6561 6429 5.9041s (thread) │ │ │ │ +00003b10: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +00003b20: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00003b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003b40: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ -00003b50: 203d 2033 3838 3339 3032 3532 3820 2020 = 3883902528 │ │ │ │ -00003b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003b80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00003b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003b60: 2020 7c0a 7c6f 3135 203d 2033 3838 3339 |.|o15 = 38839 │ │ │ │ +00003b70: 3032 3532 3820 2020 2020 2020 2020 2020 02528 │ │ │ │ +00003b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003ba0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00003bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003bc0: 2020 2020 2020 2020 7c0a 7c6f 3135 203a |.|o15 : │ │ │ │ -00003bd0: 2051 5120 2020 2020 2020 2020 2020 2020 QQ │ │ │ │ -00003be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003be0: 7c0a 7c6f 3135 203a 2051 5120 2020 2020 |.|o15 : QQ │ │ │ │ 00003bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003c00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00003c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003c10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00003c20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00003c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003c40: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2074 ------+.|i16 : t │ │ │ │ -00003c50: 696d 6520 7261 7469 6f6e 616c 4375 7276 ime rationalCurv │ │ │ │ -00003c60: 6528 342c 7b33 2c33 7d29 202d 2072 6174 e(4,{3,3}) - rat │ │ │ │ -00003c70: 696f 6e61 6c43 7572 7665 2832 2c7b 332c ionalCurve(2,{3, │ │ │ │ -00003c80: 337d 292f 387c 0a7c 202d 2d20 7573 6564 3})/8|.| -- used │ │ │ │ -00003c90: 2037 2e32 3537 3739 7320 2863 7075 293b 7.25779s (cpu); │ │ │ │ -00003ca0: 2035 2e37 3833 3273 2028 7468 7265 6164 5.7832s (thread │ │ │ │ -00003cb0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ -00003cc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00003cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00003c60: 7c69 3136 203a 2074 696d 6520 7261 7469 |i16 : time rati │ │ │ │ +00003c70: 6f6e 616c 4375 7276 6528 342c 7b33 2c33 onalCurve(4,{3,3 │ │ │ │ +00003c80: 7d29 202d 2072 6174 696f 6e61 6c43 7572 }) - rationalCur │ │ │ │ +00003c90: 7665 2832 2c7b 332c 337d 292f 387c 0a7c ve(2,{3,3})/8|.| │ │ │ │ +00003ca0: 202d 2d20 7573 6564 2037 2e35 3633 3335 -- used 7.56335 │ │ │ │ +00003cb0: 7320 2863 7075 293b 2036 2e30 3636 3233 s (cpu); 6.06623 │ │ │ │ +00003cc0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00003cd0: 6763 2920 2020 2020 2020 2020 7c0a 7c20 gc) |.| │ │ │ │ 00003ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003d00: 2020 207c 0a7c 6f31 3620 3d20 3131 3339 |.|o16 = 1139 │ │ │ │ -00003d10: 3434 3833 3834 2020 2020 2020 2020 2020 448384 │ │ │ │ -00003d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003d10: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00003d20: 3620 3d20 3131 3339 3434 3833 3834 2020 6 = 1139448384 │ │ │ │ 00003d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003d40: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00003d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003d50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00003d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003d80: 207c 0a7c 6f31 3620 3a20 5151 2020 2020 |.|o16 : QQ │ │ │ │ -00003d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003d90: 2020 2020 2020 2020 207c 0a7c 6f31 3620 |.|o16 │ │ │ │ +00003da0: 3a20 5151 2020 2020 2020 2020 2020 2020 : QQ │ │ │ │ 00003db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00003dc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00003dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00003dd0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00003de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00003df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00003e00: 0a0a 5761 7973 2074 6f20 7573 6520 7261 ..Ways to use ra │ │ │ │ -00003e10: 7469 6f6e 616c 4375 7276 653a 0a3d 3d3d tionalCurve:.=== │ │ │ │ -00003e20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00003e30: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7261 =======.. * "ra │ │ │ │ -00003e40: 7469 6f6e 616c 4375 7276 6528 5a5a 2922 tionalCurve(ZZ)" │ │ │ │ +00003df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003e10: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 -------+..Ways t │ │ │ │ +00003e20: 6f20 7573 6520 7261 7469 6f6e 616c 4375 o use rationalCu │ │ │ │ +00003e30: 7276 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d rve:.=========== │ │ │ │ +00003e40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ 00003e50: 0a20 202a 2022 7261 7469 6f6e 616c 4375 . * "rationalCu │ │ │ │ -00003e60: 7276 6528 5a5a 2c4c 6973 7429 220a 0a46 rve(ZZ,List)"..F │ │ │ │ -00003e70: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00003e80: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00003e90: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00003ea0: 202a 6e6f 7465 2072 6174 696f 6e61 6c43 *note rationalC │ │ │ │ -00003eb0: 7572 7665 3a20 7261 7469 6f6e 616c 4375 urve: rationalCu │ │ │ │ -00003ec0: 7276 652c 2069 7320 6120 2a6e 6f74 6520 rve, is a *note │ │ │ │ -00003ed0: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ -00003ee0: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ -00003ef0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ -00003f00: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ -00003f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00003e60: 7276 6528 5a5a 2922 0a20 202a 2022 7261 rve(ZZ)". * "ra │ │ │ │ +00003e70: 7469 6f6e 616c 4375 7276 6528 5a5a 2c4c tionalCurve(ZZ,L │ │ │ │ +00003e80: 6973 7429 220a 0a46 6f72 2074 6865 2070 ist)"..For the p │ │ │ │ +00003e90: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00003ea0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00003eb0: 6520 6f62 6a65 6374 202a 6e6f 7465 2072 e object *note r │ │ │ │ +00003ec0: 6174 696f 6e61 6c43 7572 7665 3a20 7261 ationalCurve: ra │ │ │ │ +00003ed0: 7469 6f6e 616c 4375 7276 652c 2069 7320 tionalCurve, is │ │ │ │ +00003ee0: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ +00003ef0: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ +00003f00: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +00003f10: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ 00003f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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00011830: 6f73 6547 6f6f 644d 696e 6f72 7328 382c oseGoodMinors(8, │ │ │ │ 00011840: 2036 2c20 4d2c 204a 2c20 5374 7261 7465 6, M, J, Strate │ │ │ │ 00011850: 6779 3d3e 4c65 784c 6172 6765 7374 2929 gy=>LexLargest)) │ │ │ │ 00011860: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00011870: 6420 302e 3139 3234 3131 7320 2863 7075 d 0.192411s (cpu │ │ │ │ -00011880: 293b 2030 2e31 3634 3530 3273 2028 7468 ); 0.164502s (th │ │ │ │ +00011870: 6420 302e 3239 3533 3936 7320 2863 7075 d 0.295396s (cpu │ │ │ │ +00011880: 293b 2030 2e32 3134 3439 3773 2028 7468 ); 0.214497s (th │ │ │ │ 00011890: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 000118a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000118b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000118c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000118d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000118e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000118f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4506,16 +4506,16 @@ │ │ │ │ 00011990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000119a0: 2d2d 2d2d 2d2d 2b0a 7c69 3332 203a 2074 ------+.|i32 : t │ │ │ │ 000119b0: 696d 6520 6469 6d20 284a 202b 2063 686f ime dim (J + cho │ │ │ │ 000119c0: 6f73 6547 6f6f 644d 696e 6f72 7328 382c oseGoodMinors(8, │ │ │ │ 000119d0: 2036 2c20 4d2c 204a 2c20 5374 7261 7465 6, M, J, Strate │ │ │ │ 000119e0: 6779 3d3e 4752 6576 4c65 7853 6d61 6c6c gy=>GRevLexSmall │ │ │ │ 000119f0: 6573 7429 2920 7c0a 7c20 2d2d 2075 7365 est)) |.| -- use │ │ │ │ -00011a00: 6420 302e 3334 3532 3736 7320 2863 7075 d 0.345276s (cpu │ │ │ │ -00011a10: 293b 2030 2e32 3036 3337 3273 2028 7468 ); 0.206372s (th │ │ │ │ +00011a00: 6420 302e 3439 3531 3433 7320 2863 7075 d 0.495143s (cpu │ │ │ │ +00011a10: 293b 2030 2e32 3435 3038 3373 2028 7468 ); 0.245083s (th │ │ │ │ 00011a20: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00011a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00011a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4531,17 +4531,17 @@ │ │ │ │ 00011b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011b30: 2d2d 2d2d 2d2d 2b0a 7c69 3333 203a 2074 ------+.|i33 : t │ │ │ │ 00011b40: 696d 6520 6469 6d20 284a 202b 2063 686f ime dim (J + cho │ │ │ │ 00011b50: 6f73 6547 6f6f 644d 696e 6f72 7328 382c oseGoodMinors(8, │ │ │ │ 00011b60: 2036 2c20 4d2c 204a 2c20 5374 7261 7465 6, M, J, Strate │ │ │ │ 00011b70: 6779 3d3e 2020 2020 2020 2020 2020 2020 gy=> │ │ │ │ 00011b80: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00011b90: 6420 302e 3333 3834 3673 2028 6370 7529 d 0.33846s (cpu) │ │ │ │ -00011ba0: 3b20 302e 3232 3436 3537 7320 2874 6872 ; 0.224657s (thr │ │ │ │ -00011bb0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00011b90: 6420 302e 3432 3536 7320 2863 7075 293b d 0.4256s (cpu); │ │ │ │ +00011ba0: 2030 2e32 3638 3338 3473 2028 7468 7265 0.268384s (thre │ │ │ │ +00011bb0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00011bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011bd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00011be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c20: 2020 2020 2020 7c0a 7c6f 3333 203d 2033 |.|o33 = 3 │ │ │ │ @@ -4566,17 +4566,17 @@ │ │ │ │ 00011d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011d60: 2d2d 2d2d 2d2d 2b0a 7c69 3334 203a 2074 ------+.|i34 : t │ │ │ │ 00011d70: 696d 6520 6469 6d20 284a 202b 2063 686f ime dim (J + cho │ │ │ │ 00011d80: 6f73 6547 6f6f 644d 696e 6f72 7328 382c oseGoodMinors(8, │ │ │ │ 00011d90: 2036 2c20 4d2c 204a 2c20 5374 7261 7465 6, M, J, Strate │ │ │ │ 00011da0: 6779 3d3e 4752 6576 4c65 784c 6172 6765 gy=>GRevLexLarge │ │ │ │ 00011db0: 7374 2929 2020 7c0a 7c20 2d2d 2075 7365 st)) |.| -- use │ │ │ │ -00011dc0: 6420 302e 3237 3236 3534 7320 2863 7075 d 0.272654s (cpu │ │ │ │ -00011dd0: 293b 2030 2e31 3839 3037 7320 2874 6872 ); 0.18907s (thr │ │ │ │ -00011de0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00011dc0: 6420 302e 3339 3530 3632 7320 2863 7075 d 0.395062s (cpu │ │ │ │ +00011dd0: 293b 2030 2e32 3336 3133 3273 2028 7468 ); 0.236132s (th │ │ │ │ +00011de0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00011df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e00: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00011e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e50: 2020 2020 2020 7c0a 7c6f 3334 203d 2033 |.|o34 = 3 │ │ │ │ @@ -4591,17 +4591,17 @@ │ │ │ │ 00011ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011ef0: 2d2d 2d2d 2d2d 2b0a 7c69 3335 203a 2074 ------+.|i35 : t │ │ │ │ 00011f00: 696d 6520 6469 6d20 284a 202b 2063 686f ime dim (J + cho │ │ │ │ 00011f10: 6f73 6547 6f6f 644d 696e 6f72 7328 382c oseGoodMinors(8, │ │ │ │ 00011f20: 2036 2c20 4d2c 204a 2c20 5374 7261 7465 6, M, J, Strate │ │ │ │ 00011f30: 6779 3d3e 506f 696e 7473 2929 2020 2020 gy=>Points)) │ │ │ │ 00011f40: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00011f50: 6420 3135 2e36 3435 3173 2028 6370 7529 d 15.6451s (cpu) │ │ │ │ -00011f60: 3b20 3131 2e30 3438 7320 2874 6872 6561 ; 11.048s (threa │ │ │ │ -00011f70: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +00011f50: 6420 3230 2e31 3932 3573 2028 6370 7529 d 20.1925s (cpu) │ │ │ │ +00011f60: 3b20 3132 2e34 3131 3373 2028 7468 7265 ; 12.4113s (thre │ │ │ │ +00011f70: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00011f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00011fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011fe0: 2020 2020 2020 7c0a 7c6f 3335 203d 2031 |.|o35 = 1 │ │ │ │ @@ -4715,16 +4715,16 @@ │ │ │ │ 000126a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000126b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3337 203a --------+.|i37 : │ │ │ │ 000126c0: 2074 696d 6520 6368 6f6f 7365 476f 6f64 time chooseGood │ │ │ │ 000126d0: 4d69 6e6f 7273 2832 302c 2036 2c20 4d2c Minors(20, 6, M, │ │ │ │ 000126e0: 204a 2c20 5374 7261 7465 6779 3d3e 5374 J, Strategy=>St │ │ │ │ 000126f0: 7261 7465 6779 4465 6661 756c 742c 2020 rategyDefault, │ │ │ │ 00012700: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00012710: 7365 6420 302e 3336 3537 3273 2028 6370 sed 0.36572s (cp │ │ │ │ -00012720: 7529 3b20 302e 3333 3039 3334 7320 2874 u); 0.330934s (t │ │ │ │ +00012710: 7365 6420 302e 3437 3933 3573 2028 6370 sed 0.47935s (cp │ │ │ │ +00012720: 7529 3b20 302e 3339 3138 3332 7320 2874 u); 0.391832s (t │ │ │ │ 00012730: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00012740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012750: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ 00012760: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ 00012770: 4368 6f6f 7369 6e67 2052 616e 646f 6d20 Choosing Random │ │ │ │ 00012780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5115,17 +5115,17 @@ │ │ │ │ 00013fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00013fc0: 0a7c 6934 3320 3a20 7469 6d65 2064 696d .|i43 : time dim │ │ │ │ 00013fd0: 2028 4a20 2b20 6368 6f6f 7365 476f 6f64 (J + chooseGood │ │ │ │ 00013fe0: 4d69 6e6f 7273 2831 2c20 362c 204d 2c20 Minors(1, 6, M, │ │ │ │ 00013ff0: 4a2c 2053 7472 6174 6567 793d 3e50 6f69 J, Strategy=>Poi │ │ │ │ 00014000: 6e74 732c 2020 2020 2020 2020 2020 207c nts, | │ │ │ │ -00014010: 0a7c 202d 2d20 7573 6564 2030 2e34 3939 .| -- used 0.499 │ │ │ │ -00014020: 3338 3873 2028 6370 7529 3b20 302e 3434 388s (cpu); 0.44 │ │ │ │ -00014030: 3433 3637 7320 2874 6872 6561 6429 3b20 4367s (thread); │ │ │ │ +00014010: 0a7c 202d 2d20 7573 6564 2030 2e37 3333 .| -- used 0.733 │ │ │ │ +00014020: 3134 3473 2028 6370 7529 3b20 302e 3635 144s (cpu); 0.65 │ │ │ │ +00014030: 3037 3833 7320 2874 6872 6561 6429 3b20 0783s (thread); │ │ │ │ 00014040: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00014050: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00014060: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00014070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000140a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -5250,17 +5250,17 @@ │ │ │ │ 00014810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00014830: 0a7c 6934 3620 3a20 7469 6d65 2064 696d .|i46 : time dim │ │ │ │ 00014840: 2028 4a20 2b20 6368 6f6f 7365 476f 6f64 (J + chooseGood │ │ │ │ 00014850: 4d69 6e6f 7273 2831 2c20 362c 204d 2c20 Minors(1, 6, M, │ │ │ │ 00014860: 4a2c 2053 7472 6174 6567 793d 3e50 6f69 J, Strategy=>Poi │ │ │ │ 00014870: 6e74 732c 2020 2020 2020 2020 2020 207c nts, | │ │ │ │ -00014880: 0a7c 202d 2d20 7573 6564 2030 2e34 3536 .| -- used 0.456 │ │ │ │ -00014890: 3635 3873 2028 6370 7529 3b20 302e 3336 658s (cpu); 0.36 │ │ │ │ -000148a0: 3139 3536 7320 2874 6872 6561 6429 3b20 1956s (thread); │ │ │ │ +00014880: 0a7c 202d 2d20 7573 6564 2030 2e36 3733 .| -- used 0.673 │ │ │ │ +00014890: 3337 3573 2028 6370 7529 3b20 302e 3438 375s (cpu); 0.48 │ │ │ │ +000148a0: 3931 3137 7320 2874 6872 6561 6429 3b20 9117s (thread); │ │ │ │ 000148b0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 000148c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 000148d0: 0a7c 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 2020 │ │ │ │ 00014910: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -5343,16 +5343,16 @@ │ │ │ │ 00014de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014df0: 2d2d 2d2d 2d2b 0a7c 6934 3720 3a20 7469 -----+.|i47 : ti │ │ │ │ 00014e00: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 00014e10: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20 mension(1, S/J, │ │ │ │ 00014e20: 4d61 784d 696e 6f72 7320 3d3e 2031 3030 MaxMinors => 100 │ │ │ │ 00014e30: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ 00014e40: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00014e50: 2033 2e33 3931 3532 7320 2863 7075 293b 3.39152s (cpu); │ │ │ │ -00014e60: 2033 2e30 3537 3639 7320 2874 6872 6561 3.05769s (threa │ │ │ │ +00014e50: 2034 2e34 3037 3136 7320 2863 7075 293b 4.40716s (cpu); │ │ │ │ +00014e60: 2033 2e38 3639 3638 7320 2874 6872 6561 3.86968s (threa │ │ │ │ 00014e70: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00014e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e90: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d |.|-------- │ │ │ │ 00014ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5368,16 +5368,16 @@ │ │ │ │ 00014f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f80: 2d2d 2d2d 2d2b 0a7c 6934 3820 3a20 7469 -----+.|i48 : ti │ │ │ │ 00014f90: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 00014fa0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20 mension(1, S/J, │ │ │ │ 00014fb0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030 MaxMinors => 100 │ │ │ │ 00014fc0: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ 00014fd0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00014fe0: 2030 2e37 3731 3039 3573 2028 6370 7529 0.771095s (cpu) │ │ │ │ -00014ff0: 3b20 302e 3638 3133 3737 7320 2874 6872 ; 0.681377s (thr │ │ │ │ +00014fe0: 2030 2e39 3839 3436 3673 2028 6370 7529 0.989466s (cpu) │ │ │ │ +00014ff0: 3b20 302e 3831 3439 3936 7320 2874 6872 ; 0.814996s (thr │ │ │ │ 00015000: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00015010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015020: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00015030: 2020 2020 2020 2020 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 │ │ │ │ @@ -5403,32 +5403,32 @@ │ │ │ │ 000151a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000151b0: 2d2d 2d2d 2d2b 0a7c 6934 3920 3a20 7469 -----+.|i49 : ti │ │ │ │ 000151c0: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 000151d0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20 mension(1, S/J, │ │ │ │ 000151e0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030 MaxMinors => 100 │ │ │ │ 000151f0: 2c20 5374 7261 7465 6779 3d3e 5261 6e64 , Strategy=>Rand │ │ │ │ 00015200: 6f6d 2920 207c 0a7c 202d 2d20 7573 6564 om) |.| -- used │ │ │ │ -00015210: 2032 2e38 3432 3936 7320 2863 7075 293b 2.84296s (cpu); │ │ │ │ -00015220: 2032 2e36 3230 3737 7320 2874 6872 6561 2.62077s (threa │ │ │ │ +00015210: 2033 2e34 3831 3032 7320 2863 7075 293b 3.48102s (cpu); │ │ │ │ +00015220: 2033 2e32 3037 3336 7320 2874 6872 6561 3.20736s (threa │ │ │ │ 00015230: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00015240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015250: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00015260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000152a0: 2d2d 2d2d 2d2b 0a7c 6935 3020 3a20 7469 -----+.|i50 : ti │ │ │ │ 000152b0: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 000152c0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20 mension(1, S/J, │ │ │ │ 000152d0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030 MaxMinors => 100 │ │ │ │ 000152e0: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ 000152f0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00015300: 2032 2e32 3930 3438 7320 2863 7075 293b 2.29048s (cpu); │ │ │ │ -00015310: 2031 2e39 3531 3673 2028 7468 7265 6164 1.9516s (thread │ │ │ │ -00015320: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +00015300: 2033 2e30 3631 3435 7320 2863 7075 293b 3.06145s (cpu); │ │ │ │ +00015310: 2032 2e34 3231 3735 7320 2874 6872 6561 2.42175s (threa │ │ │ │ +00015320: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00015330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015340: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d |.|-------- │ │ │ │ 00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015390: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779 -----|.|Strategy │ │ │ │ @@ -5443,16 +5443,16 @@ │ │ │ │ 00015420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015430: 2d2d 2d2d 2d2b 0a7c 6935 3120 3a20 7469 -----+.|i51 : ti │ │ │ │ 00015440: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 00015450: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20 mension(1, S/J, │ │ │ │ 00015460: 4d61 784d 696e 6f72 7320 3d3e 2031 3030 MaxMinors => 100 │ │ │ │ 00015470: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ 00015480: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00015490: 2030 2e38 3639 3337 3473 2028 6370 7529 0.869374s (cpu) │ │ │ │ -000154a0: 3b20 302e 3739 3039 3931 7320 2874 6872 ; 0.790991s (thr │ │ │ │ +00015490: 2030 2e39 3635 3838 3573 2028 6370 7529 0.965885s (cpu) │ │ │ │ +000154a0: 3b20 302e 3838 3033 3639 7320 2874 6872 ; 0.880369s (thr │ │ │ │ 000154b0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 000154c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000154d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000154e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000154f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5478,16 +5478,16 @@ │ │ │ │ 00015650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015660: 2d2d 2d2d 2d2b 0a7c 6935 3220 3a20 7469 -----+.|i52 : ti │ │ │ │ 00015670: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 00015680: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20 mension(1, S/J, │ │ │ │ 00015690: 4d61 784d 696e 6f72 7320 3d3e 2031 3030 MaxMinors => 100 │ │ │ │ 000156a0: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ 000156b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000156c0: 2032 2e37 3330 3236 7320 2863 7075 293b 2.73026s (cpu); │ │ │ │ -000156d0: 2032 2e33 3033 3339 7320 2874 6872 6561 2.30339s (threa │ │ │ │ +000156c0: 2033 2e34 3637 3236 7320 2863 7075 293b 3.46726s (cpu); │ │ │ │ +000156d0: 2032 2e37 3733 3333 7320 2874 6872 6561 2.77333s (threa │ │ │ │ 000156e0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 000156f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015700: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d |.|-------- │ │ │ │ 00015710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5503,17 +5503,17 @@ │ │ │ │ 000157e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000157f0: 2d2d 2d2d 2d2b 0a7c 6935 3320 3a20 7469 -----+.|i53 : ti │ │ │ │ 00015800: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 00015810: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20 mension(1, S/J, │ │ │ │ 00015820: 4d61 784d 696e 6f72 7320 3d3e 2031 3030 MaxMinors => 100 │ │ │ │ 00015830: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ 00015840: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00015850: 2033 2e31 3638 3933 7320 2863 7075 293b 3.16893s (cpu); │ │ │ │ -00015860: 2032 2e37 3730 3473 2028 7468 7265 6164 2.7704s (thread │ │ │ │ -00015870: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +00015850: 2034 2e31 3238 3738 7320 2863 7075 293b 4.12878s (cpu); │ │ │ │ +00015860: 2033 2e33 3939 3636 7320 2874 6872 6561 3.39966s (threa │ │ │ │ +00015870: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00015880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015890: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d |.|-------- │ │ │ │ 000158a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000158b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000158c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000158d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000158e0: 2d2d 2d2d 2d7c 0a7c 5374 7261 7465 6779 -----|.|Strategy │ │ │ │ @@ -5528,16 +5528,16 @@ │ │ │ │ 00015970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015980: 2d2d 2d2d 2d2b 0a7c 6935 3420 3a20 7469 -----+.|i54 : ti │ │ │ │ 00015990: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 000159a0: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20 mension(1, S/J, │ │ │ │ 000159b0: 4d61 784d 696e 6f72 7320 3d3e 2031 3030 MaxMinors => 100 │ │ │ │ 000159c0: 2c20 5374 7261 7465 6779 3d3e 506f 696e , Strategy=>Poin │ │ │ │ 000159d0: 7473 2920 207c 0a7c 202d 2d20 7573 6564 ts) |.| -- used │ │ │ │ -000159e0: 2039 2e30 3836 3136 7320 2863 7075 293b 9.08616s (cpu); │ │ │ │ -000159f0: 2037 2e36 3637 3832 7320 2874 6872 6561 7.66782s (threa │ │ │ │ +000159e0: 2031 322e 3337 3538 7320 2863 7075 293b 12.3758s (cpu); │ │ │ │ +000159f0: 2039 2e39 3139 3835 7320 2874 6872 6561 9.91985s (threa │ │ │ │ 00015a00: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00015a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015a20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00015a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5553,16 +5553,16 @@ │ │ │ │ 00015b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015b10: 2d2d 2d2d 2d2b 0a7c 6935 3520 3a20 7469 -----+.|i55 : ti │ │ │ │ 00015b20: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 00015b30: 6d65 6e73 696f 6e28 312c 2053 2f4a 2c20 mension(1, S/J, │ │ │ │ 00015b40: 4d61 784d 696e 6f72 7320 3d3e 2031 3030 MaxMinors => 100 │ │ │ │ 00015b50: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ 00015b60: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00015b70: 2037 2e32 3338 3735 7320 2863 7075 293b 7.23875s (cpu); │ │ │ │ -00015b80: 2036 2e30 3230 3435 7320 2874 6872 6561 6.02045s (threa │ │ │ │ +00015b70: 2039 2e35 3438 3232 7320 2863 7075 293b 9.54822s (cpu); │ │ │ │ +00015b80: 2037 2e36 3131 3132 7320 2874 6872 6561 7.61112s (threa │ │ │ │ 00015b90: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00015ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00015bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6217,18 +6217,18 @@ │ │ │ │ 00018480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000184a0: 6937 203a 2074 696d 6520 6973 436f 6469 i7 : time isCodi │ │ │ │ 000184b0: 6d41 744c 6561 7374 2833 2c20 4a29 2020 mAtLeast(3, J) │ │ │ │ 000184c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000184d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000184e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000184f0: 202d 2d20 7573 6564 2030 2e30 3030 3333 -- used 0.00033 │ │ │ │ -00018500: 3639 3032 7320 2863 7075 293b 2030 2e30 6902s (cpu); 0.0 │ │ │ │ -00018510: 3032 3433 3333 3473 2028 7468 7265 6164 0243334s (thread │ │ │ │ -00018520: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +000184f0: 202d 2d20 7573 6564 2030 2e30 3034 3032 -- used 0.00402 │ │ │ │ +00018500: 3636 3473 2028 6370 7529 3b20 302e 3030 664s (cpu); 0.00 │ │ │ │ +00018510: 3333 3634 3335 7320 2874 6872 6561 6429 336435s (thread) │ │ │ │ +00018520: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00018530: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00018540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018580: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00018590: 6f37 203d 2074 7275 6520 2020 2020 2020 o7 = true │ │ │ │ @@ -6427,18 +6427,18 @@ │ │ │ │ 000191a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000191b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000191c0: 7c69 3920 3a20 7469 6d65 2069 7343 6f64 |i9 : time isCod │ │ │ │ 000191d0: 696d 4174 4c65 6173 7428 352c 2049 2c20 imAtLeast(5, I, │ │ │ │ 000191e0: 5061 6972 4c69 6d69 7420 3d3e 2035 2c20 PairLimit => 5, │ │ │ │ 000191f0: 5665 7262 6f73 653d 3e74 7275 6529 2020 Verbose=>true) │ │ │ │ 00019200: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00019210: 7c20 2d2d 2075 7365 6420 302e 3030 3035 | -- used 0.0005 │ │ │ │ -00019220: 3232 3037 3973 2028 6370 7529 3b20 302e 22079s (cpu); 0. │ │ │ │ -00019230: 3030 3233 3833 3931 7320 2874 6872 6561 00238391s (threa │ │ │ │ -00019240: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +00019210: 7c20 2d2d 2075 7365 6420 302e 3030 3333 | -- used 0.0033 │ │ │ │ +00019220: 3130 3938 7320 2863 7075 293b 2030 2e30 1098s (cpu); 0.0 │ │ │ │ +00019230: 3033 3030 3539 3973 2028 7468 7265 6164 0300599s (thread │ │ │ │ +00019240: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00019250: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00019260: 7c69 7343 6f64 696d 4174 4c65 6173 743a |isCodimAtLeast: │ │ │ │ 00019270: 2043 6f6d 7075 7469 6e67 2063 6f64 696d Computing codim │ │ │ │ 00019280: 206f 6620 6d6f 6e6f 6d69 616c 7320 6261 of monomials ba │ │ │ │ 00019290: 7365 6420 6f6e 2069 6465 616c 2067 656e sed on ideal gen │ │ │ │ 000192a0: 6572 6174 6f72 732e 2020 2020 2020 7c0a erators. |. │ │ │ │ 000192b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ @@ -6457,17 +6457,17 @@ │ │ │ │ 00019380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000193a0: 7c69 3130 203a 2074 696d 6520 6973 436f |i10 : time isCo │ │ │ │ 000193b0: 6469 6d41 744c 6561 7374 2835 2c20 492c dimAtLeast(5, I, │ │ │ │ 000193c0: 2050 6169 724c 696d 6974 203d 3e20 3230 PairLimit => 20 │ │ │ │ 000193d0: 302c 2056 6572 626f 7365 3d3e 6661 6c73 0, Verbose=>fals │ │ │ │ 000193e0: 6529 2020 2020 2020 2020 2020 2020 7c0a e) |. │ │ │ │ -000193f0: 7c20 2d2d 2075 7365 6420 302e 3030 3131 | -- used 0.0011 │ │ │ │ -00019400: 3434 3035 7320 2863 7075 293b 2030 2e30 4405s (cpu); 0.0 │ │ │ │ -00019410: 3032 3232 3235 3573 2028 7468 7265 6164 0222255s (thread │ │ │ │ +000193f0: 7c20 2d2d 2075 7365 6420 302e 3030 3232 | -- used 0.0022 │ │ │ │ +00019400: 3331 3338 7320 2863 7075 293b 2030 2e30 3138s (cpu); 0.0 │ │ │ │ +00019410: 3032 3932 3138 3373 2028 7468 7265 6164 0292183s (thread │ │ │ │ 00019420: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00019430: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00019440: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00019450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019480: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -7694,17 +7694,17 @@ │ │ │ │ 0001e0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e0f0: 2d2d 2d2b 0a7c 6934 203a 2074 696d 6520 ---+.|i4 : time │ │ │ │ 0001e100: 7072 6f6a 4469 6d28 6d6f 6475 6c65 2049 projDim(module I │ │ │ │ 0001e110: 2c20 5374 7261 7465 6779 3d3e 5374 7261 , Strategy=>Stra │ │ │ │ 0001e120: 7465 6779 5261 6e64 6f6d 2920 2020 2020 tegyRandom) │ │ │ │ 0001e130: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001e140: 7c20 2d2d 2075 7365 6420 302e 3232 3130 | -- used 0.2210 │ │ │ │ -0001e150: 3135 7320 2863 7075 293b 2030 2e31 3535 15s (cpu); 0.155 │ │ │ │ -0001e160: 3230 3273 2028 7468 7265 6164 293b 2030 202s (thread); 0 │ │ │ │ +0001e140: 7c20 2d2d 2075 7365 6420 302e 3338 3932 | -- used 0.3892 │ │ │ │ +0001e150: 3238 7320 2863 7075 293b 2030 2e32 3331 28s (cpu); 0.231 │ │ │ │ +0001e160: 3032 3173 2028 7468 7265 6164 293b 2030 021s (thread); 0 │ │ │ │ 0001e170: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0001e180: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0001e190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1d0: 2020 2020 7c0a 7c6f 3420 3d20 3120 2020 |.|o4 = 1 │ │ │ │ @@ -7718,16 +7718,16 @@ │ │ │ │ 0001e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ 0001e270: 3a20 7469 6d65 2070 726f 6a44 696d 286d : time projDim(m │ │ │ │ 0001e280: 6f64 756c 6520 492c 2053 7472 6174 6567 odule I, Strateg │ │ │ │ 0001e290: 793d 3e53 7472 6174 6567 7952 616e 646f y=>StrategyRando │ │ │ │ 0001e2a0: 6d2c 204d 696e 4469 6d65 6e73 696f 6e20 m, MinDimension │ │ │ │ 0001e2b0: 3d3e 2031 297c 0a7c 202d 2d20 7573 6564 => 1)|.| -- used │ │ │ │ -0001e2c0: 2030 2e30 3131 3135 3435 7320 2863 7075 0.0111545s (cpu │ │ │ │ -0001e2d0: 293b 2030 2e30 3133 3935 3434 7320 2874 ); 0.0139544s (t │ │ │ │ +0001e2c0: 2030 2e30 3133 3238 3833 7320 2863 7075 0.0132883s (cpu │ │ │ │ +0001e2d0: 293b 2030 2e30 3135 3436 3932 7320 2874 ); 0.0154692s (t │ │ │ │ 0001e2e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0001e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e300: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001e310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e340: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ @@ -7936,16 +7936,16 @@ │ │ │ │ 0001eff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f020: 2d2b 0a7c 6933 203a 2074 696d 6520 4932 -+.|i3 : time I2 │ │ │ │ 0001f030: 203d 2072 6563 7572 7369 7665 4d69 6e6f = recursiveMino │ │ │ │ 0001f040: 7273 2834 2c20 4d2c 2054 6872 6561 6473 rs(4, M, Threads │ │ │ │ 0001f050: 3d3e 3029 3b20 2020 207c 0a7c 202d 2d20 =>0); |.| -- │ │ │ │ -0001f060: 7573 6564 2030 2e37 3933 3133 3273 2028 used 0.793132s ( │ │ │ │ -0001f070: 6370 7529 3b20 302e 3734 3638 3332 7320 cpu); 0.746832s │ │ │ │ +0001f060: 7573 6564 2030 2e35 3535 3036 3373 2028 used 0.555063s ( │ │ │ │ +0001f070: 6370 7529 3b20 302e 3438 3032 3138 7320 cpu); 0.480218s │ │ │ │ 0001f080: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 0001f090: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ 0001f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f0c0: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ 0001f0d0: 2049 6465 616c 206f 6620 5220 2020 2020 Ideal of R │ │ │ │ 0001f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7954,17 +7954,17 @@ │ │ │ │ 0001f110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f130: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ 0001f140: 2074 696d 6520 4931 203d 206d 696e 6f72 time I1 = minor │ │ │ │ 0001f150: 7328 342c 204d 2c20 5374 7261 7465 6779 s(4, M, Strategy │ │ │ │ 0001f160: 3d3e 436f 6661 6374 6f72 293b 2020 2020 =>Cofactor); │ │ │ │ 0001f170: 207c 0a7c 202d 2d20 7573 6564 2031 2e34 |.| -- used 1.4 │ │ │ │ -0001f180: 3736 3931 7320 2863 7075 293b 2031 2e32 7691s (cpu); 1.2 │ │ │ │ -0001f190: 3736 3137 7320 2874 6872 6561 6429 3b20 7617s (thread); │ │ │ │ -0001f1a0: 3073 2028 6763 2920 207c 0a7c 2020 2020 0s (gc) |.| │ │ │ │ +0001f180: 3336 3673 2028 6370 7529 3b20 312e 3237 366s (cpu); 1.27 │ │ │ │ +0001f190: 3530 3473 2028 7468 7265 6164 293b 2030 504s (thread); 0 │ │ │ │ +0001f1a0: 7320 2867 6329 2020 207c 0a7c 2020 2020 s (gc) |.| │ │ │ │ 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: 207c 0a7c 6f34 203a 2049 6465 616c 206f |.|o4 : Ideal o │ │ │ │ 0001f1f0: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ 0001f200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f210: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ @@ -8381,16 +8381,16 @@ │ │ │ │ 00020bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ 00020be0: 203a 2074 696d 6520 7265 6775 6c61 7249 : time regularI │ │ │ │ 00020bf0: 6e43 6f64 696d 656e 7369 6f6e 2831 2c20 nCodimension(1, │ │ │ │ 00020c00: 5329 2020 2020 2020 2020 2020 2020 2020 S) │ │ │ │ 00020c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020c20: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00020c30: 2d20 7573 6564 2030 2e36 3934 3937 3373 - used 0.694973s │ │ │ │ -00020c40: 2028 6370 7529 3b20 302e 3534 3138 3035 (cpu); 0.541805 │ │ │ │ +00020c30: 2d20 7573 6564 2030 2e38 3334 3235 3573 - used 0.834255s │ │ │ │ +00020c40: 2028 6370 7529 3b20 302e 3631 3634 3331 (cpu); 0.616431 │ │ │ │ 00020c50: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00020c60: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00020c70: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8406,16 +8406,16 @@ │ │ │ │ 00020d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 -----------+.|i9 │ │ │ │ 00020d70: 203a 2074 696d 6520 7265 6775 6c61 7249 : time regularI │ │ │ │ 00020d80: 6e43 6f64 696d 656e 7369 6f6e 2832 2c20 nCodimension(2, │ │ │ │ 00020d90: 5329 2020 2020 2020 2020 2020 2020 2020 S) │ │ │ │ 00020da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020db0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00020dc0: 2d20 7573 6564 2036 2e38 3939 3438 7320 - used 6.89948s │ │ │ │ -00020dd0: 2863 7075 293b 2035 2e32 3739 3532 7320 (cpu); 5.27952s │ │ │ │ +00020dc0: 2d20 7573 6564 2038 2e31 3635 3738 7320 - used 8.16578s │ │ │ │ +00020dd0: 2863 7075 293b 2035 2e36 3636 3139 7320 (cpu); 5.66619s │ │ │ │ 00020de0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00020df0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00020e00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00020e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -8536,16 +8536,16 @@ │ │ │ │ 00021570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021580: 2d2d 2d2d 2b0a 7c69 3132 203a 2074 696d ----+.|i12 : tim │ │ │ │ 00021590: 6520 2864 696d 2073 696e 6775 6c61 724c e (dim singularL │ │ │ │ 000215a0: 6f63 7573 2028 5229 2920 2020 2020 2020 ocus (R)) │ │ │ │ 000215b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000215c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000215d0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000215e0: 302e 3032 3130 3538 3673 2028 6370 7529 0.0210586s (cpu) │ │ │ │ -000215f0: 3b20 302e 3032 3032 3033 3473 2028 7468 ; 0.0202034s (th │ │ │ │ +000215e0: 302e 3032 3030 3338 3173 2028 6370 7529 0.0200381s (cpu) │ │ │ │ +000215f0: 3b20 302e 3032 3137 3836 3373 2028 7468 ; 0.0217863s (th │ │ │ │ 00021600: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00021610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021620: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00021630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8561,16 +8561,16 @@ │ │ │ │ 00021700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021710: 2d2d 2d2d 2b0a 7c69 3133 203a 2074 696d ----+.|i13 : tim │ │ │ │ 00021720: 6520 7265 6775 6c61 7249 6e43 6f64 696d e regularInCodim │ │ │ │ 00021730: 656e 7369 6f6e 2832 2c20 5229 2020 2020 ension(2, R) │ │ │ │ 00021740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021760: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00021770: 302e 3138 3238 3936 7320 2863 7075 293b 0.182896s (cpu); │ │ │ │ -00021780: 2030 2e31 3337 3731 3573 2028 7468 7265 0.137715s (thre │ │ │ │ +00021770: 302e 3233 3531 3531 7320 2863 7075 293b 0.235151s (cpu); │ │ │ │ +00021780: 2030 2e31 3537 3632 3873 2028 7468 7265 0.157628s (thre │ │ │ │ 00021790: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000217a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000217b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 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 2020 │ │ │ │ 000217f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8586,17 +8586,17 @@ │ │ │ │ 00021890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000218a0: 2d2d 2d2d 2b0a 7c69 3134 203a 2074 696d ----+.|i14 : tim │ │ │ │ 000218b0: 6520 7265 6775 6c61 7249 6e43 6f64 696d e regularInCodim │ │ │ │ 000218c0: 656e 7369 6f6e 2832 2c20 5229 2020 2020 ension(2, R) │ │ │ │ 000218d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000218e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000218f0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00021900: 302e 3938 3235 3635 7320 2863 7075 293b 0.982565s (cpu); │ │ │ │ -00021910: 2030 2e36 3930 3638 3173 2028 7468 7265 0.690681s (thre │ │ │ │ -00021920: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +00021900: 312e 3235 3235 3773 2028 6370 7529 3b20 1.25257s (cpu); │ │ │ │ +00021910: 302e 3734 3032 3236 7320 2874 6872 6561 0.740226s (threa │ │ │ │ +00021920: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00021930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021940: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00021950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021990: 2020 2020 7c0a 7c6f 3134 203d 2074 7275 |.|o14 = tru │ │ │ │ @@ -8611,17 +8611,17 @@ │ │ │ │ 00021a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021a30: 2d2d 2d2d 2b0a 7c69 3135 203a 2074 696d ----+.|i15 : tim │ │ │ │ 00021a40: 6520 7265 6775 6c61 7249 6e43 6f64 696d e regularInCodim │ │ │ │ 00021a50: 656e 7369 6f6e 2832 2c20 5229 2020 2020 ension(2, R) │ │ │ │ 00021a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021a80: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00021a90: 312e 3338 3831 3873 2028 6370 7529 3b20 1.38818s (cpu); │ │ │ │ -00021aa0: 302e 3938 3835 3234 7320 2874 6872 6561 0.988524s (threa │ │ │ │ -00021ab0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +00021a90: 312e 3731 3535 3973 2028 6370 7529 3b20 1.71559s (cpu); │ │ │ │ +00021aa0: 312e 3035 3530 3673 2028 7468 7265 6164 1.05506s (thread │ │ │ │ +00021ab0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00021ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021ad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00021ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021b20: 2020 2020 7c0a 7c6f 3135 203d 2074 7275 |.|o15 = tru │ │ │ │ @@ -10333,17 +10333,17 @@ │ │ │ │ 000285c0: 2020 2020 2020 2020 2020 207c 0a7c 696e |.|in │ │ │ │ 000285d0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f ternalChooseMino │ │ │ │ 000285e0: 723a 2043 686f 6f73 696e 6720 4c65 7853 r: Choosing LexS │ │ │ │ 000285f0: 6d61 6c6c 6573 7454 6572 6d20 2020 2020 mallestTerm │ │ │ │ 00028600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028610: 2020 2020 2020 2020 2020 207c 0a7c 696e |.|in │ │ │ │ 00028620: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f ternalChooseMino │ │ │ │ -00028630: 723a 2043 6820 2d2d 2075 7365 6420 362e r: Ch -- used 6. │ │ │ │ -00028640: 3636 3337 3173 2028 6370 7529 3b20 352e 66371s (cpu); 5. │ │ │ │ -00028650: 3238 3332 3273 2028 7468 7265 6164 293b 28322s (thread); │ │ │ │ +00028630: 723a 2043 6820 2d2d 2075 7365 6420 382e r: Ch -- used 8. │ │ │ │ +00028640: 3630 3334 3373 2028 6370 7529 3b20 362e 60343s (cpu); 6. │ │ │ │ +00028650: 3036 3935 3573 2028 7468 7265 6164 293b 06955s (thread); │ │ │ │ 00028660: 2030 7320 2867 6329 2020 207c 0a7c 6f6f 0s (gc) |.|oo │ │ │ │ 00028670: 7369 6e67 2047 5265 764c 6578 536d 616c sing GRevLexSmal │ │ │ │ 00028680: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ 00028690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000286a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000286b0: 2020 2020 2020 2020 2020 207c 0a7c 696e |.|in │ │ │ │ 000286c0: 7465 726e 616c 4368 6f6f 7365 4d69 6e6f ternalChooseMino │ │ │ │ @@ -12481,16 +12481,16 @@ │ │ │ │ 00030c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00030c20: 6931 3720 3a20 7469 6d65 2072 6567 756c i17 : time regul │ │ │ │ 00030c30: 6172 496e 436f 6469 6d65 6e73 696f 6e28 arInCodimension( │ │ │ │ 00030c40: 322c 2053 2c20 5665 7262 6f73 653d 3e74 2, S, Verbose=>t │ │ │ │ 00030c50: 7275 652c 204d 6178 4d69 6e6f 7273 3d3e rue, MaxMinors=> │ │ │ │ 00030c60: 3330 2920 2020 2020 2020 2020 207c 0a7c 30) |.| │ │ │ │ -00030c70: 202d 2d20 7573 6564 2031 2e32 3739 3231 -- used 1.27921 │ │ │ │ -00030c80: 7320 2863 7075 293b 2031 2e30 3236 3438 s (cpu); 1.02648 │ │ │ │ +00030c70: 202d 2d20 7573 6564 2031 2e37 3234 3834 -- used 1.72484 │ │ │ │ +00030c80: 7320 2863 7075 293b 2031 2e32 3430 3438 s (cpu); 1.24048 │ │ │ │ 00030c90: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00030ca0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00030cb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00030cc0: 7265 6775 6c61 7249 6e43 6f64 696d 656e regularInCodimen │ │ │ │ 00030cd0: 7369 6f6e 3a20 7269 6e67 2064 696d 656e sion: ring dimen │ │ │ │ 00030ce0: 7369 6f6e 203d 332c 2074 6865 7265 2061 sion =3, there a │ │ │ │ 00030cf0: 7265 2031 3733 3235 2070 6f73 7369 626c re 17325 possibl │ │ │ │ @@ -13093,17 +13093,17 @@ │ │ │ │ 00033240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00033270: 7c69 3231 203a 2074 696d 6520 7265 6775 |i21 : time regu │ │ │ │ 00033280: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ 00033290: 2832 2c20 522c 2053 7472 6174 6567 793d (2, R, Strategy= │ │ │ │ 000332a0: 3e53 7472 6174 6567 7943 7572 7265 6e74 >StrategyCurrent │ │ │ │ -000332b0: 297c 0a7c 202d 2d20 7573 6564 2030 2e33 )|.| -- used 0.3 │ │ │ │ -000332c0: 3038 3730 3773 2028 6370 7529 3b20 302e 08707s (cpu); 0. │ │ │ │ -000332d0: 3231 3730 3873 2028 7468 7265 6164 293b 21708s (thread); │ │ │ │ +000332b0: 297c 0a7c 202d 2d20 7573 6564 2030 2e34 )|.| -- used 0.4 │ │ │ │ +000332c0: 3434 3939 7320 2863 7075 293b 2030 2e32 4499s (cpu); 0.2 │ │ │ │ +000332d0: 3735 3432 3573 2028 7468 7265 6164 293b 75425s (thread); │ │ │ │ 000332e0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 000332f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00033300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033330: 2020 2020 2020 207c 0a7c 6f32 3120 3d20 |.|o21 = │ │ │ │ 00033340: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ @@ -13114,17 +13114,17 @@ │ │ │ │ 00033390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000333a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000333b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000333c0: 6932 3220 3a20 7469 6d65 2072 6567 756c i22 : time regul │ │ │ │ 000333d0: 6172 496e 436f 6469 6d65 6e73 696f 6e28 arInCodimension( │ │ │ │ 000333e0: 322c 2052 2c20 5374 7261 7465 6779 3d3e 2, R, Strategy=> │ │ │ │ 000333f0: 5374 7261 7465 6779 4375 7272 656e 7429 StrategyCurrent) │ │ │ │ -00033400: 7c0a 7c20 2d2d 2075 7365 6420 302e 3132 |.| -- used 0.12 │ │ │ │ -00033410: 3233 3731 7320 2863 7075 293b 2030 2e30 2371s (cpu); 0.0 │ │ │ │ -00033420: 3734 3936 3532 7320 2874 6872 6561 6429 749652s (thread) │ │ │ │ +00033400: 7c0a 7c20 2d2d 2075 7365 6420 302e 3137 |.| -- used 0.17 │ │ │ │ +00033410: 3639 3736 7320 2863 7075 293b 2030 2e30 6976s (cpu); 0.0 │ │ │ │ +00033420: 3936 3238 3032 7320 2874 6872 6561 6429 962802s (thread) │ │ │ │ 00033430: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00033440: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00033450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033480: 2020 2020 2020 7c0a 7c6f 3232 203d 2074 |.|o22 = t │ │ │ │ 00033490: 7275 6520 2020 2020 2020 2020 2020 2020 rue │ │ │ │ @@ -13135,18 +13135,18 @@ │ │ │ │ 000334e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000334f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00033510: 3233 203a 2074 696d 6520 7265 6775 6c61 23 : time regula │ │ │ │ 00033520: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00033530: 2c20 532c 2053 7472 6174 6567 793d 3e53 , S, Strategy=>S │ │ │ │ 00033540: 7472 6174 6567 7943 7572 7265 6e74 297c trategyCurrent)| │ │ │ │ -00033550: 0a7c 202d 2d20 7573 6564 2030 2e33 3733 .| -- used 0.373 │ │ │ │ -00033560: 3833 3873 2028 6370 7529 3b20 302e 3238 838s (cpu); 0.28 │ │ │ │ -00033570: 3535 3538 7320 2874 6872 6561 6429 3b20 5558s (thread); │ │ │ │ -00033580: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +00033550: 0a7c 202d 2d20 7573 6564 2030 2e35 3336 .| -- used 0.536 │ │ │ │ +00033560: 3536 3973 2028 6370 7529 3b20 302e 3336 569s (cpu); 0.36 │ │ │ │ +00033570: 3738 3973 2028 7468 7265 6164 293b 2030 789s (thread); 0 │ │ │ │ +00033580: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00033590: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000335a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000335b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000335c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000335d0: 2020 2020 207c 0a7c 6f32 3320 3d20 7472 |.|o23 = tr │ │ │ │ 000335e0: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ 000335f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13156,17 +13156,17 @@ │ │ │ │ 00033630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 00033660: 3420 3a20 7469 6d65 2072 6567 756c 6172 4 : time regular │ │ │ │ 00033670: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 00033680: 2053 2c20 5374 7261 7465 6779 3d3e 5374 S, Strategy=>St │ │ │ │ 00033690: 7261 7465 6779 4375 7272 656e 7429 7c0a rategyCurrent)|. │ │ │ │ -000336a0: 7c20 2d2d 2075 7365 6420 312e 3636 3537 | -- used 1.6657 │ │ │ │ -000336b0: 3773 2028 6370 7529 3b20 312e 3235 3233 7s (cpu); 1.2523 │ │ │ │ -000336c0: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ +000336a0: 7c20 2d2d 2075 7365 6420 322e 3334 3637 | -- used 2.3467 │ │ │ │ +000336b0: 3473 2028 6370 7529 3b20 312e 3539 3839 4s (cpu); 1.5989 │ │ │ │ +000336c0: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ 000336d0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000336e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000336f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033720: 2020 2020 7c0a 7c6f 3234 203d 2074 7275 |.|o24 = tru │ │ │ │ 00033730: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ @@ -13194,30 +13194,30 @@ │ │ │ │ 00033890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000338a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000338b0: 2d2d 2d2d 2d2d 2b0a 7c69 3237 203a 2074 ------+.|i27 : t │ │ │ │ 000338c0: 696d 6520 7265 6775 6c61 7249 6e43 6f64 ime regularInCod │ │ │ │ 000338d0: 696d 656e 7369 6f6e 2832 2c20 522c 2053 imension(2, R, S │ │ │ │ 000338e0: 7472 6174 6567 793d 3e53 7472 6174 6567 trategy=>Strateg │ │ │ │ 000338f0: 7943 7572 7265 6e74 297c 0a7c 202d 2d20 yCurrent)|.| -- │ │ │ │ -00033900: 7573 6564 2032 2e32 3636 3573 2028 6370 used 2.2665s (cp │ │ │ │ -00033910: 7529 3b20 312e 3636 3230 3673 2028 7468 u); 1.66206s (th │ │ │ │ +00033900: 7573 6564 2033 2e30 3538 3973 2028 6370 used 3.0589s (cp │ │ │ │ +00033910: 7529 3b20 322e 3030 3434 3573 2028 7468 u); 2.00445s (th │ │ │ │ 00033920: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00033930: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00033940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00033980: 0a7c 6932 3820 3a20 7469 6d65 2072 6567 .|i28 : time reg │ │ │ │ 00033990: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ 000339a0: 6e28 322c 2052 2c20 5374 7261 7465 6779 n(2, R, Strategy │ │ │ │ 000339b0: 3d3e 5374 7261 7465 6779 4375 7272 656e =>StrategyCurren │ │ │ │ -000339c0: 7429 7c0a 7c20 2d2d 2075 7365 6420 322e t)|.| -- used 2. │ │ │ │ -000339d0: 3334 3635 3973 2028 6370 7529 3b20 312e 34659s (cpu); 1. │ │ │ │ -000339e0: 3730 3134 3373 2028 7468 7265 6164 293b 70143s (thread); │ │ │ │ -000339f0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +000339c0: 7429 7c0a 7c20 2d2d 2075 7365 6420 332e t)|.| -- used 3. │ │ │ │ +000339d0: 3034 3238 7320 2863 7075 293b 2031 2e39 0428s (cpu); 1.9 │ │ │ │ +000339e0: 3339 3834 7320 2874 6872 6561 6429 3b20 3984s (thread); │ │ │ │ +000339f0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00033a00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00033a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033a40: 2020 2020 2020 2020 7c0a 7c6f 3238 203d |.|o28 = │ │ │ │ 00033a50: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ 00033a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13227,18 +13227,18 @@ │ │ │ │ 00033aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00033ad0: 7c69 3239 203a 2074 696d 6520 7265 6775 |i29 : time regu │ │ │ │ 00033ae0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ 00033af0: 2831 2c20 532c 2053 7472 6174 6567 793d (1, S, Strategy= │ │ │ │ 00033b00: 3e53 7472 6174 6567 7943 7572 7265 6e74 >StrategyCurrent │ │ │ │ -00033b10: 297c 0a7c 202d 2d20 7573 6564 2030 2e34 )|.| -- used 0.4 │ │ │ │ -00033b20: 3038 3231 3773 2028 6370 7529 3b20 302e 08217s (cpu); 0. │ │ │ │ -00033b30: 3330 3737 3936 7320 2874 6872 6561 6429 307796s (thread) │ │ │ │ -00033b40: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +00033b10: 297c 0a7c 202d 2d20 7573 6564 2030 2e35 )|.| -- used 0.5 │ │ │ │ +00033b20: 3335 3539 7320 2863 7075 293b 2030 2e33 3559s (cpu); 0.3 │ │ │ │ +00033b30: 3837 3136 3673 2028 7468 7265 6164 293b 87166s (thread); │ │ │ │ +00033b40: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00033b50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00033b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033b90: 2020 2020 2020 207c 0a7c 6f32 3920 3d20 |.|o29 = │ │ │ │ 00033ba0: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ 00033bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13248,17 +13248,17 @@ │ │ │ │ 00033bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00033c20: 6933 3020 3a20 7469 6d65 2072 6567 756c i30 : time regul │ │ │ │ 00033c30: 6172 496e 436f 6469 6d65 6e73 696f 6e28 arInCodimension( │ │ │ │ 00033c40: 312c 2053 2c20 5374 7261 7465 6779 3d3e 1, S, Strategy=> │ │ │ │ 00033c50: 5374 7261 7465 6779 4375 7272 656e 7429 StrategyCurrent) │ │ │ │ -00033c60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3731 |.| -- used 0.71 │ │ │ │ -00033c70: 3138 3435 7320 2863 7075 293b 2030 2e35 1845s (cpu); 0.5 │ │ │ │ -00033c80: 3630 3530 3373 2028 7468 7265 6164 293b 60503s (thread); │ │ │ │ +00033c60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3933 |.| -- used 0.93 │ │ │ │ +00033c70: 3433 3431 7320 2863 7075 293b 2030 2e36 4341s (cpu); 0.6 │ │ │ │ +00033c80: 3839 3233 3873 2028 7468 7265 6164 293b 89238s (thread); │ │ │ │ 00033c90: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00033ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00033cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033ce0: 2020 2020 2020 7c0a 7c6f 3330 203d 2074 |.|o30 = t │ │ │ │ 00033cf0: 7275 6520 2020 2020 2020 2020 2020 2020 rue │ │ │ │ @@ -13269,18 +13269,18 @@ │ │ │ │ 00033d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00033d70: 3331 203a 2074 696d 6520 7265 6775 6c61 31 : time regula │ │ │ │ 00033d80: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00033d90: 2c20 532c 2053 7472 6174 6567 793d 3e53 , S, Strategy=>S │ │ │ │ 00033da0: 7472 6174 6567 7952 616e 646f 6d29 207c trategyRandom) | │ │ │ │ -00033db0: 0a7c 202d 2d20 7573 6564 2030 2e39 3934 .| -- used 0.994 │ │ │ │ -00033dc0: 3736 3173 2028 6370 7529 3b20 302e 3831 761s (cpu); 0.81 │ │ │ │ -00033dd0: 3235 3873 2028 7468 7265 6164 293b 2030 258s (thread); 0 │ │ │ │ -00033de0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ +00033db0: 0a7c 202d 2d20 7573 6564 2031 2e33 3232 .| -- used 1.322 │ │ │ │ +00033dc0: 3834 7320 2863 7075 293b 2031 2e30 3032 84s (cpu); 1.002 │ │ │ │ +00033dd0: 3437 7320 2874 6872 6561 6429 3b20 3073 47s (thread); 0s │ │ │ │ +00033de0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00033df0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00033e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033e30: 2020 2020 207c 0a7c 6f33 3120 3d20 7472 |.|o31 = tr │ │ │ │ 00033e40: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ 00033e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13290,17 +13290,17 @@ │ │ │ │ 00033e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ 00033ec0: 3220 3a20 7469 6d65 2072 6567 756c 6172 2 : time regular │ │ │ │ 00033ed0: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 00033ee0: 2053 2c20 5374 7261 7465 6779 3d3e 5374 S, Strategy=>St │ │ │ │ 00033ef0: 7261 7465 6779 5261 6e64 6f6d 2920 7c0a rategyRandom) |. │ │ │ │ -00033f00: 7c20 2d2d 2075 7365 6420 312e 3636 3534 | -- used 1.6654 │ │ │ │ -00033f10: 3773 2028 6370 7529 3b20 312e 3333 3934 7s (cpu); 1.3394 │ │ │ │ -00033f20: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ +00033f00: 7c20 2d2d 2075 7365 6420 322e 3135 3631 | -- used 2.1561 │ │ │ │ +00033f10: 3173 2028 6370 7529 3b20 312e 3630 3234 1s (cpu); 1.6024 │ │ │ │ +00033f20: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ 00033f30: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00033f40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00033f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033f80: 2020 2020 7c0a 7c6f 3332 203d 2074 7275 |.|o32 = tru │ │ │ │ 00033f90: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ @@ -13557,3464 +13557,3464 @@ │ │ │ │ 00034f40: 5e38 242e 2020 496e 2070 6172 7469 6375 ^8$. In particu │ │ │ │ 00034f50: 6c61 722c 2074 6869 7320 6578 616d 706c lar, this exampl │ │ │ │ 00034f60: 650a 6973 2065 7665 6e20 7265 6775 6c61 e.is even regula │ │ │ │ 00034f70: 7220 696e 2063 6f64 696d 656e 7369 6f6e r in codimension │ │ │ │ 00034f80: 2033 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 3...+---------- │ │ │ │ 00034f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00034fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00034fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -00034fc0: 203a 2074 696d 6520 7265 6775 6c61 7249 : time regularI │ │ │ │ -00034fd0: 6e43 6f64 696d 656e 7369 6f6e 2831 2c20 nCodimension(1, │ │ │ │ -00034fe0: 532f 4a29 2020 2020 2020 2020 2020 2020 S/J) │ │ │ │ -00034ff0: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -00035000: 2e38 3634 3233 3173 2028 6370 7529 3b20 .864231s (cpu); │ │ │ │ -00035010: 302e 3631 3130 3731 7320 2874 6872 6561 0.611071s (threa │ │ │ │ -00035020: 6429 3b20 3073 2028 6763 297c 0a7c 2020 d); 0s (gc)|.| │ │ │ │ +00034fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +00034fc0: 3a20 7469 6d65 2072 6567 756c 6172 496e : time regularIn │ │ │ │ +00034fd0: 436f 6469 6d65 6e73 696f 6e28 312c 2053 Codimension(1, S │ │ │ │ +00034fe0: 2f4a 2920 2020 2020 2020 2020 2020 2020 /J) │ │ │ │ +00034ff0: 207c 0a7c 202d 2d20 7573 6564 2031 2e33 |.| -- used 1.3 │ │ │ │ +00035000: 3231 3935 7320 2863 7075 293b 2030 2e38 2195s (cpu); 0.8 │ │ │ │ +00035010: 3432 3836 3273 2028 7468 7265 6164 293b 42862s (thread); │ │ │ │ +00035020: 2030 7320 2867 6329 7c0a 7c20 2020 2020 0s (gc)|.| │ │ │ │ 00035030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035060: 2020 207c 0a7c 6f34 203d 2074 7275 6520 |.|o4 = true │ │ │ │ +00035050: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00035060: 0a7c 6f34 203d 2074 7275 6520 2020 2020 .|o4 = true │ │ │ │ 00035070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035090: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00035090: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 000350a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000350b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000350c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000350d0: 2d2d 2d2b 0a7c 6935 203a 2074 696d 6520 ---+.|i5 : time │ │ │ │ -000350e0: 7265 6775 6c61 7249 6e43 6f64 696d 656e regularInCodimen │ │ │ │ -000350f0: 7369 6f6e 2832 2c20 532f 4a29 2020 2020 sion(2, S/J) │ │ │ │ -00035100: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00035110: 2d20 7573 6564 2031 312e 3038 3335 7320 - used 11.0835s │ │ │ │ -00035120: 2863 7075 293b 2038 2e33 3330 3731 7320 (cpu); 8.33071s │ │ │ │ -00035130: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00035140: 2920 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ) |.+---------- │ │ │ │ +000350c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000350d0: 6935 203a 2074 696d 6520 7265 6775 6c61 i5 : time regula │ │ │ │ +000350e0: 7249 6e43 6f64 696d 656e 7369 6f6e 2832 rInCodimension(2 │ │ │ │ +000350f0: 2c20 532f 4a29 2020 2020 2020 2020 2020 , S/J) │ │ │ │ +00035100: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ +00035110: 3134 2e31 3439 7320 2863 7075 293b 2039 14.149s (cpu); 9 │ │ │ │ +00035120: 2e34 3437 3932 7320 2874 6872 6561 6429 .44792s (thread) │ │ │ │ +00035130: 3b20 3073 2028 6763 2920 207c 0a2b 2d2d ; 0s (gc) |.+-- │ │ │ │ +00035140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00035150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00035160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00035170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5765 -----------+..We │ │ │ │ -00035180: 2074 7279 2074 6f20 7665 7269 6679 2074 try to verify t │ │ │ │ -00035190: 6861 7420 2453 2f4a 2420 6973 2072 6567 hat $S/J$ is reg │ │ │ │ -000351a0: 756c 6172 2069 6e20 636f 6469 6d65 6e73 ular in codimens │ │ │ │ -000351b0: 696f 6e20 3120 6f72 2032 2062 7920 636f ion 1 or 2 by co │ │ │ │ -000351c0: 6d70 7574 696e 6720 7468 650a 6964 6561 mputing the.idea │ │ │ │ -000351d0: 6c20 6d61 6465 2075 7020 6f66 2061 2073 l made up of a s │ │ │ │ -000351e0: 6d61 6c6c 206e 756d 6265 7220 6f66 206d mall number of m │ │ │ │ -000351f0: 696e 6f72 7320 6f66 2074 6865 204a 6163 inors of the Jac │ │ │ │ -00035200: 6f62 6961 6e20 6d61 7472 6978 2e20 496e obian matrix. In │ │ │ │ -00035210: 2074 6869 730a 6578 616d 706c 652c 2069 this.example, i │ │ │ │ -00035220: 6e73 7465 6164 206f 6620 636f 6d70 7574 nstead of comput │ │ │ │ -00035230: 696e 6720 616c 6c20 7265 6c65 7661 6e74 ing all relevant │ │ │ │ -00035240: 2031 3436 3531 3238 206d 696e 6f72 7320 1465128 minors │ │ │ │ -00035250: 746f 2063 6f6d 7075 7465 2074 6865 0a73 to compute the.s │ │ │ │ -00035260: 696e 6775 6c61 7220 6c6f 6375 732c 2061 ingular locus, a │ │ │ │ -00035270: 6e64 2074 6865 6e20 7472 7969 6e67 2074 nd then trying t │ │ │ │ -00035280: 6f20 636f 6d70 7574 6520 7468 6520 6469 o compute the di │ │ │ │ -00035290: 6d65 6e73 696f 6e20 6f66 2074 6865 2069 mension of the i │ │ │ │ -000352a0: 6465 616c 2074 6865 790a 6765 6e65 7261 deal they.genera │ │ │ │ -000352b0: 7465 2c20 7765 2069 6e73 7465 6164 2063 te, we instead c │ │ │ │ -000352c0: 6f6d 7075 7465 2061 2066 6577 206f 6620 ompute a few of │ │ │ │ -000352d0: 7468 656d 2e20 2072 6567 756c 6172 496e them. regularIn │ │ │ │ -000352e0: 436f 6469 6d65 6e73 696f 6e20 7265 7475 Codimension retu │ │ │ │ -000352f0: 726e 7320 7472 7565 0a69 6620 6974 2076 rns true.if it v │ │ │ │ -00035300: 6572 6966 6965 6420 7468 6174 2074 6865 erified that the │ │ │ │ -00035310: 2072 696e 6720 6973 2072 6567 756c 6172 ring is regular │ │ │ │ -00035320: 2069 6e20 636f 6469 6d20 3120 6f72 2032 in codim 1 or 2 │ │ │ │ -00035330: 2028 7265 7370 6563 7469 7665 6c79 2920 (respectively) │ │ │ │ -00035340: 616e 6420 6e75 6c6c 0a69 6620 6e6f 742e and null.if not. │ │ │ │ -00035350: 2020 4265 6361 7573 6520 6f66 2074 6865 Because of the │ │ │ │ -00035360: 2072 616e 646f 6d6e 6573 7320 7468 6174 randomness that │ │ │ │ -00035370: 2065 7869 7374 7320 696e 2074 6572 6d73 exists in terms │ │ │ │ -00035380: 206f 6620 7365 6c65 6374 696e 6720 6d69 of selecting mi │ │ │ │ -00035390: 6e6f 7273 2c0a 7468 6520 6578 6563 7574 nors,.the execut │ │ │ │ -000353a0: 696f 6e20 7469 6d65 2063 616e 2061 6374 ion time can act │ │ │ │ -000353b0: 7561 6c6c 7920 7661 7279 2071 7569 7465 ually vary quite │ │ │ │ -000353c0: 2061 2062 6974 2e20 2020 4c65 7427 7320 a bit. Let's │ │ │ │ -000353d0: 7461 6b65 2061 206c 6f6f 6b20 6174 2077 take a look at w │ │ │ │ -000353e0: 6861 740a 6973 206f 6363 7572 7269 6e67 hat.is occurring │ │ │ │ -000353f0: 2062 7920 7573 696e 6720 7468 6520 5665 by using the Ve │ │ │ │ -00035400: 7262 6f73 6520 6f70 7469 6f6e 2e20 2057 rbose option. W │ │ │ │ -00035410: 6520 676f 2074 6872 6f75 6768 2074 6865 e go through the │ │ │ │ -00035420: 206f 7574 7075 7420 616e 6420 6578 706c output and expl │ │ │ │ -00035430: 6169 6e0a 7768 6174 2065 6163 6820 6c69 ain.what each li │ │ │ │ -00035440: 6e65 2069 7320 7465 6c6c 696e 6720 7573 ne is telling us │ │ │ │ -00035450: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ +00035170: 2d2d 2b0a 0a57 6520 7472 7920 746f 2076 --+..We try to v │ │ │ │ +00035180: 6572 6966 7920 7468 6174 2024 532f 4a24 erify that $S/J$ │ │ │ │ +00035190: 2069 7320 7265 6775 6c61 7220 696e 2063 is regular in c │ │ │ │ +000351a0: 6f64 696d 656e 7369 6f6e 2031 206f 7220 odimension 1 or │ │ │ │ +000351b0: 3220 6279 2063 6f6d 7075 7469 6e67 2074 2 by computing t │ │ │ │ +000351c0: 6865 0a69 6465 616c 206d 6164 6520 7570 he.ideal made up │ │ │ │ +000351d0: 206f 6620 6120 736d 616c 6c20 6e75 6d62 of a small numb │ │ │ │ +000351e0: 6572 206f 6620 6d69 6e6f 7273 206f 6620 er of minors of │ │ │ │ +000351f0: 7468 6520 4a61 636f 6269 616e 206d 6174 the Jacobian mat │ │ │ │ +00035200: 7269 782e 2049 6e20 7468 6973 0a65 7861 rix. In this.exa │ │ │ │ +00035210: 6d70 6c65 2c20 696e 7374 6561 6420 6f66 mple, instead of │ │ │ │ +00035220: 2063 6f6d 7075 7469 6e67 2061 6c6c 2072 computing all r │ │ │ │ +00035230: 656c 6576 616e 7420 3134 3635 3132 3820 elevant 1465128 │ │ │ │ +00035240: 6d69 6e6f 7273 2074 6f20 636f 6d70 7574 minors to comput │ │ │ │ +00035250: 6520 7468 650a 7369 6e67 756c 6172 206c e the.singular l │ │ │ │ +00035260: 6f63 7573 2c20 616e 6420 7468 656e 2074 ocus, and then t │ │ │ │ +00035270: 7279 696e 6720 746f 2063 6f6d 7075 7465 rying to compute │ │ │ │ +00035280: 2074 6865 2064 696d 656e 7369 6f6e 206f the dimension o │ │ │ │ +00035290: 6620 7468 6520 6964 6561 6c20 7468 6579 f the ideal they │ │ │ │ +000352a0: 0a67 656e 6572 6174 652c 2077 6520 696e .generate, we in │ │ │ │ +000352b0: 7374 6561 6420 636f 6d70 7574 6520 6120 stead compute a │ │ │ │ +000352c0: 6665 7720 6f66 2074 6865 6d2e 2020 7265 few of them. re │ │ │ │ +000352d0: 6775 6c61 7249 6e43 6f64 696d 656e 7369 gularInCodimensi │ │ │ │ +000352e0: 6f6e 2072 6574 7572 6e73 2074 7275 650a on returns true. │ │ │ │ +000352f0: 6966 2069 7420 7665 7269 6669 6564 2074 if it verified t │ │ │ │ +00035300: 6861 7420 7468 6520 7269 6e67 2069 7320 hat the ring is │ │ │ │ +00035310: 7265 6775 6c61 7220 696e 2063 6f64 696d regular in codim │ │ │ │ +00035320: 2031 206f 7220 3220 2872 6573 7065 6374 1 or 2 (respect │ │ │ │ +00035330: 6976 656c 7929 2061 6e64 206e 756c 6c0a ively) and null. │ │ │ │ +00035340: 6966 206e 6f74 2e20 2042 6563 6175 7365 if not. Because │ │ │ │ +00035350: 206f 6620 7468 6520 7261 6e64 6f6d 6e65 of the randomne │ │ │ │ +00035360: 7373 2074 6861 7420 6578 6973 7473 2069 ss that exists i │ │ │ │ +00035370: 6e20 7465 726d 7320 6f66 2073 656c 6563 n terms of selec │ │ │ │ +00035380: 7469 6e67 206d 696e 6f72 732c 0a74 6865 ting minors,.the │ │ │ │ +00035390: 2065 7865 6375 7469 6f6e 2074 696d 6520 execution time │ │ │ │ +000353a0: 6361 6e20 6163 7475 616c 6c79 2076 6172 can actually var │ │ │ │ +000353b0: 7920 7175 6974 6520 6120 6269 742e 2020 y quite a bit. │ │ │ │ +000353c0: 204c 6574 2773 2074 616b 6520 6120 6c6f Let's take a lo │ │ │ │ +000353d0: 6f6b 2061 7420 7768 6174 0a69 7320 6f63 ok at what.is oc │ │ │ │ +000353e0: 6375 7272 696e 6720 6279 2075 7369 6e67 curring by using │ │ │ │ +000353f0: 2074 6865 2056 6572 626f 7365 206f 7074 the Verbose opt │ │ │ │ +00035400: 696f 6e2e 2020 5765 2067 6f20 7468 726f ion. We go thro │ │ │ │ +00035410: 7567 6820 7468 6520 6f75 7470 7574 2061 ugh the output a │ │ │ │ +00035420: 6e64 2065 7870 6c61 696e 0a77 6861 7420 nd explain.what │ │ │ │ +00035430: 6561 6368 206c 696e 6520 6973 2074 656c each line is tel │ │ │ │ +00035440: 6c69 6e67 2075 732e 0a0a 2b2d 2d2d 2d2d ling us...+----- │ │ │ │ +00035450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00035460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00035470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00035480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00035490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000354a0: 2d2b 0a7c 6936 203a 2074 696d 6520 7265 -+.|i6 : time re │ │ │ │ -000354b0: 6775 6c61 7249 6e43 6f64 696d 656e 7369 gularInCodimensi │ │ │ │ -000354c0: 6f6e 2831 2c20 532f 4a2c 2056 6572 626f on(1, S/J, Verbo │ │ │ │ -000354d0: 7365 3d3e 7472 7565 2920 2020 2020 2020 se=>true) │ │ │ │ -000354e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000354f0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00035500: 696d 656e 7369 6f6e 3a20 7269 6e67 2064 imension: ring d │ │ │ │ -00035510: 696d 656e 7369 6f6e 203d 342c 2074 6865 imension =4, the │ │ │ │ -00035520: 7265 2061 7265 2031 3436 3531 3238 2070 re are 1465128 p │ │ │ │ -00035530: 6f73 7369 626c 6520 3520 6279 2035 206d ossible 5 by 5 m │ │ │ │ -00035540: 697c 0a7c 7265 6775 6c61 7249 6e43 6f64 i|.|regularInCod │ │ │ │ -00035550: 696d 656e 7369 6f6e 3a20 4162 6f75 7420 imension: About │ │ │ │ -00035560: 746f 2065 6e74 6572 206c 6f6f 7020 2020 to enter loop │ │ │ │ +00035490: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ +000354a0: 7469 6d65 2072 6567 756c 6172 496e 436f time regularInCo │ │ │ │ +000354b0: 6469 6d65 6e73 696f 6e28 312c 2053 2f4a dimension(1, S/J │ │ │ │ +000354c0: 2c20 5665 7262 6f73 653d 3e74 7275 6529 , Verbose=>true) │ │ │ │ +000354d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000354e0: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +000354f0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035500: 2072 696e 6720 6469 6d65 6e73 696f 6e20 ring dimension │ │ │ │ +00035510: 3d34 2c20 7468 6572 6520 6172 6520 3134 =4, there are 14 │ │ │ │ +00035520: 3635 3132 3820 706f 7373 6962 6c65 2035 65128 possible 5 │ │ │ │ +00035530: 2062 7920 3520 6d69 7c0a 7c72 6567 756c by 5 mi|.|regul │ │ │ │ +00035540: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035550: 2041 626f 7574 2074 6f20 656e 7465 7220 About to enter │ │ │ │ +00035560: 6c6f 6f70 2020 2020 2020 2020 2020 2020 loop │ │ │ │ 00035570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035590: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000355a0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000355b0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ -000355c0: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ -000355d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000355e0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000355f0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035600: 6720 5261 6e64 6f6d 2020 2020 2020 2020 g Random │ │ │ │ +00035580: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035590: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000355a0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +000355b0: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ +000355c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000355d0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000355e0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000355f0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20 Choosing Random │ │ │ │ +00035600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035630: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035640: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035650: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -00035660: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00035670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035680: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035690: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000356a0: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +00035620: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035630: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035640: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00035650: 536d 616c 6c65 7374 2020 2020 2020 2020 Smallest │ │ │ │ +00035660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00035670: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035680: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035690: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +000356a0: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 000356b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000356c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000356d0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000356e0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000356f0: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +000356c0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000356d0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000356e0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +000356f0: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 00035700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035720: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035730: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035740: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -00035750: 7454 6572 6d20 2020 2020 2020 2020 2020 tTerm │ │ │ │ -00035760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035770: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035780: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035790: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -000357a0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -000357b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000357c0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -000357d0: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ -000357e0: 7374 6570 2c20 6162 6f75 7420 746f 2063 step, about to c │ │ │ │ -000357f0: 6f6d 7075 7465 2064 696d 656e 7369 6f6e ompute dimension │ │ │ │ -00035800: 2e20 2053 7562 6d61 7472 6963 6573 2063 . Submatrices c │ │ │ │ -00035810: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64 o|.|regularInCod │ │ │ │ -00035820: 696d 656e 7369 6f6e 3a20 2069 7343 6f64 imension: isCod │ │ │ │ -00035830: 696d 4174 4c65 6173 7420 6661 696c 6564 imAtLeast failed │ │ │ │ -00035840: 2c20 636f 6d70 7574 696e 6720 636f 6469 , computing codi │ │ │ │ -00035850: 6d2e 2020 2020 2020 2020 2020 2020 2020 m. │ │ │ │ -00035860: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00035870: 696d 656e 7369 6f6e 3a20 2070 6172 7469 imension: parti │ │ │ │ -00035880: 616c 2073 696e 6775 6c61 7220 6c6f 6375 al singular locu │ │ │ │ -00035890: 7320 6469 6d65 6e73 696f 6e20 636f 6d70 s dimension comp │ │ │ │ -000358a0: 7574 6564 2c20 3d20 3320 2020 2020 2020 uted, = 3 │ │ │ │ -000358b0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000358c0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000358d0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ -000358e0: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ -000358f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035900: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035910: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035920: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +00035710: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035720: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035730: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00035740: 536d 616c 6c65 7374 5465 726d 2020 2020 SmallestTerm │ │ │ │ +00035750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00035760: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035770: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035780: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00035790: 536d 616c 6c65 7374 2020 2020 2020 2020 Smallest │ │ │ │ +000357a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000357b0: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +000357c0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +000357d0: 2020 4c6f 6f70 2073 7465 702c 2061 626f Loop step, abo │ │ │ │ +000357e0: 7574 2074 6f20 636f 6d70 7574 6520 6469 ut to compute di │ │ │ │ +000357f0: 6d65 6e73 696f 6e2e 2020 5375 626d 6174 mension. Submat │ │ │ │ +00035800: 7269 6365 7320 636f 7c0a 7c72 6567 756c rices co|.|regul │ │ │ │ +00035810: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035820: 2020 6973 436f 6469 6d41 744c 6561 7374 isCodimAtLeast │ │ │ │ +00035830: 2066 6169 6c65 642c 2063 6f6d 7075 7469 failed, computi │ │ │ │ +00035840: 6e67 2063 6f64 696d 2e20 2020 2020 2020 ng codim. │ │ │ │ +00035850: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +00035860: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035870: 2020 7061 7274 6961 6c20 7369 6e67 756c partial singul │ │ │ │ +00035880: 6172 206c 6f63 7573 2064 696d 656e 7369 ar locus dimensi │ │ │ │ +00035890: 6f6e 2063 6f6d 7075 7465 642c 203d 2033 on computed, = 3 │ │ │ │ +000358a0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000358b0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000358c0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +000358d0: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ +000358e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000358f0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035900: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035910: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00035920: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 00035930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035950: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035960: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035970: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ +00035940: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035950: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035960: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e Choosing RandomN │ │ │ │ +00035970: 6f6e 5a65 726f 2020 2020 2020 2020 2020 onZero │ │ │ │ 00035980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000359a0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000359b0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000359c0: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -000359d0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -000359e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000359f0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00035a00: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ -00035a10: 7374 6570 2c20 6162 6f75 7420 746f 2063 step, about to c │ │ │ │ -00035a20: 6f6d 7075 7465 2064 696d 656e 7369 6f6e ompute dimension │ │ │ │ -00035a30: 2e20 2053 7562 6d61 7472 6963 6573 2063 . Submatrices c │ │ │ │ -00035a40: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64 o|.|regularInCod │ │ │ │ -00035a50: 696d 656e 7369 6f6e 3a20 2069 7343 6f64 imension: isCod │ │ │ │ -00035a60: 696d 4174 4c65 6173 7420 6661 696c 6564 imAtLeast failed │ │ │ │ -00035a70: 2c20 636f 6d70 7574 696e 6720 636f 6469 , computing codi │ │ │ │ -00035a80: 6d2e 2020 2020 2020 2020 2020 2020 2020 m. │ │ │ │ -00035a90: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00035aa0: 696d 656e 7369 6f6e 3a20 2070 6172 7469 imension: parti │ │ │ │ -00035ab0: 616c 2073 696e 6775 6c61 7220 6c6f 6375 al singular locu │ │ │ │ -00035ac0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70 s dimension comp │ │ │ │ -00035ad0: 7574 6564 2c20 3d20 3320 2020 2020 2020 uted, = 3 │ │ │ │ -00035ae0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035af0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035b00: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +00035990: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000359a0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000359b0: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +000359c0: 536d 616c 6c65 7374 2020 2020 2020 2020 Smallest │ │ │ │ +000359d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000359e0: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +000359f0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035a00: 2020 4c6f 6f70 2073 7465 702c 2061 626f Loop step, abo │ │ │ │ +00035a10: 7574 2074 6f20 636f 6d70 7574 6520 6469 ut to compute di │ │ │ │ +00035a20: 6d65 6e73 696f 6e2e 2020 5375 626d 6174 mension. Submat │ │ │ │ +00035a30: 7269 6365 7320 636f 7c0a 7c72 6567 756c rices co|.|regul │ │ │ │ +00035a40: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035a50: 2020 6973 436f 6469 6d41 744c 6561 7374 isCodimAtLeast │ │ │ │ +00035a60: 2066 6169 6c65 642c 2063 6f6d 7075 7469 failed, computi │ │ │ │ +00035a70: 6e67 2063 6f64 696d 2e20 2020 2020 2020 ng codim. │ │ │ │ +00035a80: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +00035a90: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035aa0: 2020 7061 7274 6961 6c20 7369 6e67 756c partial singul │ │ │ │ +00035ab0: 6172 206c 6f63 7573 2064 696d 656e 7369 ar locus dimensi │ │ │ │ +00035ac0: 6f6e 2063 6f6d 7075 7465 642c 203d 2033 on computed, = 3 │ │ │ │ +00035ad0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035ae0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035af0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00035b00: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 00035b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035b30: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035b40: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035b50: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ -00035b60: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ -00035b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035b80: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035b90: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035ba0: 6720 5261 6e64 6f6d 2020 2020 2020 2020 g Random │ │ │ │ +00035b20: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035b30: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035b40: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00035b50: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ +00035b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00035b70: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035b80: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035b90: 4368 6f6f 7369 6e67 2052 616e 646f 6d20 Choosing Random │ │ │ │ +00035ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035bd0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035be0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035bf0: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -00035c00: 7454 6572 6d20 2020 2020 2020 2020 2020 tTerm │ │ │ │ -00035c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035c20: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00035c30: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ -00035c40: 7374 6570 2c20 6162 6f75 7420 746f 2063 step, about to c │ │ │ │ -00035c50: 6f6d 7075 7465 2064 696d 656e 7369 6f6e ompute dimension │ │ │ │ -00035c60: 2e20 2053 7562 6d61 7472 6963 6573 2063 . Submatrices c │ │ │ │ -00035c70: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64 o|.|regularInCod │ │ │ │ -00035c80: 696d 656e 7369 6f6e 3a20 2069 7343 6f64 imension: isCod │ │ │ │ -00035c90: 696d 4174 4c65 6173 7420 6661 696c 6564 imAtLeast failed │ │ │ │ -00035ca0: 2c20 636f 6d70 7574 696e 6720 636f 6469 , computing codi │ │ │ │ -00035cb0: 6d2e 2020 2020 2020 2020 2020 2020 2020 m. │ │ │ │ -00035cc0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00035cd0: 696d 656e 7369 6f6e 3a20 2070 6172 7469 imension: parti │ │ │ │ -00035ce0: 616c 2073 696e 6775 6c61 7220 6c6f 6375 al singular locu │ │ │ │ -00035cf0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70 s dimension comp │ │ │ │ -00035d00: 7574 6564 2c20 3d20 3320 2020 2020 2020 uted, = 3 │ │ │ │ -00035d10: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035d20: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035d30: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ +00035bc0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035bd0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035be0: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00035bf0: 536d 616c 6c65 7374 5465 726d 2020 2020 SmallestTerm │ │ │ │ +00035c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00035c10: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +00035c20: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035c30: 2020 4c6f 6f70 2073 7465 702c 2061 626f Loop step, abo │ │ │ │ +00035c40: 7574 2074 6f20 636f 6d70 7574 6520 6469 ut to compute di │ │ │ │ +00035c50: 6d65 6e73 696f 6e2e 2020 5375 626d 6174 mension. Submat │ │ │ │ +00035c60: 7269 6365 7320 636f 7c0a 7c72 6567 756c rices co|.|regul │ │ │ │ +00035c70: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035c80: 2020 6973 436f 6469 6d41 744c 6561 7374 isCodimAtLeast │ │ │ │ +00035c90: 2066 6169 6c65 642c 2063 6f6d 7075 7469 failed, computi │ │ │ │ +00035ca0: 6e67 2063 6f64 696d 2e20 2020 2020 2020 ng codim. │ │ │ │ +00035cb0: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +00035cc0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035cd0: 2020 7061 7274 6961 6c20 7369 6e67 756c partial singul │ │ │ │ +00035ce0: 6172 206c 6f63 7573 2064 696d 656e 7369 ar locus dimensi │ │ │ │ +00035cf0: 6f6e 2063 6f6d 7075 7465 642c 203d 2033 on computed, = 3 │ │ │ │ +00035d00: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035d10: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035d20: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e Choosing RandomN │ │ │ │ +00035d30: 6f6e 5a65 726f 2020 2020 2020 2020 2020 onZero │ │ │ │ 00035d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035d60: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035d70: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035d80: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ +00035d50: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035d60: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035d70: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e Choosing RandomN │ │ │ │ +00035d80: 6f6e 5a65 726f 2020 2020 2020 2020 2020 onZero │ │ │ │ 00035d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035db0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035dc0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035dd0: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ +00035da0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035db0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035dc0: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e Choosing RandomN │ │ │ │ +00035dd0: 6f6e 5a65 726f 2020 2020 2020 2020 2020 onZero │ │ │ │ 00035de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035e00: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035e10: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035e20: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -00035e30: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00035e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035e50: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035e60: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035e70: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ -00035e80: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ -00035e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035ea0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035eb0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00035ec0: 6720 5261 6e64 6f6d 2020 2020 2020 2020 g Random │ │ │ │ +00035df0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035e00: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035e10: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00035e20: 536d 616c 6c65 7374 2020 2020 2020 2020 Smallest │ │ │ │ +00035e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00035e40: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035e50: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035e60: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00035e70: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ +00035e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00035e90: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035ea0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035eb0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20 Choosing Random │ │ │ │ +00035ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035ef0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00035f00: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ -00035f10: 7374 6570 2c20 6162 6f75 7420 746f 2063 step, about to c │ │ │ │ -00035f20: 6f6d 7075 7465 2064 696d 656e 7369 6f6e ompute dimension │ │ │ │ -00035f30: 2e20 2053 7562 6d61 7472 6963 6573 2063 . Submatrices c │ │ │ │ -00035f40: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64 o|.|regularInCod │ │ │ │ -00035f50: 696d 656e 7369 6f6e 3a20 2069 7343 6f64 imension: isCod │ │ │ │ -00035f60: 696d 4174 4c65 6173 7420 6661 696c 6564 imAtLeast failed │ │ │ │ -00035f70: 2c20 636f 6d70 7574 696e 6720 636f 6469 , computing codi │ │ │ │ -00035f80: 6d2e 2020 2020 2020 2020 2020 2020 2020 m. │ │ │ │ -00035f90: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00035fa0: 696d 656e 7369 6f6e 3a20 2070 6172 7469 imension: parti │ │ │ │ -00035fb0: 616c 2073 696e 6775 6c61 7220 6c6f 6375 al singular locu │ │ │ │ -00035fc0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70 s dimension comp │ │ │ │ -00035fd0: 7574 6564 2c20 3d20 3320 2020 2020 2020 uted, = 3 │ │ │ │ -00035fe0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00035ff0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036000: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ +00035ee0: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +00035ef0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035f00: 2020 4c6f 6f70 2073 7465 702c 2061 626f Loop step, abo │ │ │ │ +00035f10: 7574 2074 6f20 636f 6d70 7574 6520 6469 ut to compute di │ │ │ │ +00035f20: 6d65 6e73 696f 6e2e 2020 5375 626d 6174 mension. Submat │ │ │ │ +00035f30: 7269 6365 7320 636f 7c0a 7c72 6567 756c rices co|.|regul │ │ │ │ +00035f40: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035f50: 2020 6973 436f 6469 6d41 744c 6561 7374 isCodimAtLeast │ │ │ │ +00035f60: 2066 6169 6c65 642c 2063 6f6d 7075 7469 failed, computi │ │ │ │ +00035f70: 6e67 2063 6f64 696d 2e20 2020 2020 2020 ng codim. │ │ │ │ +00035f80: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +00035f90: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00035fa0: 2020 7061 7274 6961 6c20 7369 6e67 756c partial singul │ │ │ │ +00035fb0: 6172 206c 6f63 7573 2064 696d 656e 7369 ar locus dimensi │ │ │ │ +00035fc0: 6f6e 2063 6f6d 7075 7465 642c 203d 2033 on computed, = 3 │ │ │ │ +00035fd0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00035fe0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00035ff0: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e Choosing RandomN │ │ │ │ +00036000: 6f6e 5a65 726f 2020 2020 2020 2020 2020 onZero │ │ │ │ 00036010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036030: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036040: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036050: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -00036060: 7454 6572 6d20 2020 2020 2020 2020 2020 tTerm │ │ │ │ -00036070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036080: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036090: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000360a0: 6720 5261 6e64 6f6d 2020 2020 2020 2020 g Random │ │ │ │ +00036020: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036030: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036040: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00036050: 536d 616c 6c65 7374 5465 726d 2020 2020 SmallestTerm │ │ │ │ +00036060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036070: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036080: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036090: 4368 6f6f 7369 6e67 2052 616e 646f 6d20 Choosing Random │ │ │ │ +000360a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000360b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000360c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000360d0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000360e0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000360f0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ -00036100: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ -00036110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036120: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036130: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036140: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -00036150: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00036160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036170: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036180: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036190: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +000360c0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000360d0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000360e0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +000360f0: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ +00036100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036110: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036120: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036130: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00036140: 536d 616c 6c65 7374 2020 2020 2020 2020 Smallest │ │ │ │ +00036150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036160: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036170: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036180: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00036190: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 000361a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000361b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000361c0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000361d0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000361e0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ -000361f0: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ -00036200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036210: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00036220: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ -00036230: 7374 6570 2c20 6162 6f75 7420 746f 2063 step, about to c │ │ │ │ -00036240: 6f6d 7075 7465 2064 696d 656e 7369 6f6e ompute dimension │ │ │ │ -00036250: 2e20 2053 7562 6d61 7472 6963 6573 2063 . Submatrices c │ │ │ │ -00036260: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64 o|.|regularInCod │ │ │ │ -00036270: 696d 656e 7369 6f6e 3a20 2069 7343 6f64 imension: isCod │ │ │ │ -00036280: 696d 4174 4c65 6173 7420 6661 696c 6564 imAtLeast failed │ │ │ │ -00036290: 2c20 636f 6d70 7574 696e 6720 636f 6469 , computing codi │ │ │ │ -000362a0: 6d2e 2020 2020 2020 2020 2020 2020 2020 m. │ │ │ │ -000362b0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -000362c0: 696d 656e 7369 6f6e 3a20 2070 6172 7469 imension: parti │ │ │ │ -000362d0: 616c 2073 696e 6775 6c61 7220 6c6f 6375 al singular locu │ │ │ │ -000362e0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70 s dimension comp │ │ │ │ -000362f0: 7574 6564 2c20 3d20 3320 2020 2020 2020 uted, = 3 │ │ │ │ -00036300: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036310: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036320: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -00036330: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00036340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036350: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036360: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036370: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +000361b0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000361c0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000361d0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +000361e0: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ +000361f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036200: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +00036210: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00036220: 2020 4c6f 6f70 2073 7465 702c 2061 626f Loop step, abo │ │ │ │ +00036230: 7574 2074 6f20 636f 6d70 7574 6520 6469 ut to compute di │ │ │ │ +00036240: 6d65 6e73 696f 6e2e 2020 5375 626d 6174 mension. Submat │ │ │ │ +00036250: 7269 6365 7320 636f 7c0a 7c72 6567 756c rices co|.|regul │ │ │ │ +00036260: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00036270: 2020 6973 436f 6469 6d41 744c 6561 7374 isCodimAtLeast │ │ │ │ +00036280: 2066 6169 6c65 642c 2063 6f6d 7075 7469 failed, computi │ │ │ │ +00036290: 6e67 2063 6f64 696d 2e20 2020 2020 2020 ng codim. │ │ │ │ +000362a0: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +000362b0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +000362c0: 2020 7061 7274 6961 6c20 7369 6e67 756c partial singul │ │ │ │ +000362d0: 6172 206c 6f63 7573 2064 696d 656e 7369 ar locus dimensi │ │ │ │ +000362e0: 6f6e 2063 6f6d 7075 7465 642c 203d 2033 on computed, = 3 │ │ │ │ +000362f0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036300: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036310: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00036320: 536d 616c 6c65 7374 2020 2020 2020 2020 Smallest │ │ │ │ +00036330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036340: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036350: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036360: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00036370: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 00036380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000363a0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000363b0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000363c0: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ +00036390: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000363a0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000363b0: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e Choosing RandomN │ │ │ │ +000363c0: 6f6e 5a65 726f 2020 2020 2020 2020 2020 onZero │ │ │ │ 000363d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000363e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000363f0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036400: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036410: 6720 5261 6e64 6f6d 2020 2020 2020 2020 g Random │ │ │ │ +000363e0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000363f0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036400: 4368 6f6f 7369 6e67 2052 616e 646f 6d20 Choosing Random │ │ │ │ +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: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036450: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036460: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ +00036430: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036440: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036450: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e Choosing RandomN │ │ │ │ +00036460: 6f6e 5a65 726f 2020 2020 2020 2020 2020 onZero │ │ │ │ 00036470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036490: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000364a0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000364b0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ -000364c0: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ -000364d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000364e0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000364f0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036500: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +00036480: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036490: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000364a0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +000364b0: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ +000364c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000364d0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000364e0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000364f0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00036500: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 00036510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036530: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036540: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036550: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -00036560: 7454 6572 6d20 2020 2020 2020 2020 2020 tTerm │ │ │ │ -00036570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036580: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036590: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000365a0: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -000365b0: 7454 6572 6d20 2020 2020 2020 2020 2020 tTerm │ │ │ │ -000365c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000365d0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -000365e0: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ -000365f0: 7374 6570 2c20 6162 6f75 7420 746f 2063 step, about to c │ │ │ │ -00036600: 6f6d 7075 7465 2064 696d 656e 7369 6f6e ompute dimension │ │ │ │ -00036610: 2e20 2053 7562 6d61 7472 6963 6573 2063 . Submatrices c │ │ │ │ -00036620: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64 o|.|regularInCod │ │ │ │ -00036630: 696d 656e 7369 6f6e 3a20 2069 7343 6f64 imension: isCod │ │ │ │ -00036640: 696d 4174 4c65 6173 7420 6661 696c 6564 imAtLeast failed │ │ │ │ -00036650: 2c20 636f 6d70 7574 696e 6720 636f 6469 , computing codi │ │ │ │ -00036660: 6d2e 2020 2020 2020 2020 2020 2020 2020 m. │ │ │ │ -00036670: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00036680: 696d 656e 7369 6f6e 3a20 2070 6172 7469 imension: parti │ │ │ │ -00036690: 616c 2073 696e 6775 6c61 7220 6c6f 6375 al singular locu │ │ │ │ -000366a0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70 s dimension comp │ │ │ │ -000366b0: 7574 6564 2c20 3d20 3320 2020 2020 2020 uted, = 3 │ │ │ │ -000366c0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000366d0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000366e0: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +00036520: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036530: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036540: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00036550: 536d 616c 6c65 7374 5465 726d 2020 2020 SmallestTerm │ │ │ │ +00036560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036570: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036580: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036590: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +000365a0: 536d 616c 6c65 7374 5465 726d 2020 2020 SmallestTerm │ │ │ │ +000365b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000365c0: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +000365d0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +000365e0: 2020 4c6f 6f70 2073 7465 702c 2061 626f Loop step, abo │ │ │ │ +000365f0: 7574 2074 6f20 636f 6d70 7574 6520 6469 ut to compute di │ │ │ │ +00036600: 6d65 6e73 696f 6e2e 2020 5375 626d 6174 mension. Submat │ │ │ │ +00036610: 7269 6365 7320 636f 7c0a 7c72 6567 756c rices co|.|regul │ │ │ │ +00036620: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00036630: 2020 6973 436f 6469 6d41 744c 6561 7374 isCodimAtLeast │ │ │ │ +00036640: 2066 6169 6c65 642c 2063 6f6d 7075 7469 failed, computi │ │ │ │ +00036650: 6e67 2063 6f64 696d 2e20 2020 2020 2020 ng codim. │ │ │ │ +00036660: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +00036670: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00036680: 2020 7061 7274 6961 6c20 7369 6e67 756c partial singul │ │ │ │ +00036690: 6172 206c 6f63 7573 2064 696d 656e 7369 ar locus dimensi │ │ │ │ +000366a0: 6f6e 2063 6f6d 7075 7465 642c 203d 2033 on computed, = 3 │ │ │ │ +000366b0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000366c0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000366d0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +000366e0: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 000366f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036710: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036720: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036730: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ -00036740: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ -00036750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036760: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036770: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036780: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ -00036790: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ -000367a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000367b0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000367c0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000367d0: 6720 5261 6e64 6f6d 2020 2020 2020 2020 g Random │ │ │ │ +00036700: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036710: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036720: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00036730: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ +00036740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036750: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036760: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036770: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00036780: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ +00036790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000367a0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000367b0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000367c0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20 Choosing Random │ │ │ │ +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: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036810: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036820: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ +000367f0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036800: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036810: 4368 6f6f 7369 6e67 2052 616e 646f 6d4e Choosing RandomN │ │ │ │ +00036820: 6f6e 5a65 726f 2020 2020 2020 2020 2020 onZero │ │ │ │ 00036830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036850: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036860: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036870: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -00036880: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00036890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000368a0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000368b0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000368c0: 6720 5261 6e64 6f6d 2020 2020 2020 2020 g Random │ │ │ │ +00036840: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036850: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036860: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00036870: 536d 616c 6c65 7374 2020 2020 2020 2020 Smallest │ │ │ │ +00036880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036890: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000368a0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000368b0: 4368 6f6f 7369 6e67 2052 616e 646f 6d20 Choosing Random │ │ │ │ +000368c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000368d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000368e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000368f0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036900: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036910: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ -00036920: 7454 6572 6d20 2020 2020 2020 2020 2020 tTerm │ │ │ │ -00036930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036940: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036950: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036960: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +000368e0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000368f0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036900: 4368 6f6f 7369 6e67 2047 5265 764c 6578 Choosing GRevLex │ │ │ │ +00036910: 536d 616c 6c65 7374 5465 726d 2020 2020 SmallestTerm │ │ │ │ +00036920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036930: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036940: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036950: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00036960: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 00036970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036990: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000369a0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -000369b0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ -000369c0: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ -000369d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000369e0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -000369f0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036a00: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +00036980: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036990: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000369a0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +000369b0: 6c65 7374 5465 726d 2020 2020 2020 2020 lestTerm │ │ │ │ +000369c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000369d0: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +000369e0: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +000369f0: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ +00036a00: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 00036a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036a30: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ -00036a40: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ -00036a50: 6720 5261 6e64 6f6d 2020 2020 2020 2020 g Random │ │ │ │ +00036a20: 2020 2020 2020 2020 7c0a 7c69 6e74 6572 |.|inter │ │ │ │ +00036a30: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ +00036a40: 4368 6f6f 7369 6e67 2052 616e 646f 6d20 Choosing Random │ │ │ │ +00036a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036a80: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00036a90: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ -00036aa0: 7374 6570 2c20 6162 6f75 7420 746f 2063 step, about to c │ │ │ │ -00036ab0: 6f6d 7075 7465 2064 696d 656e 7369 6f6e ompute dimension │ │ │ │ -00036ac0: 2e20 2053 7562 6d61 7472 6963 6573 2063 . Submatrices c │ │ │ │ -00036ad0: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64 o|.|regularInCod │ │ │ │ -00036ae0: 696d 656e 7369 6f6e 3a20 2073 696e 6775 imension: singu │ │ │ │ -00036af0: 6c61 724c 6f63 7573 2064 696d 656e 7369 larLocus dimensi │ │ │ │ -00036b00: 6f6e 2076 6572 6966 6965 6420 6279 2069 on verified by i │ │ │ │ -00036b10: 7343 6f64 696d 4174 4c65 6173 7420 2020 sCodimAtLeast │ │ │ │ -00036b20: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00036b30: 696d 656e 7369 6f6e 3a20 2070 6172 7469 imension: parti │ │ │ │ -00036b40: 616c 2073 696e 6775 6c61 7220 6c6f 6375 al singular locu │ │ │ │ -00036b50: 7320 6469 6d65 6e73 696f 6e20 636f 6d70 s dimension comp │ │ │ │ -00036b60: 7574 6564 2c20 3d20 3220 2020 2020 2020 uted, = 2 │ │ │ │ -00036b70: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ -00036b80: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ -00036b90: 636f 6d70 6c65 7465 642c 2073 7562 6d61 completed, subma │ │ │ │ -00036ba0: 7472 6963 6573 2063 6f6e 7369 6465 7265 trices considere │ │ │ │ -00036bb0: 6420 3d20 3439 2c20 616e 6420 636f 6d70 d = 49, and comp │ │ │ │ -00036bc0: 757c 0a7c 6420 3d20 3339 2e20 2073 696e u|.|d = 39. sin │ │ │ │ -00036bd0: 6775 6c61 7220 6c6f 6375 7320 6469 6d65 gular locus dime │ │ │ │ -00036be0: 6e73 696f 6e20 6170 7065 6172 7320 746f nsion appears to │ │ │ │ -00036bf0: 2062 6520 3d20 3220 2020 2020 2020 2020 be = 2 │ │ │ │ -00036c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036a70: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +00036a80: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00036a90: 2020 4c6f 6f70 2073 7465 702c 2061 626f Loop step, abo │ │ │ │ +00036aa0: 7574 2074 6f20 636f 6d70 7574 6520 6469 ut to compute di │ │ │ │ +00036ab0: 6d65 6e73 696f 6e2e 2020 5375 626d 6174 mension. Submat │ │ │ │ +00036ac0: 7269 6365 7320 636f 7c0a 7c72 6567 756c rices co|.|regul │ │ │ │ +00036ad0: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00036ae0: 2020 7369 6e67 756c 6172 4c6f 6375 7320 singularLocus │ │ │ │ +00036af0: 6469 6d65 6e73 696f 6e20 7665 7269 6669 dimension verifi │ │ │ │ +00036b00: 6564 2062 7920 6973 436f 6469 6d41 744c ed by isCodimAtL │ │ │ │ +00036b10: 6561 7374 2020 2020 7c0a 7c72 6567 756c east |.|regul │ │ │ │ +00036b20: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00036b30: 2020 7061 7274 6961 6c20 7369 6e67 756c partial singul │ │ │ │ +00036b40: 6172 206c 6f63 7573 2064 696d 656e 7369 ar locus dimensi │ │ │ │ +00036b50: 6f6e 2063 6f6d 7075 7465 642c 203d 2032 on computed, = 2 │ │ │ │ +00036b60: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ +00036b70: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ +00036b80: 2020 4c6f 6f70 2063 6f6d 706c 6574 6564 Loop completed │ │ │ │ +00036b90: 2c20 7375 626d 6174 7269 6365 7320 636f , submatrices co │ │ │ │ +00036ba0: 6e73 6964 6572 6564 203d 2034 392c 2061 nsidered = 49, a │ │ │ │ +00036bb0: 6e64 2063 6f6d 7075 7c0a 7c64 203d 2033 nd compu|.|d = 3 │ │ │ │ +00036bc0: 392e 2020 7369 6e67 756c 6172 206c 6f63 9. singular loc │ │ │ │ +00036bd0: 7573 2064 696d 656e 7369 6f6e 2061 7070 us dimension app │ │ │ │ +00036be0: 6561 7273 2074 6f20 6265 203d 2032 2020 ears to be = 2 │ │ │ │ +00036bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c60: 207c 0a7c 6f36 203d 2074 7275 6520 2020 |.|o6 = true │ │ │ │ +00036c50: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ +00036c60: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ 00036c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036cb0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ +00036ca0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d |.|----- │ │ │ │ +00036cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036d00: 2d7c 0a7c 6e6f 7273 2c20 7765 2077 696c -|.|nors, we wil │ │ │ │ -00036d10: 6c20 636f 6d70 7574 6520 7570 2074 6f20 l compute up to │ │ │ │ -00036d20: 3435 322e 3930 3820 6f66 2074 6865 6d2e 452.908 of them. │ │ │ │ +00036cf0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c6e 6f72 732c --------|.|nors, │ │ │ │ +00036d00: 2077 6520 7769 6c6c 2063 6f6d 7075 7465 we will compute │ │ │ │ +00036d10: 2075 7020 746f 2034 3532 2e39 3038 206f up to 452.908 o │ │ │ │ +00036d20: 6620 7468 656d 2e20 2020 2020 2020 2020 f them. │ │ │ │ 00036d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036d50: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036d40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036da0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036d90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036df0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036de0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036e40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036e90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036e80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036ee0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036ed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036f30: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036f20: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036f80: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036f70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036fd0: 207c 0a7c 6e73 6964 6572 6564 3a20 372c |.|nsidered: 7, │ │ │ │ -00036fe0: 2061 6e64 2063 6f6d 7075 7465 6420 3d20 and computed = │ │ │ │ -00036ff0: 3720 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ +00036fc0: 2020 2020 2020 2020 7c0a 7c6e 7369 6465 |.|nside │ │ │ │ +00036fd0: 7265 643a 2037 2c20 616e 6420 636f 6d70 red: 7, and comp │ │ │ │ +00036fe0: 7574 6564 203d 2037 2020 2020 2020 2020 uted = 7 │ │ │ │ +00036ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037020: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00037010: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00037020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037070: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00037060: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00037070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000370a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000370b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000370c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000370b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000370c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000370d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000370e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000370f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037110: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00037100: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00037110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037160: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00037150: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00037160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000371a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000371b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000371a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000371b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000371c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000371d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000371e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000371f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037200: 207c 0a7c 6e73 6964 6572 6564 3a20 3131 |.|nsidered: 11 │ │ │ │ -00037210: 2c20 616e 6420 636f 6d70 7574 6564 203d , and computed = │ │ │ │ -00037220: 2031 3020 2020 2020 2020 2020 2020 2020 10 │ │ │ │ +000371f0: 2020 2020 2020 2020 7c0a 7c6e 7369 6465 |.|nside │ │ │ │ +00037200: 7265 643a 2031 312c 2061 6e64 2063 6f6d red: 11, and com │ │ │ │ +00037210: 7075 7465 6420 3d20 3130 2020 2020 2020 puted = 10 │ │ │ │ +00037220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037250: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00037240: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00037250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000372a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00037290: 2020 2020 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00037360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037390: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00037380: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00037390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000373a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000373b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000373c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000373d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000373e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000373d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000373e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000373f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037420: 2020 2020 2020 2020 2020 2020 2020 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2020 │ │ │ │ -00038290: 207c 0a7c 6e73 6964 6572 6564 3a20 3439 |.|nsidered: 49 │ │ │ │ -000382a0: 2c20 616e 6420 636f 6d70 7574 6564 203d , and computed = │ │ │ │ -000382b0: 2033 3920 2020 2020 2020 2020 2020 2020 39 │ │ │ │ +00038280: 2020 2020 2020 2020 7c0a 7c6e 7369 6465 |.|nside │ │ │ │ +00038290: 7265 643a 2034 392c 2061 6e64 2063 6f6d red: 49, and com │ │ │ │ +000382a0: 7075 7465 6420 3d20 3339 2020 2020 2020 puted = 39 │ │ │ │ +000382b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000382c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000382d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000382e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000382d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000382e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000382f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038330: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00038320: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00038330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038380: 207c 0a7c 7465 202d 2d20 7573 6564 2031 |.|te -- used 1 │ │ │ │ -00038390: 2e33 3231 3173 2028 6370 7529 3b20 302e .3211s (cpu); 0. │ │ │ │ -000383a0: 3937 3534 3531 7320 2874 6872 6561 6429 975451s (thread) │ │ │ │ -000383b0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ -000383c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000383d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00038370: 2020 2020 2020 2020 7c0a 7c74 6520 2d2d |.|te -- │ │ │ │ +00038380: 2075 7365 6420 312e 3832 3331 3673 2028 used 1.82316s ( │ │ │ │ +00038390: 6370 7529 3b20 312e 3138 3239 3373 2028 cpu); 1.18293s ( │ │ │ │ +000383a0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000383b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000383c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000383d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000383e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000383f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038420: 2d2b 0a0a 4d61 784d 696e 6f72 732e 2020 -+..MaxMinors. │ │ │ │ -00038430: 5468 6520 6669 7273 7420 6f75 7470 7574 The first output │ │ │ │ -00038440: 2073 6179 7320 7468 6174 2077 6520 7769 says that we wi │ │ │ │ -00038450: 6c6c 2063 6f6d 7075 7465 2075 7020 746f ll compute up to │ │ │ │ -00038460: 2034 3532 2e39 206d 696e 6f72 730a 6265 452.9 minors.be │ │ │ │ -00038470: 666f 7265 2067 6976 696e 6720 7570 2e20 fore giving up. │ │ │ │ -00038480: 2057 6520 6361 6e20 636f 6e74 726f 6c20 We can control │ │ │ │ -00038490: 7468 6174 2062 7920 7365 7474 696e 6720 that by setting │ │ │ │ -000384a0: 7468 6520 6f70 7469 6f6e 204d 6178 4d69 the option MaxMi │ │ │ │ -000384b0: 6e6f 7273 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d nors...+-------- │ │ │ │ +00038410: 2d2d 2d2d 2d2d 2d2d 2b0a 0a4d 6178 4d69 --------+..MaxMi │ │ │ │ +00038420: 6e6f 7273 2e20 2054 6865 2066 6972 7374 nors. The first │ │ │ │ +00038430: 206f 7574 7075 7420 7361 7973 2074 6861 output says tha │ │ │ │ +00038440: 7420 7765 2077 696c 6c20 636f 6d70 7574 t we will comput │ │ │ │ +00038450: 6520 7570 2074 6f20 3435 322e 3920 6d69 e up to 452.9 mi │ │ │ │ +00038460: 6e6f 7273 0a62 6566 6f72 6520 6769 7669 nors.before givi │ │ │ │ +00038470: 6e67 2075 702e 2020 5765 2063 616e 2063 ng up. We can c │ │ │ │ +00038480: 6f6e 7472 6f6c 2074 6861 7420 6279 2073 ontrol that by s │ │ │ │ +00038490: 6574 7469 6e67 2074 6865 206f 7074 696f etting the optio │ │ │ │ +000384a0: 6e20 4d61 784d 696e 6f72 732e 0a0a 2b2d n MaxMinors...+- │ │ │ │ +000384b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000384c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000384d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000384e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000384f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038500: 2d2d 2d2d 2d2b 0a7c 6937 203a 2074 696d -----+.|i7 : tim │ │ │ │ -00038510: 6520 7265 6775 6c61 7249 6e43 6f64 696d e regularInCodim │ │ │ │ -00038520: 656e 7369 6f6e 2831 2c20 532f 4a2c 204d ension(1, S/J, M │ │ │ │ -00038530: 6178 4d69 6e6f 7273 3d3e 3130 2c20 5665 axMinors=>10, Ve │ │ │ │ -00038540: 7262 6f73 653d 3e74 7275 6529 2020 2020 rbose=>true) │ │ │ │ -00038550: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00038560: 2030 2e31 3832 3637 3173 2028 6370 7529 0.182671s (cpu) │ │ │ │ -00038570: 3b20 302e 3133 3331 3637 7320 2874 6872 ; 0.133167s (thr │ │ │ │ -00038580: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ -00038590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000385a0: 2020 2020 207c 0a7c 7265 6775 6c61 7249 |.|regularI │ │ │ │ -000385b0: 6e43 6f64 696d 656e 7369 6f6e 3a20 7269 nCodimension: ri │ │ │ │ -000385c0: 6e67 2064 696d 656e 7369 6f6e 203d 342c ng dimension =4, │ │ │ │ -000385d0: 2074 6865 7265 2061 7265 2031 3436 3531 there are 14651 │ │ │ │ -000385e0: 3238 2070 6f73 7369 626c 6520 3520 6279 28 possible 5 by │ │ │ │ -000385f0: 2035 206d 697c 0a7c 7265 6775 6c61 7249 5 mi|.|regularI │ │ │ │ -00038600: 6e43 6f64 696d 656e 7369 6f6e 3a20 4162 nCodimension: Ab │ │ │ │ -00038610: 6f75 7420 746f 2065 6e74 6572 206c 6f6f out to enter loo │ │ │ │ -00038620: 7020 2020 2020 2020 2020 2020 2020 2020 p │ │ │ │ -00038630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038640: 2020 2020 207c 0a7c 696e 7465 726e 616c |.|internal │ │ │ │ -00038650: 4368 6f6f 7365 4d69 6e6f 723a 2043 686f ChooseMinor: Cho │ │ │ │ -00038660: 6f73 696e 6720 5261 6e64 6f6d 2020 2020 osing Random │ │ │ │ +000384f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00038500: 3720 3a20 7469 6d65 2072 6567 756c 6172 7 : time regular │ │ │ │ +00038510: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ +00038520: 2053 2f4a 2c20 4d61 784d 696e 6f72 733d S/J, MaxMinors= │ │ │ │ +00038530: 3e31 302c 2056 6572 626f 7365 3d3e 7472 >10, Verbose=>tr │ │ │ │ +00038540: 7565 2920 2020 2020 2020 2020 7c0a 7c20 ue) |.| │ │ │ │ +00038550: 2d2d 2075 7365 6420 302e 3232 3139 3837 -- used 0.221987 │ │ │ │ +00038560: 7320 2863 7075 293b 2030 2e31 3437 3536 s (cpu); 0.14756 │ │ │ │ +00038570: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ +00038580: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00038590: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ +000385a0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +000385b0: 696f 6e3a 2072 696e 6720 6469 6d65 6e73 ion: ring dimens │ │ │ │ +000385c0: 696f 6e20 3d34 2c20 7468 6572 6520 6172 ion =4, there ar │ │ │ │ +000385d0: 6520 3134 3635 3132 3820 706f 7373 6962 e 1465128 possib │ │ │ │ +000385e0: 6c65 2035 2062 7920 3520 6d69 7c0a 7c72 le 5 by 5 mi|.|r │ │ │ │ +000385f0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +00038600: 696f 6e3a 2041 626f 7574 2074 6f20 656e ion: About to en │ │ │ │ +00038610: 7465 7220 6c6f 6f70 2020 2020 2020 2020 ter loop │ │ │ │ +00038620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038630: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ +00038640: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ +00038650: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e or: Choosing Ran │ │ │ │ +00038660: 646f 6d20 2020 2020 2020 2020 2020 2020 dom │ │ │ │ 00038670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038690: 2020 2020 207c 0a7c 696e 7465 726e 616c |.|internal │ │ │ │ -000386a0: 4368 6f6f 7365 4d69 6e6f 723a 2043 686f ChooseMinor: Cho │ │ │ │ -000386b0: 6f73 696e 6720 5261 6e64 6f6d 4e6f 6e5a osing RandomNonZ │ │ │ │ -000386c0: 6572 6f20 2020 2020 2020 2020 2020 2020 ero │ │ │ │ -000386d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000386e0: 2020 2020 207c 0a7c 696e 7465 726e 616c |.|internal │ │ │ │ -000386f0: 4368 6f6f 7365 4d69 6e6f 723a 2043 686f ChooseMinor: Cho │ │ │ │ -00038700: 6f73 696e 6720 4752 6576 4c65 7853 6d61 osing GRevLexSma │ │ │ │ -00038710: 6c6c 6573 7454 6572 6d20 2020 2020 2020 llestTerm │ │ │ │ -00038720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038730: 2020 2020 207c 0a7c 696e 7465 726e 616c |.|internal │ │ │ │ -00038740: 4368 6f6f 7365 4d69 6e6f 723a 2043 686f ChooseMinor: Cho │ │ │ │ -00038750: 6f73 696e 6720 5261 6e64 6f6d 2020 2020 osing Random │ │ │ │ +00038680: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ +00038690: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ +000386a0: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e or: Choosing Ran │ │ │ │ +000386b0: 646f 6d4e 6f6e 5a65 726f 2020 2020 2020 domNonZero │ │ │ │ +000386c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000386d0: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ +000386e0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ +000386f0: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265 or: Choosing GRe │ │ │ │ +00038700: 764c 6578 536d 616c 6c65 7374 5465 726d vLexSmallestTerm │ │ │ │ +00038710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038720: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ +00038730: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ +00038740: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e or: Choosing Ran │ │ │ │ +00038750: 646f 6d20 2020 2020 2020 2020 2020 2020 dom │ │ │ │ 00038760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038780: 2020 2020 207c 0a7c 696e 7465 726e 616c |.|internal │ │ │ │ -00038790: 4368 6f6f 7365 4d69 6e6f 723a 2043 686f ChooseMinor: Cho │ │ │ │ -000387a0: 6f73 696e 6720 5261 6e64 6f6d 2020 2020 osing Random │ │ │ │ +00038770: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ +00038780: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ +00038790: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e or: Choosing Ran │ │ │ │ +000387a0: 646f 6d20 2020 2020 2020 2020 2020 2020 dom │ │ │ │ 000387b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000387c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000387d0: 2020 2020 207c 0a7c 696e 7465 726e 616c |.|internal │ │ │ │ -000387e0: 4368 6f6f 7365 4d69 6e6f 723a 2043 686f ChooseMinor: Cho │ │ │ │ -000387f0: 6f73 696e 6720 5261 6e64 6f6d 2020 2020 osing Random │ │ │ │ +000387c0: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ +000387d0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ +000387e0: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e or: Choosing Ran │ │ │ │ +000387f0: 646f 6d20 2020 2020 2020 2020 2020 2020 dom │ │ │ │ 00038800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038820: 2020 2020 207c 0a7c 696e 7465 726e 616c |.|internal │ │ │ │ -00038830: 4368 6f6f 7365 4d69 6e6f 723a 2043 686f ChooseMinor: Cho │ │ │ │ -00038840: 6f73 696e 6720 4c65 7853 6d61 6c6c 6573 osing LexSmalles │ │ │ │ -00038850: 7454 6572 6d20 2020 2020 2020 2020 2020 tTerm │ │ │ │ -00038860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038870: 2020 2020 207c 0a7c 7265 6775 6c61 7249 |.|regularI │ │ │ │ -00038880: 6e43 6f64 696d 656e 7369 6f6e 3a20 204c nCodimension: L │ │ │ │ -00038890: 6f6f 7020 7374 6570 2c20 6162 6f75 7420 oop step, about │ │ │ │ -000388a0: 746f 2063 6f6d 7075 7465 2064 696d 656e to compute dimen │ │ │ │ -000388b0: 7369 6f6e 2e20 2053 7562 6d61 7472 6963 sion. Submatric │ │ │ │ -000388c0: 6573 2063 6f7c 0a7c 7265 6775 6c61 7249 es co|.|regularI │ │ │ │ -000388d0: 6e43 6f64 696d 656e 7369 6f6e 3a20 2069 nCodimension: i │ │ │ │ -000388e0: 7343 6f64 696d 4174 4c65 6173 7420 6661 sCodimAtLeast fa │ │ │ │ -000388f0: 696c 6564 2c20 636f 6d70 7574 696e 6720 iled, computing │ │ │ │ -00038900: 636f 6469 6d2e 2020 2020 2020 2020 2020 codim. │ │ │ │ -00038910: 2020 2020 207c 0a7c 7265 6775 6c61 7249 |.|regularI │ │ │ │ -00038920: 6e43 6f64 696d 656e 7369 6f6e 3a20 2070 nCodimension: p │ │ │ │ -00038930: 6172 7469 616c 2073 696e 6775 6c61 7220 artial singular │ │ │ │ -00038940: 6c6f 6375 7320 6469 6d65 6e73 696f 6e20 locus dimension │ │ │ │ -00038950: 636f 6d70 7574 6564 2c20 3d20 3320 2020 computed, = 3 │ │ │ │ -00038960: 2020 2020 207c 0a7c 696e 7465 726e 616c |.|internal │ │ │ │ -00038970: 4368 6f6f 7365 4d69 6e6f 723a 2043 686f ChooseMinor: Cho │ │ │ │ -00038980: 6f73 696e 6720 5261 6e64 6f6d 2020 2020 osing Random │ │ │ │ +00038810: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ +00038820: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ +00038830: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578 or: Choosing Lex │ │ │ │ +00038840: 536d 616c 6c65 7374 5465 726d 2020 2020 SmallestTerm │ │ │ │ +00038850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038860: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ +00038870: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +00038880: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c ion: Loop step, │ │ │ │ +00038890: 2061 626f 7574 2074 6f20 636f 6d70 7574 about to comput │ │ │ │ +000388a0: 6520 6469 6d65 6e73 696f 6e2e 2020 5375 e dimension. Su │ │ │ │ +000388b0: 626d 6174 7269 6365 7320 636f 7c0a 7c72 bmatrices co|.|r │ │ │ │ +000388c0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +000388d0: 696f 6e3a 2020 6973 436f 6469 6d41 744c ion: isCodimAtL │ │ │ │ +000388e0: 6561 7374 2066 6169 6c65 642c 2063 6f6d east failed, com │ │ │ │ +000388f0: 7075 7469 6e67 2063 6f64 696d 2e20 2020 puting codim. │ │ │ │ +00038900: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ +00038910: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +00038920: 696f 6e3a 2020 7061 7274 6961 6c20 7369 ion: partial si │ │ │ │ +00038930: 6e67 756c 6172 206c 6f63 7573 2064 696d ngular locus dim │ │ │ │ +00038940: 656e 7369 6f6e 2063 6f6d 7075 7465 642c ension computed, │ │ │ │ +00038950: 203d 2033 2020 2020 2020 2020 7c0a 7c69 = 3 |.|i │ │ │ │ +00038960: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ +00038970: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e or: Choosing Ran │ │ │ │ +00038980: 646f 6d20 2020 2020 2020 2020 2020 2020 dom │ │ │ │ 00038990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000389a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000389b0: 2020 2020 207c 0a7c 696e 7465 726e 616c |.|internal │ │ │ │ -000389c0: 4368 6f6f 7365 4d69 6e6f 723a 2043 686f ChooseMinor: Cho │ │ │ │ -000389d0: 6f73 696e 6720 4752 6576 4c65 7853 6d61 osing GRevLexSma │ │ │ │ -000389e0: 6c6c 6573 7420 2020 2020 2020 2020 2020 llest │ │ │ │ -000389f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038a00: 2020 2020 207c 0a7c 696e 7465 726e 616c |.|internal │ │ │ │ -00038a10: 4368 6f6f 7365 4d69 6e6f 723a 2043 686f ChooseMinor: Cho │ │ │ │ -00038a20: 6f73 696e 6720 4c65 7853 6d61 6c6c 6573 osing LexSmalles │ │ │ │ -00038a30: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00038a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038a50: 2020 2020 207c 0a7c 7265 6775 6c61 7249 |.|regularI │ │ │ │ -00038a60: 6e43 6f64 696d 656e 7369 6f6e 3a20 204c nCodimension: L │ │ │ │ -00038a70: 6f6f 7020 7374 6570 2c20 6162 6f75 7420 oop step, about │ │ │ │ -00038a80: 746f 2063 6f6d 7075 7465 2064 696d 656e to compute dimen │ │ │ │ -00038a90: 7369 6f6e 2e20 2053 7562 6d61 7472 6963 sion. Submatric │ │ │ │ -00038aa0: 6573 2063 6f7c 0a7c 7265 6775 6c61 7249 es co|.|regularI │ │ │ │ -00038ab0: 6e43 6f64 696d 656e 7369 6f6e 3a20 2069 nCodimension: i │ │ │ │ -00038ac0: 7343 6f64 696d 4174 4c65 6173 7420 6661 sCodimAtLeast fa │ │ │ │ -00038ad0: 696c 6564 2c20 636f 6d70 7574 696e 6720 iled, computing │ │ │ │ -00038ae0: 636f 6469 6d2e 2020 2020 2020 2020 2020 codim. │ │ │ │ -00038af0: 2020 2020 207c 0a7c 7265 6775 6c61 7249 |.|regularI │ │ │ │ -00038b00: 6e43 6f64 696d 656e 7369 6f6e 3a20 2070 nCodimension: p │ │ │ │ -00038b10: 6172 7469 616c 2073 696e 6775 6c61 7220 artial singular │ │ │ │ -00038b20: 6c6f 6375 7320 6469 6d65 6e73 696f 6e20 locus dimension │ │ │ │ -00038b30: 636f 6d70 7574 6564 2c20 3d20 3320 2020 computed, = 3 │ │ │ │ -00038b40: 2020 2020 207c 0a7c 7265 6775 6c61 7249 |.|regularI │ │ │ │ -00038b50: 6e43 6f64 696d 656e 7369 6f6e 3a20 204c nCodimension: L │ │ │ │ -00038b60: 6f6f 7020 636f 6d70 6c65 7465 642c 2073 oop completed, s │ │ │ │ -00038b70: 7562 6d61 7472 6963 6573 2063 6f6e 7369 ubmatrices consi │ │ │ │ -00038b80: 6465 7265 6420 3d20 3130 2c20 616e 6420 dered = 10, and │ │ │ │ -00038b90: 636f 6d70 757c 0a7c 2d2d 2d2d 2d2d 2d2d compu|.|-------- │ │ │ │ +000389a0: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ +000389b0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ +000389c0: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265 or: Choosing GRe │ │ │ │ +000389d0: 764c 6578 536d 616c 6c65 7374 2020 2020 vLexSmallest │ │ │ │ +000389e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000389f0: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ +00038a00: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ +00038a10: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578 or: Choosing Lex │ │ │ │ +00038a20: 536d 616c 6c65 7374 2020 2020 2020 2020 Smallest │ │ │ │ +00038a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038a40: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ +00038a50: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +00038a60: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c ion: Loop step, │ │ │ │ +00038a70: 2061 626f 7574 2074 6f20 636f 6d70 7574 about to comput │ │ │ │ +00038a80: 6520 6469 6d65 6e73 696f 6e2e 2020 5375 e dimension. Su │ │ │ │ +00038a90: 626d 6174 7269 6365 7320 636f 7c0a 7c72 bmatrices co|.|r │ │ │ │ +00038aa0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +00038ab0: 696f 6e3a 2020 6973 436f 6469 6d41 744c ion: isCodimAtL │ │ │ │ +00038ac0: 6561 7374 2066 6169 6c65 642c 2063 6f6d east failed, com │ │ │ │ +00038ad0: 7075 7469 6e67 2063 6f64 696d 2e20 2020 puting codim. │ │ │ │ +00038ae0: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ +00038af0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +00038b00: 696f 6e3a 2020 7061 7274 6961 6c20 7369 ion: partial si │ │ │ │ +00038b10: 6e67 756c 6172 206c 6f63 7573 2064 696d ngular locus dim │ │ │ │ +00038b20: 656e 7369 6f6e 2063 6f6d 7075 7465 642c ension computed, │ │ │ │ +00038b30: 203d 2033 2020 2020 2020 2020 7c0a 7c72 = 3 |.|r │ │ │ │ +00038b40: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +00038b50: 696f 6e3a 2020 4c6f 6f70 2063 6f6d 706c ion: Loop compl │ │ │ │ +00038b60: 6574 6564 2c20 7375 626d 6174 7269 6365 eted, submatrice │ │ │ │ +00038b70: 7320 636f 6e73 6964 6572 6564 203d 2031 s considered = 1 │ │ │ │ +00038b80: 302c 2061 6e64 2063 6f6d 7075 7c0a 7c2d 0, and compu|.|- │ │ │ │ +00038b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038be0: 2d2d 2d2d 2d7c 0a7c 6e6f 7273 2c20 7765 -----|.|nors, we │ │ │ │ -00038bf0: 2077 696c 6c20 636f 6d70 7574 6520 7570 will compute up │ │ │ │ -00038c00: 2074 6f20 3130 206f 6620 7468 656d 2e20 to 10 of them. │ │ │ │ +00038bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6e ------------|.|n │ │ │ │ +00038be0: 6f72 732c 2077 6520 7769 6c6c 2063 6f6d ors, we will com │ │ │ │ +00038bf0: 7075 7465 2075 7020 746f 2031 3020 6f66 pute up to 10 of │ │ │ │ +00038c00: 2074 6865 6d2e 2020 2020 2020 2020 2020 them. │ │ │ │ 00038c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038c30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038c20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038c80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038d20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038d70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038d60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038e10: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 2020 2020 │ │ │ │ -00038e60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038e50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038eb0: 2020 2020 207c 0a7c 6e73 6964 6572 6564 |.|nsidered │ │ │ │ -00038ec0: 3a20 372c 2061 6e64 2063 6f6d 7075 7465 : 7, and compute │ │ │ │ -00038ed0: 6420 3d20 3720 2020 2020 2020 2020 2020 d = 7 │ │ │ │ +00038ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c6e |.|n │ │ │ │ +00038eb0: 7369 6465 7265 643a 2037 2c20 616e 6420 sidered: 7, and │ │ │ │ +00038ec0: 636f 6d70 7574 6564 203d 2037 2020 2020 computed = 7 │ │ │ │ +00038ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038f00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038f50: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038f40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038fa0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038f90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038ff0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00038ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039040: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00039030: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00039040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039090: 2020 2020 207c 0a7c 6e73 6964 6572 6564 |.|nsidered │ │ │ │ -000390a0: 3a20 3130 2c20 616e 6420 636f 6d70 7574 : 10, and comput │ │ │ │ -000390b0: 6564 203d 2031 3020 2020 2020 2020 2020 ed = 10 │ │ │ │ +00039080: 2020 2020 2020 2020 2020 2020 7c0a 7c6e |.|n │ │ │ │ +00039090: 7369 6465 7265 643a 2031 302c 2061 6e64 sidered: 10, and │ │ │ │ +000390a0: 2063 6f6d 7075 7465 6420 3d20 3130 2020 computed = 10 │ │ │ │ +000390b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000390c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000390d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000390e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000390d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000390e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000390f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039130: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00039120: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00039130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039180: 2020 2020 207c 0a7c 7465 6420 3d20 3130 |.|ted = 10 │ │ │ │ -00039190: 2e20 2073 696e 6775 6c61 7220 6c6f 6375 . singular locu │ │ │ │ -000391a0: 7320 6469 6d65 6e73 696f 6e20 6170 7065 s dimension appe │ │ │ │ -000391b0: 6172 7320 746f 2062 6520 3d20 3320 2020 ars to be = 3 │ │ │ │ -000391c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000391d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00039170: 2020 2020 2020 2020 2020 2020 7c0a 7c74 |.|t │ │ │ │ +00039180: 6564 203d 2031 302e 2020 7369 6e67 756c ed = 10. singul │ │ │ │ +00039190: 6172 206c 6f63 7573 2064 696d 656e 7369 ar locus dimensi │ │ │ │ +000391a0: 6f6e 2061 7070 6561 7273 2074 6f20 6265 on appears to be │ │ │ │ +000391b0: 203d 2033 2020 2020 2020 2020 2020 2020 = 3 │ │ │ │ +000391c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000391d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000391e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000391f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00039200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00039210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00039220: 2d2d 2d2d 2d2b 0a0a 5468 6572 6520 6172 -----+..There ar │ │ │ │ -00039230: 6520 6f74 6865 7220 6669 6e65 7220 7761 e other finer wa │ │ │ │ -00039240: 7973 2074 6f20 636f 6e74 726f 6c20 7468 ys to control th │ │ │ │ -00039250: 6520 4d61 784d 696e 6f72 7320 6f70 7469 e MaxMinors opti │ │ │ │ -00039260: 6f6e 2c20 6275 7420 7468 6579 2077 696c on, but they wil │ │ │ │ -00039270: 6c20 6e6f 740a 6265 2064 6973 6375 7373 l not.be discuss │ │ │ │ -00039280: 6564 2069 6e20 7468 6973 2074 7574 6f72 ed in this tutor │ │ │ │ -00039290: 6961 6c2e 2020 5365 6520 2a6e 6f74 6520 ial. See *note │ │ │ │ -000392a0: 7265 6775 6c61 7249 6e43 6f64 696d 656e regularInCodimen │ │ │ │ -000392b0: 7369 6f6e 3a0a 7265 6775 6c61 7249 6e43 sion:.regularInC │ │ │ │ -000392c0: 6f64 696d 656e 7369 6f6e 2c2e 0a0a 5365 odimension,...Se │ │ │ │ -000392d0: 6c65 6374 696e 6720 7375 626d 6174 7269 lecting submatri │ │ │ │ -000392e0: 6365 7320 6f66 2074 6865 204a 6163 6f62 ces of the Jacob │ │ │ │ -000392f0: 6961 6e2e 2020 5765 2061 6c73 6f20 7365 ian. We also se │ │ │ │ -00039300: 6520 6f75 7470 7574 206c 696b 653a 2060 e output like: ` │ │ │ │ -00039310: 6043 686f 6f73 696e 670a 4c65 7853 6d61 `Choosing.LexSma │ │ │ │ -00039320: 6c6c 6573 7427 2720 6f72 2060 6043 686f llest'' or ``Cho │ │ │ │ -00039330: 6f73 696e 6720 5261 6e64 6f6d 2727 2e20 osing Random''. │ │ │ │ -00039340: 2054 6869 7320 6973 2073 6179 696e 6720 This is saying │ │ │ │ -00039350: 686f 7720 7765 2061 7265 2073 656c 6563 how we are selec │ │ │ │ -00039360: 7469 6e67 2061 0a67 6976 656e 2073 7562 ting a.given sub │ │ │ │ -00039370: 6d61 7472 6978 2e20 2046 6f72 2069 6e73 matrix. For ins │ │ │ │ -00039380: 7461 6e63 652c 2077 6520 6361 6e20 7275 tance, we can ru │ │ │ │ -00039390: 6e3a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d n:..+----------- │ │ │ │ +00039210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 ------------+..T │ │ │ │ +00039220: 6865 7265 2061 7265 206f 7468 6572 2066 here are other f │ │ │ │ +00039230: 696e 6572 2077 6179 7320 746f 2063 6f6e iner ways to con │ │ │ │ +00039240: 7472 6f6c 2074 6865 204d 6178 4d69 6e6f trol the MaxMino │ │ │ │ +00039250: 7273 206f 7074 696f 6e2c 2062 7574 2074 rs option, but t │ │ │ │ +00039260: 6865 7920 7769 6c6c 206e 6f74 0a62 6520 hey will not.be │ │ │ │ +00039270: 6469 7363 7573 7365 6420 696e 2074 6869 discussed in thi │ │ │ │ +00039280: 7320 7475 746f 7269 616c 2e20 2053 6565 s tutorial. See │ │ │ │ +00039290: 202a 6e6f 7465 2072 6567 756c 6172 496e *note regularIn │ │ │ │ +000392a0: 436f 6469 6d65 6e73 696f 6e3a 0a72 6567 Codimension:.reg │ │ │ │ +000392b0: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ +000392c0: 6e2c 2e0a 0a53 656c 6563 7469 6e67 2073 n,...Selecting s │ │ │ │ +000392d0: 7562 6d61 7472 6963 6573 206f 6620 7468 ubmatrices of th │ │ │ │ +000392e0: 6520 4a61 636f 6269 616e 2e20 2057 6520 e Jacobian. We │ │ │ │ +000392f0: 616c 736f 2073 6565 206f 7574 7075 7420 also see output │ │ │ │ +00039300: 6c69 6b65 3a20 6060 4368 6f6f 7369 6e67 like: ``Choosing │ │ │ │ +00039310: 0a4c 6578 536d 616c 6c65 7374 2727 206f .LexSmallest'' o │ │ │ │ +00039320: 7220 6060 4368 6f6f 7369 6e67 2052 616e r ``Choosing Ran │ │ │ │ +00039330: 646f 6d27 272e 2020 5468 6973 2069 7320 dom''. This is │ │ │ │ +00039340: 7361 7969 6e67 2068 6f77 2077 6520 6172 saying how we ar │ │ │ │ +00039350: 6520 7365 6c65 6374 696e 6720 610a 6769 e selecting a.gi │ │ │ │ +00039360: 7665 6e20 7375 626d 6174 7269 782e 2020 ven submatrix. │ │ │ │ +00039370: 466f 7220 696e 7374 616e 6365 2c20 7765 For instance, we │ │ │ │ +00039380: 2063 616e 2072 756e 3a0a 0a2b 2d2d 2d2d can run:..+---- │ │ │ │ +00039390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000393a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000393b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000393c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000393d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000393e0: 2d2d 2b0a 7c69 3820 3a20 7469 6d65 2072 --+.|i8 : time r │ │ │ │ -000393f0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -00039400: 696f 6e28 312c 2053 2f4a 2c20 4d61 784d ion(1, S/J, MaxM │ │ │ │ -00039410: 696e 6f72 733d 3e31 302c 2053 7472 6174 inors=>10, Strat │ │ │ │ -00039420: 6567 793d 3e53 7472 6174 6567 7952 616e egy=>StrategyRan │ │ │ │ -00039430: 646f 7c0a 7c20 2d2d 2075 7365 6420 302e do|.| -- used 0. │ │ │ │ -00039440: 3133 3530 3933 7320 2863 7075 293b 2030 135093s (cpu); 0 │ │ │ │ -00039450: 2e31 3135 3434 3773 2028 7468 7265 6164 .115447s (thread │ │ │ │ -00039460: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ -00039470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039480: 2020 7c0a 7c72 6567 756c 6172 496e 436f |.|regularInCo │ │ │ │ -00039490: 6469 6d65 6e73 696f 6e3a 2072 696e 6720 dimension: ring │ │ │ │ -000394a0: 6469 6d65 6e73 696f 6e20 3d34 2c20 7468 dimension =4, th │ │ │ │ -000394b0: 6572 6520 6172 6520 3134 3635 3132 3820 ere are 1465128 │ │ │ │ -000394c0: 706f 7373 6962 6c65 2035 2062 7920 3520 possible 5 by 5 │ │ │ │ -000394d0: 6d69 7c0a 7c72 6567 756c 6172 496e 436f mi|.|regularInCo │ │ │ │ -000394e0: 6469 6d65 6e73 696f 6e3a 2041 626f 7574 dimension: About │ │ │ │ -000394f0: 2074 6f20 656e 7465 7220 6c6f 6f70 2020 to enter loop │ │ │ │ +000393d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a ---------+.|i8 : │ │ │ │ +000393e0: 2074 696d 6520 7265 6775 6c61 7249 6e43 time regularInC │ │ │ │ +000393f0: 6f64 696d 656e 7369 6f6e 2831 2c20 532f odimension(1, S/ │ │ │ │ +00039400: 4a2c 204d 6178 4d69 6e6f 7273 3d3e 3130 J, MaxMinors=>10 │ │ │ │ +00039410: 2c20 5374 7261 7465 6779 3d3e 5374 7261 , Strategy=>Stra │ │ │ │ +00039420: 7465 6779 5261 6e64 6f7c 0a7c 202d 2d20 tegyRando|.| -- │ │ │ │ +00039430: 7573 6564 2030 2e33 3039 3639 3273 2028 used 0.309692s ( │ │ │ │ +00039440: 6370 7529 3b20 302e 3233 3638 3435 7320 cpu); 0.236845s │ │ │ │ +00039450: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +00039460: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00039470: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ +00039480: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ +00039490: 3a20 7269 6e67 2064 696d 656e 7369 6f6e : ring dimension │ │ │ │ +000394a0: 203d 342c 2074 6865 7265 2061 7265 2031 =4, there are 1 │ │ │ │ +000394b0: 3436 3531 3238 2070 6f73 7369 626c 6520 465128 possible │ │ │ │ +000394c0: 3520 6279 2035 206d 697c 0a7c 7265 6775 5 by 5 mi|.|regu │ │ │ │ +000394d0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ +000394e0: 3a20 4162 6f75 7420 746f 2065 6e74 6572 : About to enter │ │ │ │ +000394f0: 206c 6f6f 7020 2020 2020 2020 2020 2020 loop │ │ │ │ 00039500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039520: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f |.|internalCho │ │ │ │ -00039530: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ -00039540: 6e67 2052 616e 646f 6d20 2020 2020 2020 ng Random │ │ │ │ +00039510: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ +00039520: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ +00039530: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +00039540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039570: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f |.|internalCho │ │ │ │ -00039580: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ -00039590: 6e67 2052 616e 646f 6d20 2020 2020 2020 ng Random │ │ │ │ +00039560: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ +00039570: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ +00039580: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +00039590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000395a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000395b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000395c0: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f |.|internalCho │ │ │ │ -000395d0: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ -000395e0: 6e67 2052 616e 646f 6d20 2020 2020 2020 ng Random │ │ │ │ +000395b0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ +000395c0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ +000395d0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +000395e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000395f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039610: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f |.|internalCho │ │ │ │ -00039620: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ -00039630: 6e67 2052 616e 646f 6d20 2020 2020 2020 ng Random │ │ │ │ +00039600: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ +00039610: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ +00039620: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +00039630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039660: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f |.|internalCho │ │ │ │ -00039670: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ -00039680: 6e67 2052 616e 646f 6d20 2020 2020 2020 ng Random │ │ │ │ +00039650: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ +00039660: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ +00039670: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +00039680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000396a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000396b0: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f |.|internalCho │ │ │ │ -000396c0: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ -000396d0: 6e67 2052 616e 646f 6d20 2020 2020 2020 ng Random │ │ │ │ +000396a0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ +000396b0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ +000396c0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +000396d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000396e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000396f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039700: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f |.|internalCho │ │ │ │ -00039710: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ -00039720: 6e67 2052 616e 646f 6d20 2020 2020 2020 ng Random │ │ │ │ +000396f0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ +00039700: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ +00039710: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +00039720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039750: 2020 7c0a 7c72 6567 756c 6172 496e 436f |.|regularInCo │ │ │ │ -00039760: 6469 6d65 6e73 696f 6e3a 2020 4c6f 6f70 dimension: Loop │ │ │ │ -00039770: 2073 7465 702c 2061 626f 7574 2074 6f20 step, about to │ │ │ │ -00039780: 636f 6d70 7574 6520 6469 6d65 6e73 696f compute dimensio │ │ │ │ -00039790: 6e2e 2020 5375 626d 6174 7269 6365 7320 n. Submatrices │ │ │ │ -000397a0: 636f 7c0a 7c72 6567 756c 6172 496e 436f co|.|regularInCo │ │ │ │ -000397b0: 6469 6d65 6e73 696f 6e3a 2020 6973 436f dimension: isCo │ │ │ │ -000397c0: 6469 6d41 744c 6561 7374 2066 6169 6c65 dimAtLeast faile │ │ │ │ -000397d0: 642c 2063 6f6d 7075 7469 6e67 2063 6f64 d, computing cod │ │ │ │ -000397e0: 696d 2e20 2020 2020 2020 2020 2020 2020 im. │ │ │ │ -000397f0: 2020 7c0a 7c72 6567 756c 6172 496e 436f |.|regularInCo │ │ │ │ -00039800: 6469 6d65 6e73 696f 6e3a 2020 7061 7274 dimension: part │ │ │ │ -00039810: 6961 6c20 7369 6e67 756c 6172 206c 6f63 ial singular loc │ │ │ │ -00039820: 7573 2064 696d 656e 7369 6f6e 2063 6f6d us dimension com │ │ │ │ -00039830: 7075 7465 642c 203d 2033 2020 2020 2020 puted, = 3 │ │ │ │ -00039840: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f |.|internalCho │ │ │ │ -00039850: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ -00039860: 6e67 2052 616e 646f 6d20 2020 2020 2020 ng Random │ │ │ │ +00039740: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ +00039750: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ +00039760: 3a20 204c 6f6f 7020 7374 6570 2c20 6162 : Loop step, ab │ │ │ │ +00039770: 6f75 7420 746f 2063 6f6d 7075 7465 2064 out to compute d │ │ │ │ +00039780: 696d 656e 7369 6f6e 2e20 2053 7562 6d61 imension. Subma │ │ │ │ +00039790: 7472 6963 6573 2063 6f7c 0a7c 7265 6775 trices co|.|regu │ │ │ │ +000397a0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ +000397b0: 3a20 2069 7343 6f64 696d 4174 4c65 6173 : isCodimAtLeas │ │ │ │ +000397c0: 7420 6661 696c 6564 2c20 636f 6d70 7574 t failed, comput │ │ │ │ +000397d0: 696e 6720 636f 6469 6d2e 2020 2020 2020 ing codim. │ │ │ │ +000397e0: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ +000397f0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ +00039800: 3a20 2070 6172 7469 616c 2073 696e 6775 : partial singu │ │ │ │ +00039810: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73 lar locus dimens │ │ │ │ +00039820: 696f 6e20 636f 6d70 7574 6564 2c20 3d20 ion computed, = │ │ │ │ +00039830: 3320 2020 2020 2020 207c 0a7c 696e 7465 3 |.|inte │ │ │ │ +00039840: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ +00039850: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +00039860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039890: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f |.|internalCho │ │ │ │ -000398a0: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ -000398b0: 6e67 2052 616e 646f 6d20 2020 2020 2020 ng Random │ │ │ │ +00039880: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ +00039890: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ +000398a0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +000398b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000398c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000398d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000398e0: 2020 7c0a 7c69 6e74 6572 6e61 6c43 686f |.|internalCho │ │ │ │ -000398f0: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ -00039900: 6e67 2052 616e 646f 6d20 2020 2020 2020 ng Random │ │ │ │ +000398d0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ +000398e0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ +000398f0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +00039900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039930: 2020 7c0a 7c72 6567 756c 6172 496e 436f |.|regularInCo │ │ │ │ -00039940: 6469 6d65 6e73 696f 6e3a 2020 4c6f 6f70 dimension: Loop │ │ │ │ -00039950: 2073 7465 702c 2061 626f 7574 2074 6f20 step, about to │ │ │ │ -00039960: 636f 6d70 7574 6520 6469 6d65 6e73 696f compute dimensio │ │ │ │ -00039970: 6e2e 2020 5375 626d 6174 7269 6365 7320 n. Submatrices │ │ │ │ -00039980: 636f 7c0a 7c72 6567 756c 6172 496e 436f co|.|regularInCo │ │ │ │ -00039990: 6469 6d65 6e73 696f 6e3a 2020 6973 436f dimension: isCo │ │ │ │ -000399a0: 6469 6d41 744c 6561 7374 2066 6169 6c65 dimAtLeast faile │ │ │ │ -000399b0: 642c 2063 6f6d 7075 7469 6e67 2063 6f64 d, computing cod │ │ │ │ -000399c0: 696d 2e20 2020 2020 2020 2020 2020 2020 im. │ │ │ │ -000399d0: 2020 7c0a 7c72 6567 756c 6172 496e 436f |.|regularInCo │ │ │ │ -000399e0: 6469 6d65 6e73 696f 6e3a 2020 7061 7274 dimension: part │ │ │ │ -000399f0: 6961 6c20 7369 6e67 756c 6172 206c 6f63 ial singular loc │ │ │ │ -00039a00: 7573 2064 696d 656e 7369 6f6e 2063 6f6d us dimension com │ │ │ │ -00039a10: 7075 7465 642c 203d 2033 2020 2020 2020 puted, = 3 │ │ │ │ -00039a20: 2020 7c0a 7c72 6567 756c 6172 496e 436f |.|regularInCo │ │ │ │ -00039a30: 6469 6d65 6e73 696f 6e3a 2020 4c6f 6f70 dimension: Loop │ │ │ │ -00039a40: 2063 6f6d 706c 6574 6564 2c20 7375 626d completed, subm │ │ │ │ -00039a50: 6174 7269 6365 7320 636f 6e73 6964 6572 atrices consider │ │ │ │ -00039a60: 6564 203d 2031 302c 2061 6e64 2063 6f6d ed = 10, and com │ │ │ │ -00039a70: 7075 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d pu|.|----------- │ │ │ │ +00039920: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ +00039930: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ +00039940: 3a20 204c 6f6f 7020 7374 6570 2c20 6162 : Loop step, ab │ │ │ │ +00039950: 6f75 7420 746f 2063 6f6d 7075 7465 2064 out to compute d │ │ │ │ +00039960: 696d 656e 7369 6f6e 2e20 2053 7562 6d61 imension. Subma │ │ │ │ +00039970: 7472 6963 6573 2063 6f7c 0a7c 7265 6775 trices co|.|regu │ │ │ │ +00039980: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ +00039990: 3a20 2069 7343 6f64 696d 4174 4c65 6173 : isCodimAtLeas │ │ │ │ +000399a0: 7420 6661 696c 6564 2c20 636f 6d70 7574 t failed, comput │ │ │ │ +000399b0: 696e 6720 636f 6469 6d2e 2020 2020 2020 ing codim. │ │ │ │ +000399c0: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ +000399d0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ +000399e0: 3a20 2070 6172 7469 616c 2073 696e 6775 : partial singu │ │ │ │ +000399f0: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73 lar locus dimens │ │ │ │ +00039a00: 696f 6e20 636f 6d70 7574 6564 2c20 3d20 ion computed, = │ │ │ │ +00039a10: 3320 2020 2020 2020 207c 0a7c 7265 6775 3 |.|regu │ │ │ │ +00039a20: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ +00039a30: 3a20 204c 6f6f 7020 636f 6d70 6c65 7465 : Loop complete │ │ │ │ +00039a40: 642c 2073 7562 6d61 7472 6963 6573 2063 d, submatrices c │ │ │ │ +00039a50: 6f6e 7369 6465 7265 6420 3d20 3130 2c20 onsidered = 10, │ │ │ │ +00039a60: 616e 6420 636f 6d70 757c 0a7c 2d2d 2d2d and compu|.|---- │ │ │ │ +00039a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00039a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00039a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00039aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00039ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00039ac0: 2d2d 7c0a 7c6d 2c20 5665 7262 6f73 653d --|.|m, Verbose= │ │ │ │ -00039ad0: 3e74 7275 6529 2020 2020 2020 2020 2020 >true) │ │ │ │ +00039ab0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6d2c 2056 ---------|.|m, V │ │ │ │ +00039ac0: 6572 626f 7365 3d3e 7472 7565 2920 2020 erbose=>true) │ │ │ │ +00039ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039b10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039b00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039b60: 2020 7c0a 7c6e 6f72 732c 2077 6520 7769 |.|nors, we wi │ │ │ │ -00039b70: 6c6c 2063 6f6d 7075 7465 2075 7020 746f ll compute up to │ │ │ │ -00039b80: 2031 3020 6f66 2074 6865 6d2e 2020 2020 10 of them. │ │ │ │ +00039b50: 2020 2020 2020 2020 207c 0a7c 6e6f 7273 |.|nors │ │ │ │ +00039b60: 2c20 7765 2077 696c 6c20 636f 6d70 7574 , we will comput │ │ │ │ +00039b70: 6520 7570 2074 6f20 3130 206f 6620 7468 e up to 10 of th │ │ │ │ +00039b80: 656d 2e20 2020 2020 2020 2020 2020 2020 em. │ │ │ │ 00039b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039bb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039ba0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039c00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039bf0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039c50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039c40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039ca0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039c90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039cf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039ce0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039d40: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039d30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039d90: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039d80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039de0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039dd0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039e30: 2020 7c0a 7c6e 7369 6465 7265 643a 2037 |.|nsidered: 7 │ │ │ │ -00039e40: 2c20 616e 6420 636f 6d70 7574 6564 203d , and computed = │ │ │ │ -00039e50: 2037 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ +00039e20: 2020 2020 2020 2020 207c 0a7c 6e73 6964 |.|nsid │ │ │ │ +00039e30: 6572 6564 3a20 372c 2061 6e64 2063 6f6d ered: 7, and com │ │ │ │ +00039e40: 7075 7465 6420 3d20 3720 2020 2020 2020 puted = 7 │ │ │ │ +00039e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039e80: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039e70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039ed0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039ec0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039f20: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039f10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039f70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039f60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00039fc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00039fb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00039fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00039ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a010: 2020 7c0a 7c6e 7369 6465 7265 643a 2031 |.|nsidered: 1 │ │ │ │ -0003a020: 302c 2061 6e64 2063 6f6d 7075 7465 6420 0, and computed │ │ │ │ -0003a030: 3d20 3130 2020 2020 2020 2020 2020 2020 = 10 │ │ │ │ +0003a000: 2020 2020 2020 2020 207c 0a7c 6e73 6964 |.|nsid │ │ │ │ +0003a010: 6572 6564 3a20 3130 2c20 616e 6420 636f ered: 10, and co │ │ │ │ +0003a020: 6d70 7574 6564 203d 2031 3020 2020 2020 mputed = 10 │ │ │ │ +0003a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a060: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003a050: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a0b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003a0a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a100: 2020 7c0a 7c74 6564 203d 2031 302e 2020 |.|ted = 10. │ │ │ │ -0003a110: 7369 6e67 756c 6172 206c 6f63 7573 2064 singular locus d │ │ │ │ -0003a120: 696d 656e 7369 6f6e 2061 7070 6561 7273 imension appears │ │ │ │ -0003a130: 2074 6f20 6265 203d 2033 2020 2020 2020 to be = 3 │ │ │ │ -0003a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a150: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0003a0f0: 2020 2020 2020 2020 207c 0a7c 7465 6420 |.|ted │ │ │ │ +0003a100: 3d20 3130 2e20 2073 696e 6775 6c61 7220 = 10. singular │ │ │ │ +0003a110: 6c6f 6375 7320 6469 6d65 6e73 696f 6e20 locus dimension │ │ │ │ +0003a120: 6170 7065 6172 7320 746f 2062 6520 3d20 appears to be = │ │ │ │ +0003a130: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0003a140: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003a150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003a160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003a1a0: 2d2d 2b0a 0a61 6e64 206f 6e6c 7920 7261 --+..and only ra │ │ │ │ -0003a1b0: 6e64 6f6d 2073 7562 6d61 7472 6963 6573 ndom submatrices │ │ │ │ -0003a1c0: 2061 7265 2063 686f 7365 6e2e 2020 5765 are chosen. We │ │ │ │ -0003a1d0: 2064 6973 6375 7373 2073 7472 6174 6567 discuss strateg │ │ │ │ -0003a1e0: 6965 7320 666f 7220 6368 6f6f 7369 6e67 ies for choosing │ │ │ │ -0003a1f0: 0a73 7562 6d61 7472 6963 6573 206d 7563 .submatrices muc │ │ │ │ -0003a200: 6820 6d6f 7265 2067 656e 6572 616c 6c79 h more generally │ │ │ │ -0003a210: 2069 6e20 7468 6520 2a6e 6f74 6520 4661 in the *note Fa │ │ │ │ -0003a220: 7374 4d69 6e6f 7273 5374 7261 7465 6779 stMinorsStrategy │ │ │ │ -0003a230: 5475 746f 7269 616c 3a0a 4661 7374 4d69 Tutorial:.FastMi │ │ │ │ -0003a240: 6e6f 7273 5374 7261 7465 6779 5475 746f norsStrategyTuto │ │ │ │ -0003a250: 7269 616c 2c2e 2052 6567 6172 646c 6573 rial,. Regardles │ │ │ │ -0003a260: 732c 2061 6674 6572 2061 2063 6572 7461 s, after a certa │ │ │ │ -0003a270: 696e 206e 756d 6265 7220 6f66 206d 696e in number of min │ │ │ │ -0003a280: 6f72 7320 6861 7665 0a62 6565 6e20 6c6f ors have.been lo │ │ │ │ -0003a290: 6f6b 6564 2061 742c 2077 6520 7365 6520 oked at, we see │ │ │ │ -0003a2a0: 6f75 7470 7574 206c 696e 6573 206c 696b output lines lik │ │ │ │ -0003a2b0: 653a 2020 6060 4c6f 6f70 2073 7465 702c e: ``Loop step, │ │ │ │ -0003a2c0: 2061 626f 7574 2074 6f20 636f 6d70 7574 about to comput │ │ │ │ -0003a2d0: 650a 6469 6d65 6e73 696f 6e2e 2020 5375 e.dimension. Su │ │ │ │ -0003a2e0: 626d 6174 7269 6365 7320 636f 6e73 6964 bmatrices consid │ │ │ │ -0003a2f0: 6572 6564 3a20 372c 2061 6e64 2063 6f6d ered: 7, and com │ │ │ │ -0003a300: 7075 7465 6420 3d20 3727 272e 2020 5765 puted = 7''. We │ │ │ │ -0003a310: 206f 6e6c 7920 636f 6d70 7574 650a 6d69 only compute.mi │ │ │ │ -0003a320: 6e6f 7273 2077 6520 6861 7665 6e27 7420 nors we haven't │ │ │ │ -0003a330: 636f 6e73 6964 6572 6564 2062 6566 6f72 considered befor │ │ │ │ -0003a340: 652e 2020 536f 2061 7320 7765 2063 6f6d e. So as we com │ │ │ │ -0003a350: 7075 7465 206d 6f72 6520 6d69 6e6f 7273 pute more minors │ │ │ │ -0003a360: 2c20 7468 6572 6520 6361 6e0a 6265 2061 , there can.be a │ │ │ │ -0003a370: 2064 6973 7469 6e63 7469 6f6e 2062 6574 distinction bet │ │ │ │ -0003a380: 7765 656e 2063 6f6e 7369 6465 7265 6420 ween considered │ │ │ │ -0003a390: 616e 6420 636f 6d70 7574 6564 2e0a 0a43 and computed...C │ │ │ │ -0003a3a0: 6f6d 7075 7469 6e67 206d 696e 6f72 7320 omputing minors │ │ │ │ -0003a3b0: 7673 2063 6f6e 7369 6465 7269 6e67 2074 vs considering t │ │ │ │ -0003a3c0: 6865 2064 696d 656e 7369 6f6e 206f 6620 he dimension of │ │ │ │ -0003a3d0: 7768 6174 2068 6173 2062 6565 6e20 636f what has been co │ │ │ │ -0003a3e0: 6d70 7574 6564 2e0a 5065 7269 6f64 6963 mputed..Periodic │ │ │ │ -0003a3f0: 616c 6c79 2077 6520 636f 6d70 7574 6520 ally we compute │ │ │ │ -0003a400: 7468 6520 636f 6469 6d65 6e73 696f 6e20 the codimension │ │ │ │ -0003a410: 6f66 2074 6865 2070 6172 7469 616c 2069 of the partial i │ │ │ │ -0003a420: 6465 616c 206f 6620 6d69 6e6f 7273 2077 deal of minors w │ │ │ │ -0003a430: 6520 6861 7665 0a63 6f6d 7075 7465 6420 e have.computed │ │ │ │ -0003a440: 736f 2066 6172 2e20 2054 6865 7265 2061 so far. There a │ │ │ │ -0003a450: 7265 2074 776f 206f 7074 696f 6e73 2074 re two options t │ │ │ │ -0003a460: 6f20 636f 6e74 726f 6c20 7468 6973 2e20 o control this. │ │ │ │ -0003a470: 2046 6972 7374 2c20 7765 2063 616e 2074 First, we can t │ │ │ │ -0003a480: 656c 6c0a 7468 6520 6675 6e63 7469 6f6e ell.the function │ │ │ │ -0003a490: 2077 6865 6e20 746f 2066 6972 7374 2063 when to first c │ │ │ │ -0003a4a0: 6f6d 7075 7465 2074 6865 2064 696d 656e ompute the dimen │ │ │ │ -0003a4b0: 7369 6f6e 206f 6620 7468 6520 776f 726b sion of the work │ │ │ │ -0003a4c0: 696e 6720 7061 7274 6961 6c20 6964 6561 ing partial idea │ │ │ │ -0003a4d0: 6c0a 6f66 206d 696e 6f72 732e 0a0a 2b2d l.of minors...+- │ │ │ │ +0003a190: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 616e 6420 ---------+..and │ │ │ │ +0003a1a0: 6f6e 6c79 2072 616e 646f 6d20 7375 626d only random subm │ │ │ │ +0003a1b0: 6174 7269 6365 7320 6172 6520 6368 6f73 atrices are chos │ │ │ │ +0003a1c0: 656e 2e20 2057 6520 6469 7363 7573 7320 en. We discuss │ │ │ │ +0003a1d0: 7374 7261 7465 6769 6573 2066 6f72 2063 strategies for c │ │ │ │ +0003a1e0: 686f 6f73 696e 670a 7375 626d 6174 7269 hoosing.submatri │ │ │ │ +0003a1f0: 6365 7320 6d75 6368 206d 6f72 6520 6765 ces much more ge │ │ │ │ +0003a200: 6e65 7261 6c6c 7920 696e 2074 6865 202a nerally in the * │ │ │ │ +0003a210: 6e6f 7465 2046 6173 744d 696e 6f72 7353 note FastMinorsS │ │ │ │ +0003a220: 7472 6174 6567 7954 7574 6f72 6961 6c3a trategyTutorial: │ │ │ │ +0003a230: 0a46 6173 744d 696e 6f72 7353 7472 6174 .FastMinorsStrat │ │ │ │ +0003a240: 6567 7954 7574 6f72 6961 6c2c 2e20 5265 egyTutorial,. Re │ │ │ │ +0003a250: 6761 7264 6c65 7373 2c20 6166 7465 7220 gardless, after │ │ │ │ +0003a260: 6120 6365 7274 6169 6e20 6e75 6d62 6572 a certain number │ │ │ │ +0003a270: 206f 6620 6d69 6e6f 7273 2068 6176 650a of minors have. │ │ │ │ +0003a280: 6265 656e 206c 6f6f 6b65 6420 6174 2c20 been looked at, │ │ │ │ +0003a290: 7765 2073 6565 206f 7574 7075 7420 6c69 we see output li │ │ │ │ +0003a2a0: 6e65 7320 6c69 6b65 3a20 2060 604c 6f6f nes like: ``Loo │ │ │ │ +0003a2b0: 7020 7374 6570 2c20 6162 6f75 7420 746f p step, about to │ │ │ │ +0003a2c0: 2063 6f6d 7075 7465 0a64 696d 656e 7369 compute.dimensi │ │ │ │ +0003a2d0: 6f6e 2e20 2053 7562 6d61 7472 6963 6573 on. Submatrices │ │ │ │ +0003a2e0: 2063 6f6e 7369 6465 7265 643a 2037 2c20 considered: 7, │ │ │ │ +0003a2f0: 616e 6420 636f 6d70 7574 6564 203d 2037 and computed = 7 │ │ │ │ +0003a300: 2727 2e20 2057 6520 6f6e 6c79 2063 6f6d ''. We only com │ │ │ │ +0003a310: 7075 7465 0a6d 696e 6f72 7320 7765 2068 pute.minors we h │ │ │ │ +0003a320: 6176 656e 2774 2063 6f6e 7369 6465 7265 aven't considere │ │ │ │ +0003a330: 6420 6265 666f 7265 2e20 2053 6f20 6173 d before. So as │ │ │ │ +0003a340: 2077 6520 636f 6d70 7574 6520 6d6f 7265 we compute more │ │ │ │ +0003a350: 206d 696e 6f72 732c 2074 6865 7265 2063 minors, there c │ │ │ │ +0003a360: 616e 0a62 6520 6120 6469 7374 696e 6374 an.be a distinct │ │ │ │ +0003a370: 696f 6e20 6265 7477 6565 6e20 636f 6e73 ion between cons │ │ │ │ +0003a380: 6964 6572 6564 2061 6e64 2063 6f6d 7075 idered and compu │ │ │ │ +0003a390: 7465 642e 0a0a 436f 6d70 7574 696e 6720 ted...Computing │ │ │ │ +0003a3a0: 6d69 6e6f 7273 2076 7320 636f 6e73 6964 minors vs consid │ │ │ │ +0003a3b0: 6572 696e 6720 7468 6520 6469 6d65 6e73 ering the dimens │ │ │ │ +0003a3c0: 696f 6e20 6f66 2077 6861 7420 6861 7320 ion of what has │ │ │ │ +0003a3d0: 6265 656e 2063 6f6d 7075 7465 642e 0a50 been computed..P │ │ │ │ +0003a3e0: 6572 696f 6469 6361 6c6c 7920 7765 2063 eriodically we c │ │ │ │ +0003a3f0: 6f6d 7075 7465 2074 6865 2063 6f64 696d ompute the codim │ │ │ │ +0003a400: 656e 7369 6f6e 206f 6620 7468 6520 7061 ension of the pa │ │ │ │ +0003a410: 7274 6961 6c20 6964 6561 6c20 6f66 206d rtial ideal of m │ │ │ │ +0003a420: 696e 6f72 7320 7765 2068 6176 650a 636f inors we have.co │ │ │ │ +0003a430: 6d70 7574 6564 2073 6f20 6661 722e 2020 mputed so far. │ │ │ │ +0003a440: 5468 6572 6520 6172 6520 7477 6f20 6f70 There are two op │ │ │ │ +0003a450: 7469 6f6e 7320 746f 2063 6f6e 7472 6f6c tions to control │ │ │ │ +0003a460: 2074 6869 732e 2020 4669 7273 742c 2077 this. First, w │ │ │ │ +0003a470: 6520 6361 6e20 7465 6c6c 0a74 6865 2066 e can tell.the f │ │ │ │ +0003a480: 756e 6374 696f 6e20 7768 656e 2074 6f20 unction when to │ │ │ │ +0003a490: 6669 7273 7420 636f 6d70 7574 6520 7468 first compute th │ │ │ │ +0003a4a0: 6520 6469 6d65 6e73 696f 6e20 6f66 2074 e dimension of t │ │ │ │ +0003a4b0: 6865 2077 6f72 6b69 6e67 2070 6172 7469 he working parti │ │ │ │ +0003a4c0: 616c 2069 6465 616c 0a6f 6620 6d69 6e6f al ideal.of mino │ │ │ │ +0003a4d0: 7273 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d rs...+---------- │ │ │ │ 0003a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003a500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003a530: 3920 3a20 7469 6d65 2072 6567 756c 6172 9 : time regular │ │ │ │ -0003a540: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ -0003a550: 2053 2f4a 2c20 4d61 784d 696e 6f72 733d S/J, MaxMinors= │ │ │ │ -0003a560: 3e31 302c 204d 696e 4d69 6e6f 7273 4675 >10, MinMinorsFu │ │ │ │ -0003a570: 6e63 7469 6f6e 203d 3e20 742d 7c0a 7c20 nction => t-|.| │ │ │ │ -0003a580: 2d2d 2075 7365 6420 302e 3538 3637 3635 -- used 0.586765 │ │ │ │ -0003a590: 7320 2863 7075 293b 2030 2e34 3431 3135 s (cpu); 0.44115 │ │ │ │ -0003a5a0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -0003a5b0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ -0003a5c0: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ -0003a5d0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003a5e0: 696f 6e3a 2072 696e 6720 6469 6d65 6e73 ion: ring dimens │ │ │ │ -0003a5f0: 696f 6e20 3d34 2c20 7468 6572 6520 6172 ion =4, there ar │ │ │ │ -0003a600: 6520 3134 3635 3132 3820 706f 7373 6962 e 1465128 possib │ │ │ │ -0003a610: 6c65 2035 2062 7920 3520 6d69 7c0a 7c72 le 5 by 5 mi|.|r │ │ │ │ -0003a620: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003a630: 696f 6e3a 2041 626f 7574 2074 6f20 656e ion: About to en │ │ │ │ -0003a640: 7465 7220 6c6f 6f70 2020 2020 2020 2020 ter loop │ │ │ │ +0003a520: 2d2d 2d2b 0a7c 6939 203a 2074 696d 6520 ---+.|i9 : time │ │ │ │ +0003a530: 7265 6775 6c61 7249 6e43 6f64 696d 656e regularInCodimen │ │ │ │ +0003a540: 7369 6f6e 2831 2c20 532f 4a2c 204d 6178 sion(1, S/J, Max │ │ │ │ +0003a550: 4d69 6e6f 7273 3d3e 3130 2c20 4d69 6e4d Minors=>10, MinM │ │ │ │ +0003a560: 696e 6f72 7346 756e 6374 696f 6e20 3d3e inorsFunction => │ │ │ │ +0003a570: 2074 2d7c 0a7c 202d 2d20 7573 6564 2030 t-|.| -- used 0 │ │ │ │ +0003a580: 2e37 3636 3932 3973 2028 6370 7529 3b20 .766929s (cpu); │ │ │ │ +0003a590: 302e 3532 3532 3834 7320 2874 6872 6561 0.525284s (threa │ │ │ │ +0003a5a0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0003a5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003a5c0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ +0003a5d0: 6f64 696d 656e 7369 6f6e 3a20 7269 6e67 odimension: ring │ │ │ │ +0003a5e0: 2064 696d 656e 7369 6f6e 203d 342c 2074 dimension =4, t │ │ │ │ +0003a5f0: 6865 7265 2061 7265 2031 3436 3531 3238 here are 1465128 │ │ │ │ +0003a600: 2070 6f73 7369 626c 6520 3520 6279 2035 possible 5 by 5 │ │ │ │ +0003a610: 206d 697c 0a7c 7265 6775 6c61 7249 6e43 mi|.|regularInC │ │ │ │ +0003a620: 6f64 696d 656e 7369 6f6e 3a20 4162 6f75 odimension: Abou │ │ │ │ +0003a630: 7420 746f 2065 6e74 6572 206c 6f6f 7020 t to enter loop │ │ │ │ +0003a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a660: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ -0003a670: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ -0003a680: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e or: Choosing Ran │ │ │ │ -0003a690: 646f 6d4e 6f6e 5a65 726f 2020 2020 2020 domNonZero │ │ │ │ +0003a660: 2020 207c 0a7c 696e 7465 726e 616c 4368 |.|internalCh │ │ │ │ +0003a670: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ +0003a680: 696e 6720 5261 6e64 6f6d 4e6f 6e5a 6572 ing RandomNonZer │ │ │ │ +0003a690: 6f20 2020 2020 2020 2020 2020 2020 2020 o │ │ │ │ 0003a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ -0003a6c0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ -0003a6d0: 6f72 3a20 4368 6f6f 7369 6e67 2052 616e or: Choosing Ran │ │ │ │ -0003a6e0: 646f 6d20 2020 2020 2020 2020 2020 2020 dom │ │ │ │ +0003a6b0: 2020 207c 0a7c 696e 7465 726e 616c 4368 |.|internalCh │ │ │ │ +0003a6c0: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ +0003a6d0: 696e 6720 5261 6e64 6f6d 2020 2020 2020 ing Random │ │ │ │ +0003a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a700: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ -0003a710: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ -0003a720: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578 or: Choosing Lex │ │ │ │ -0003a730: 536d 616c 6c65 7374 2020 2020 2020 2020 Smallest │ │ │ │ +0003a700: 2020 207c 0a7c 696e 7465 726e 616c 4368 |.|internalCh │ │ │ │ +0003a710: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ +0003a720: 696e 6720 4c65 7853 6d61 6c6c 6573 7420 ing LexSmallest │ │ │ │ +0003a730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a750: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ -0003a760: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003a770: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c ion: Loop step, │ │ │ │ -0003a780: 2061 626f 7574 2074 6f20 636f 6d70 7574 about to comput │ │ │ │ -0003a790: 6520 6469 6d65 6e73 696f 6e2e 2020 5375 e dimension. Su │ │ │ │ -0003a7a0: 626d 6174 7269 6365 7320 636f 7c0a 7c72 bmatrices co|.|r │ │ │ │ -0003a7b0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003a7c0: 696f 6e3a 2020 6973 436f 6469 6d41 744c ion: isCodimAtL │ │ │ │ -0003a7d0: 6561 7374 2066 6169 6c65 642c 2063 6f6d east failed, com │ │ │ │ -0003a7e0: 7075 7469 6e67 2063 6f64 696d 2e20 2020 puting codim. │ │ │ │ -0003a7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ -0003a800: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003a810: 696f 6e3a 2020 7061 7274 6961 6c20 7369 ion: partial si │ │ │ │ -0003a820: 6e67 756c 6172 206c 6f63 7573 2064 696d ngular locus dim │ │ │ │ -0003a830: 656e 7369 6f6e 2063 6f6d 7075 7465 642c ension computed, │ │ │ │ -0003a840: 203d 2033 2020 2020 2020 2020 7c0a 7c69 = 3 |.|i │ │ │ │ -0003a850: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ -0003a860: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578 or: Choosing Lex │ │ │ │ -0003a870: 536d 616c 6c65 7374 5465 726d 2020 2020 SmallestTerm │ │ │ │ +0003a750: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ +0003a760: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f odimension: Loo │ │ │ │ +0003a770: 7020 7374 6570 2c20 6162 6f75 7420 746f p step, about to │ │ │ │ +0003a780: 2063 6f6d 7075 7465 2064 696d 656e 7369 compute dimensi │ │ │ │ +0003a790: 6f6e 2e20 2053 7562 6d61 7472 6963 6573 on. Submatrices │ │ │ │ +0003a7a0: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43 co|.|regularInC │ │ │ │ +0003a7b0: 6f64 696d 656e 7369 6f6e 3a20 2069 7343 odimension: isC │ │ │ │ +0003a7c0: 6f64 696d 4174 4c65 6173 7420 6661 696c odimAtLeast fail │ │ │ │ +0003a7d0: 6564 2c20 636f 6d70 7574 696e 6720 636f ed, computing co │ │ │ │ +0003a7e0: 6469 6d2e 2020 2020 2020 2020 2020 2020 dim. │ │ │ │ +0003a7f0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ +0003a800: 6f64 696d 656e 7369 6f6e 3a20 2070 6172 odimension: par │ │ │ │ +0003a810: 7469 616c 2073 696e 6775 6c61 7220 6c6f tial singular lo │ │ │ │ +0003a820: 6375 7320 6469 6d65 6e73 696f 6e20 636f cus dimension co │ │ │ │ +0003a830: 6d70 7574 6564 2c20 3d20 3320 2020 2020 mputed, = 3 │ │ │ │ +0003a840: 2020 207c 0a7c 696e 7465 726e 616c 4368 |.|internalCh │ │ │ │ +0003a850: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ +0003a860: 696e 6720 4c65 7853 6d61 6c6c 6573 7454 ing LexSmallestT │ │ │ │ +0003a870: 6572 6d20 2020 2020 2020 2020 2020 2020 erm │ │ │ │ 0003a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a890: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ -0003a8a0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ -0003a8b0: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265 or: Choosing GRe │ │ │ │ -0003a8c0: 764c 6578 536d 616c 6c65 7374 5465 726d vLexSmallestTerm │ │ │ │ +0003a890: 2020 207c 0a7c 696e 7465 726e 616c 4368 |.|internalCh │ │ │ │ +0003a8a0: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ +0003a8b0: 696e 6720 4752 6576 4c65 7853 6d61 6c6c ing GRevLexSmall │ │ │ │ +0003a8c0: 6573 7454 6572 6d20 2020 2020 2020 2020 estTerm │ │ │ │ 0003a8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a8e0: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ -0003a8f0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ -0003a900: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265 or: Choosing GRe │ │ │ │ -0003a910: 764c 6578 536d 616c 6c65 7374 2020 2020 vLexSmallest │ │ │ │ +0003a8e0: 2020 207c 0a7c 696e 7465 726e 616c 4368 |.|internalCh │ │ │ │ +0003a8f0: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ +0003a900: 696e 6720 4752 6576 4c65 7853 6d61 6c6c ing GRevLexSmall │ │ │ │ +0003a910: 6573 7420 2020 2020 2020 2020 2020 2020 est │ │ │ │ 0003a920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003a930: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ -0003a940: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003a950: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c ion: Loop step, │ │ │ │ -0003a960: 2061 626f 7574 2074 6f20 636f 6d70 7574 about to comput │ │ │ │ -0003a970: 6520 6469 6d65 6e73 696f 6e2e 2020 5375 e dimension. Su │ │ │ │ -0003a980: 626d 6174 7269 6365 7320 636f 7c0a 7c72 bmatrices co|.|r │ │ │ │ -0003a990: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003a9a0: 696f 6e3a 2020 6973 436f 6469 6d41 744c ion: isCodimAtL │ │ │ │ -0003a9b0: 6561 7374 2066 6169 6c65 642c 2063 6f6d east failed, com │ │ │ │ -0003a9c0: 7075 7469 6e67 2063 6f64 696d 2e20 2020 puting codim. │ │ │ │ -0003a9d0: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ -0003a9e0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003a9f0: 696f 6e3a 2020 7061 7274 6961 6c20 7369 ion: partial si │ │ │ │ -0003aa00: 6e67 756c 6172 206c 6f63 7573 2064 696d ngular locus dim │ │ │ │ -0003aa10: 656e 7369 6f6e 2063 6f6d 7075 7465 642c ension computed, │ │ │ │ -0003aa20: 203d 2033 2020 2020 2020 2020 7c0a 7c69 = 3 |.|i │ │ │ │ -0003aa30: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ -0003aa40: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265 or: Choosing GRe │ │ │ │ -0003aa50: 764c 6578 536d 616c 6c65 7374 5465 726d vLexSmallestTerm │ │ │ │ +0003a930: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ +0003a940: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f odimension: Loo │ │ │ │ +0003a950: 7020 7374 6570 2c20 6162 6f75 7420 746f p step, about to │ │ │ │ +0003a960: 2063 6f6d 7075 7465 2064 696d 656e 7369 compute dimensi │ │ │ │ +0003a970: 6f6e 2e20 2053 7562 6d61 7472 6963 6573 on. Submatrices │ │ │ │ +0003a980: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43 co|.|regularInC │ │ │ │ +0003a990: 6f64 696d 656e 7369 6f6e 3a20 2069 7343 odimension: isC │ │ │ │ +0003a9a0: 6f64 696d 4174 4c65 6173 7420 6661 696c odimAtLeast fail │ │ │ │ +0003a9b0: 6564 2c20 636f 6d70 7574 696e 6720 636f ed, computing co │ │ │ │ +0003a9c0: 6469 6d2e 2020 2020 2020 2020 2020 2020 dim. │ │ │ │ +0003a9d0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ +0003a9e0: 6f64 696d 656e 7369 6f6e 3a20 2070 6172 odimension: par │ │ │ │ +0003a9f0: 7469 616c 2073 696e 6775 6c61 7220 6c6f tial singular lo │ │ │ │ +0003aa00: 6375 7320 6469 6d65 6e73 696f 6e20 636f cus dimension co │ │ │ │ +0003aa10: 6d70 7574 6564 2c20 3d20 3320 2020 2020 mputed, = 3 │ │ │ │ +0003aa20: 2020 207c 0a7c 696e 7465 726e 616c 4368 |.|internalCh │ │ │ │ +0003aa30: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ +0003aa40: 696e 6720 4752 6576 4c65 7853 6d61 6c6c ing GRevLexSmall │ │ │ │ +0003aa50: 6573 7454 6572 6d20 2020 2020 2020 2020 estTerm │ │ │ │ 0003aa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003aa70: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ -0003aa80: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ -0003aa90: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578 or: Choosing Lex │ │ │ │ -0003aaa0: 536d 616c 6c65 7374 5465 726d 2020 2020 SmallestTerm │ │ │ │ +0003aa70: 2020 207c 0a7c 696e 7465 726e 616c 4368 |.|internalCh │ │ │ │ +0003aa80: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ +0003aa90: 696e 6720 4c65 7853 6d61 6c6c 6573 7454 ing LexSmallestT │ │ │ │ +0003aaa0: 6572 6d20 2020 2020 2020 2020 2020 2020 erm │ │ │ │ 0003aab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003aac0: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ -0003aad0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003aae0: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c ion: Loop step, │ │ │ │ -0003aaf0: 2061 626f 7574 2074 6f20 636f 6d70 7574 about to comput │ │ │ │ -0003ab00: 6520 6469 6d65 6e73 696f 6e2e 2020 5375 e dimension. Su │ │ │ │ -0003ab10: 626d 6174 7269 6365 7320 636f 7c0a 7c72 bmatrices co|.|r │ │ │ │ -0003ab20: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003ab30: 696f 6e3a 2020 6973 436f 6469 6d41 744c ion: isCodimAtL │ │ │ │ -0003ab40: 6561 7374 2066 6169 6c65 642c 2063 6f6d east failed, com │ │ │ │ -0003ab50: 7075 7469 6e67 2063 6f64 696d 2e20 2020 puting codim. │ │ │ │ -0003ab60: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ -0003ab70: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003ab80: 696f 6e3a 2020 7061 7274 6961 6c20 7369 ion: partial si │ │ │ │ -0003ab90: 6e67 756c 6172 206c 6f63 7573 2064 696d ngular locus dim │ │ │ │ -0003aba0: 656e 7369 6f6e 2063 6f6d 7075 7465 642c ension computed, │ │ │ │ -0003abb0: 203d 2033 2020 2020 2020 2020 7c0a 7c69 = 3 |.|i │ │ │ │ -0003abc0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ -0003abd0: 6f72 3a20 4368 6f6f 7369 6e67 204c 6578 or: Choosing Lex │ │ │ │ -0003abe0: 536d 616c 6c65 7374 5465 726d 2020 2020 SmallestTerm │ │ │ │ +0003aac0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ +0003aad0: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f odimension: Loo │ │ │ │ +0003aae0: 7020 7374 6570 2c20 6162 6f75 7420 746f p step, about to │ │ │ │ +0003aaf0: 2063 6f6d 7075 7465 2064 696d 656e 7369 compute dimensi │ │ │ │ +0003ab00: 6f6e 2e20 2053 7562 6d61 7472 6963 6573 on. Submatrices │ │ │ │ +0003ab10: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43 co|.|regularInC │ │ │ │ +0003ab20: 6f64 696d 656e 7369 6f6e 3a20 2069 7343 odimension: isC │ │ │ │ +0003ab30: 6f64 696d 4174 4c65 6173 7420 6661 696c odimAtLeast fail │ │ │ │ +0003ab40: 6564 2c20 636f 6d70 7574 696e 6720 636f ed, computing co │ │ │ │ +0003ab50: 6469 6d2e 2020 2020 2020 2020 2020 2020 dim. │ │ │ │ +0003ab60: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ +0003ab70: 6f64 696d 656e 7369 6f6e 3a20 2070 6172 odimension: par │ │ │ │ +0003ab80: 7469 616c 2073 696e 6775 6c61 7220 6c6f tial singular lo │ │ │ │ +0003ab90: 6375 7320 6469 6d65 6e73 696f 6e20 636f cus dimension co │ │ │ │ +0003aba0: 6d70 7574 6564 2c20 3d20 3320 2020 2020 mputed, = 3 │ │ │ │ +0003abb0: 2020 207c 0a7c 696e 7465 726e 616c 4368 |.|internalCh │ │ │ │ +0003abc0: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ +0003abd0: 696e 6720 4c65 7853 6d61 6c6c 6573 7454 ing LexSmallestT │ │ │ │ +0003abe0: 6572 6d20 2020 2020 2020 2020 2020 2020 erm │ │ │ │ 0003abf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ac00: 2020 2020 2020 2020 2020 2020 7c0a 7c69 |.|i │ │ │ │ -0003ac10: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ -0003ac20: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265 or: Choosing GRe │ │ │ │ -0003ac30: 764c 6578 536d 616c 6c65 7374 5465 726d vLexSmallestTerm │ │ │ │ +0003ac00: 2020 207c 0a7c 696e 7465 726e 616c 4368 |.|internalCh │ │ │ │ +0003ac10: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ +0003ac20: 696e 6720 4752 6576 4c65 7853 6d61 6c6c ing GRevLexSmall │ │ │ │ +0003ac30: 6573 7454 6572 6d20 2020 2020 2020 2020 estTerm │ │ │ │ 0003ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ac50: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ -0003ac60: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003ac70: 696f 6e3a 2020 4c6f 6f70 2073 7465 702c ion: Loop step, │ │ │ │ -0003ac80: 2061 626f 7574 2074 6f20 636f 6d70 7574 about to comput │ │ │ │ -0003ac90: 6520 6469 6d65 6e73 696f 6e2e 2020 5375 e dimension. Su │ │ │ │ -0003aca0: 626d 6174 7269 6365 7320 636f 7c0a 7c72 bmatrices co|.|r │ │ │ │ -0003acb0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003acc0: 696f 6e3a 2020 6973 436f 6469 6d41 744c ion: isCodimAtL │ │ │ │ -0003acd0: 6561 7374 2066 6169 6c65 642c 2063 6f6d east failed, com │ │ │ │ -0003ace0: 7075 7469 6e67 2063 6f64 696d 2e20 2020 puting codim. │ │ │ │ -0003acf0: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ -0003ad00: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003ad10: 696f 6e3a 2020 7061 7274 6961 6c20 7369 ion: partial si │ │ │ │ -0003ad20: 6e67 756c 6172 206c 6f63 7573 2064 696d ngular locus dim │ │ │ │ -0003ad30: 656e 7369 6f6e 2063 6f6d 7075 7465 642c ension computed, │ │ │ │ -0003ad40: 203d 2033 2020 2020 2020 2020 7c0a 7c72 = 3 |.|r │ │ │ │ -0003ad50: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -0003ad60: 696f 6e3a 2020 4c6f 6f70 2063 6f6d 706c ion: Loop compl │ │ │ │ -0003ad70: 6574 6564 2c20 7375 626d 6174 7269 6365 eted, submatrice │ │ │ │ -0003ad80: 7320 636f 6e73 6964 6572 6564 203d 2031 s considered = 1 │ │ │ │ -0003ad90: 302c 2061 6e64 2063 6f6d 7075 7c0a 7c2d 0, and compu|.|- │ │ │ │ +0003ac50: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ +0003ac60: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f odimension: Loo │ │ │ │ +0003ac70: 7020 7374 6570 2c20 6162 6f75 7420 746f p step, about to │ │ │ │ +0003ac80: 2063 6f6d 7075 7465 2064 696d 656e 7369 compute dimensi │ │ │ │ +0003ac90: 6f6e 2e20 2053 7562 6d61 7472 6963 6573 on. Submatrices │ │ │ │ +0003aca0: 2063 6f7c 0a7c 7265 6775 6c61 7249 6e43 co|.|regularInC │ │ │ │ +0003acb0: 6f64 696d 656e 7369 6f6e 3a20 2069 7343 odimension: isC │ │ │ │ +0003acc0: 6f64 696d 4174 4c65 6173 7420 6661 696c odimAtLeast fail │ │ │ │ +0003acd0: 6564 2c20 636f 6d70 7574 696e 6720 636f ed, computing co │ │ │ │ +0003ace0: 6469 6d2e 2020 2020 2020 2020 2020 2020 dim. │ │ │ │ +0003acf0: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ +0003ad00: 6f64 696d 656e 7369 6f6e 3a20 2070 6172 odimension: par │ │ │ │ +0003ad10: 7469 616c 2073 696e 6775 6c61 7220 6c6f tial singular lo │ │ │ │ +0003ad20: 6375 7320 6469 6d65 6e73 696f 6e20 636f cus dimension co │ │ │ │ +0003ad30: 6d70 7574 6564 2c20 3d20 3320 2020 2020 mputed, = 3 │ │ │ │ +0003ad40: 2020 207c 0a7c 7265 6775 6c61 7249 6e43 |.|regularInC │ │ │ │ +0003ad50: 6f64 696d 656e 7369 6f6e 3a20 204c 6f6f odimension: Loo │ │ │ │ +0003ad60: 7020 636f 6d70 6c65 7465 642c 2073 7562 p completed, sub │ │ │ │ +0003ad70: 6d61 7472 6963 6573 2063 6f6e 7369 6465 matrices conside │ │ │ │ +0003ad80: 7265 6420 3d20 3130 2c20 616e 6420 636f red = 10, and co │ │ │ │ +0003ad90: 6d70 757c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d mpu|.|---------- │ │ │ │ 0003ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003adb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003adc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003add0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ade0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c3e ------------|.|> │ │ │ │ -0003adf0: 332c 2056 6572 626f 7365 3d3e 7472 7565 3, Verbose=>true │ │ │ │ -0003ae00: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0003ade0: 2d2d 2d7c 0a7c 3e33 2c20 5665 7262 6f73 ---|.|>3, Verbos │ │ │ │ +0003adf0: 653d 3e74 7275 6529 2020 2020 2020 2020 e=>true) │ │ │ │ +0003ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ae10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ae30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003ae30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ae50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ae80: 2020 2020 2020 2020 2020 2020 7c0a 7c6e |.|n │ │ │ │ -0003ae90: 6f72 732c 2077 6520 7769 6c6c 2063 6f6d ors, we will com │ │ │ │ -0003aea0: 7075 7465 2075 7020 746f 2031 3020 6f66 pute up to 10 of │ │ │ │ -0003aeb0: 2074 6865 6d2e 2020 2020 2020 2020 2020 them. │ │ │ │ +0003ae80: 2020 207c 0a7c 6e6f 7273 2c20 7765 2077 |.|nors, we w │ │ │ │ +0003ae90: 696c 6c20 636f 6d70 7574 6520 7570 2074 ill compute up t │ │ │ │ +0003aea0: 6f20 3130 206f 6620 7468 656d 2e20 2020 o 10 of them. │ │ │ │ +0003aeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003aec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003aed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003aed0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003aee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003aef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003af00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003af10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003af20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003af20: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003af30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003af40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003af70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003af70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003afc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003afc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003afd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 7c6e |.|n │ │ │ │ -0003b020: 7369 6465 7265 643a 2033 2c20 616e 6420 sidered: 3, and │ │ │ │ -0003b030: 636f 6d70 7574 6564 203d 2033 2020 2020 computed = 3 │ │ │ │ +0003b010: 2020 207c 0a7c 6e73 6964 6572 6564 3a20 |.|nsidered: │ │ │ │ +0003b020: 332c 2061 6e64 2063 6f6d 7075 7465 6420 3, and computed │ │ │ │ +0003b030: 3d20 3320 2020 2020 2020 2020 2020 2020 = 3 │ │ │ │ 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 7c20 |.| │ │ │ │ +0003b060: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b0b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b0b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b100: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b100: 2020 207c 0a7c 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 7c20 |.| │ │ │ │ +0003b150: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b1a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6e |.|n │ │ │ │ -0003b200: 7369 6465 7265 643a 2036 2c20 616e 6420 sidered: 6, and │ │ │ │ -0003b210: 636f 6d70 7574 6564 203d 2036 2020 2020 computed = 6 │ │ │ │ +0003b1f0: 2020 207c 0a7c 6e73 6964 6572 6564 3a20 |.|nsidered: │ │ │ │ +0003b200: 362c 2061 6e64 2063 6f6d 7075 7465 6420 6, and computed │ │ │ │ +0003b210: 3d20 3620 2020 2020 2020 2020 2020 2020 = 6 │ │ │ │ 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: 2020 207c 0a7c 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 |.| │ │ │ │ +0003b290: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 7c20 |.| │ │ │ │ +0003b2e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 7c20 |.| │ │ │ │ +0003b330: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b380: 2020 2020 2020 2020 2020 2020 7c0a 7c6e |.|n │ │ │ │ -0003b390: 7369 6465 7265 643a 2038 2c20 616e 6420 sidered: 8, and │ │ │ │ -0003b3a0: 636f 6d70 7574 6564 203d 2038 2020 2020 computed = 8 │ │ │ │ +0003b380: 2020 207c 0a7c 6e73 6964 6572 6564 3a20 |.|nsidered: │ │ │ │ +0003b390: 382c 2061 6e64 2063 6f6d 7075 7465 6420 8, and computed │ │ │ │ +0003b3a0: 3d20 3820 2020 2020 2020 2020 2020 2020 = 8 │ │ │ │ 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: 2020 207c 0a7c 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 7c20 |.| │ │ │ │ +0003b420: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 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: 2020 207c 0a7c 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 7c20 |.| │ │ │ │ +0003b4c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 7c6e |.|n │ │ │ │ -0003b520: 7369 6465 7265 643a 2031 302c 2061 6e64 sidered: 10, and │ │ │ │ -0003b530: 2063 6f6d 7075 7465 6420 3d20 3130 2020 computed = 10 │ │ │ │ +0003b510: 2020 207c 0a7c 6e73 6964 6572 6564 3a20 |.|nsidered: │ │ │ │ +0003b520: 3130 2c20 616e 6420 636f 6d70 7574 6564 10, and computed │ │ │ │ +0003b530: 203d 2031 3020 2020 2020 2020 2020 2020 = 10 │ │ │ │ 0003b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b560: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b560: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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: 2020 207c 0a7c 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 7c74 |.|t │ │ │ │ -0003b610: 6564 203d 2031 302e 2020 7369 6e67 756c ed = 10. singul │ │ │ │ -0003b620: 6172 206c 6f63 7573 2064 696d 656e 7369 ar locus dimensi │ │ │ │ -0003b630: 6f6e 2061 7070 6561 7273 2074 6f20 6265 on appears to be │ │ │ │ -0003b640: 203d 2033 2020 2020 2020 2020 2020 2020 = 3 │ │ │ │ -0003b650: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b600: 2020 207c 0a7c 7465 6420 3d20 3130 2e20 |.|ted = 10. │ │ │ │ +0003b610: 2073 696e 6775 6c61 7220 6c6f 6375 7320 singular locus │ │ │ │ +0003b620: 6469 6d65 6e73 696f 6e20 6170 7065 6172 dimension appear │ │ │ │ +0003b630: 7320 746f 2062 6520 3d20 3320 2020 2020 s to be = 3 │ │ │ │ +0003b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b650: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0003b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a4d ------------+..M │ │ │ │ -0003b6b0: 696e 4d69 6e6f 7273 4675 6e63 7469 6f6e inMinorsFunction │ │ │ │ -0003b6c0: 2e20 5765 2070 6173 7320 4d69 6e4d 696e . We pass MinMin │ │ │ │ -0003b6d0: 6f72 7346 756e 6374 696f 6e20 6120 6675 orsFunction a fu │ │ │ │ -0003b6e0: 6e63 7469 6f6e 2077 6869 6368 2073 656e nction which sen │ │ │ │ -0003b6f0: 6473 2074 6865 206d 696e 696d 756d 0a6e ds the minimum.n │ │ │ │ -0003b700: 756d 6265 7220 6f66 206d 696e 6f72 7320 umber of minors │ │ │ │ -0003b710: 6e65 6564 6564 2074 6f20 7665 7269 6679 needed to verify │ │ │ │ -0003b720: 2074 6861 7420 736f 6d65 7468 696e 6720 that something │ │ │ │ -0003b730: 6973 2072 6567 756c 6172 2069 6e20 636f is regular in co │ │ │ │ -0003b740: 6469 6d65 6e73 696f 6e20 246e 240a 2877 dimension $n$.(w │ │ │ │ -0003b750: 6869 6368 2069 7320 616c 7761 7973 2024 hich is always $ │ │ │ │ -0003b760: 6e2b 3124 2920 746f 2074 6865 206e 756d n+1$) to the num │ │ │ │ -0003b770: 6265 7220 6f66 206d 696e 6f72 7320 746f ber of minors to │ │ │ │ -0003b780: 2063 6f6d 7075 7465 2062 6566 6f72 6520 compute before │ │ │ │ -0003b790: 636f 6d70 7574 696e 6720 7468 650a 6469 computing the.di │ │ │ │ -0003b7a0: 6d65 6e73 696f 6e20 6f66 2074 6865 2070 mension of the p │ │ │ │ -0003b7b0: 6172 7469 616c 2069 6465 616c 206f 6620 artial ideal of │ │ │ │ -0003b7c0: 6d69 6e6f 7273 2066 6f72 2074 6865 2066 minors for the f │ │ │ │ -0003b7d0: 6972 7374 2074 696d 652e 2020 2059 6f75 irst time. You │ │ │ │ -0003b7e0: 2063 616e 2073 6565 2074 6861 740a 7468 can see that.th │ │ │ │ -0003b7f0: 7265 6520 6d69 6e6f 7273 2077 6572 6520 ree minors were │ │ │ │ -0003b800: 636f 6d70 7574 6564 2069 6e20 7468 6520 computed in the │ │ │ │ -0003b810: 6162 6f76 6520 6578 616d 706c 6520 6265 above example be │ │ │ │ -0003b820: 666f 7265 2077 6520 6174 7465 6d70 7420 fore we attempt │ │ │ │ -0003b830: 746f 2063 6f6d 7075 7465 0a63 6f64 696d to compute.codim │ │ │ │ -0003b840: 656e 7369 6f6e 2e0a 0a43 6f64 696d 4368 ension...CodimCh │ │ │ │ -0003b850: 6563 6b46 756e 6374 696f 6e2e 2054 6865 eckFunction. The │ │ │ │ -0003b860: 206f 7074 696f 6e20 436f 6469 6d43 6865 option CodimChe │ │ │ │ -0003b870: 636b 4675 6e63 7469 6f6e 2063 6f6e 7472 ckFunction contr │ │ │ │ -0003b880: 6f6c 7320 686f 7720 6672 6571 7565 6e74 ols how frequent │ │ │ │ -0003b890: 6c79 2074 6865 0a64 696d 656e 7369 6f6e ly the.dimension │ │ │ │ -0003b8a0: 206f 6620 7468 6520 7061 7274 6961 6c20 of the partial │ │ │ │ -0003b8b0: 6964 6561 6c20 6f66 206d 696e 6f72 7320 ideal of minors │ │ │ │ -0003b8c0: 6973 2063 6f6d 7075 7465 642e 2020 466f is computed. Fo │ │ │ │ -0003b8d0: 7220 696e 7374 616e 6365 2c20 7365 7474 r instance, sett │ │ │ │ -0003b8e0: 696e 670a 436f 6469 6d43 6865 636b 4675 ing.CodimCheckFu │ │ │ │ -0003b8f0: 6e63 7469 6f6e 203d 3e20 7420 2d3e 2074 nction => t -> t │ │ │ │ -0003b900: 2f35 2077 696c 6c20 7361 7920 6974 2073 /5 will say it s │ │ │ │ -0003b910: 686f 756c 6420 636f 6d70 7574 6520 6469 hould compute di │ │ │ │ -0003b920: 6d65 6e73 696f 6e20 6166 7465 7220 6576 mension after ev │ │ │ │ -0003b930: 6572 790a 3520 6d69 6e6f 7273 2061 7265 ery.5 minors are │ │ │ │ -0003b940: 2065 7861 6d69 6e65 642e 2020 496e 2067 examined. In g │ │ │ │ -0003b950: 656e 6572 616c 2c20 6166 7465 7220 7468 eneral, after th │ │ │ │ -0003b960: 6520 6f75 7470 7574 206f 6620 7468 6520 e output of the │ │ │ │ -0003b970: 436f 6469 6d43 6865 636b 4675 6e63 7469 CodimCheckFuncti │ │ │ │ -0003b980: 6f6e 0a69 6e63 7265 6173 6573 2062 7920 on.increases by │ │ │ │ -0003b990: 616e 2069 6e74 6567 6572 2077 6520 636f an integer we co │ │ │ │ -0003b9a0: 6d70 7574 6520 7468 6520 636f 6469 6d65 mpute the codime │ │ │ │ -0003b9b0: 6e73 696f 6e20 6167 6169 6e2e 2020 5468 nsion again. Th │ │ │ │ -0003b9c0: 6520 6465 6661 756c 7420 6675 6e63 7469 e default functi │ │ │ │ -0003b9d0: 6f6e 0a68 6173 2074 6865 2073 7061 6365 on.has the space │ │ │ │ -0003b9e0: 2062 6574 7765 656e 2063 6f6d 7075 7461 between computa │ │ │ │ -0003b9f0: 7469 6f6e 7320 6772 6f77 2065 7870 6f6e tions grow expon │ │ │ │ -0003ba00: 656e 7469 616c 6c79 2e0a 0a2b 2d2d 2d2d entially...+---- │ │ │ │ +0003b6a0: 2d2d 2d2b 0a0a 4d69 6e4d 696e 6f72 7346 ---+..MinMinorsF │ │ │ │ +0003b6b0: 756e 6374 696f 6e2e 2057 6520 7061 7373 unction. We pass │ │ │ │ +0003b6c0: 204d 696e 4d69 6e6f 7273 4675 6e63 7469 MinMinorsFuncti │ │ │ │ +0003b6d0: 6f6e 2061 2066 756e 6374 696f 6e20 7768 on a function wh │ │ │ │ +0003b6e0: 6963 6820 7365 6e64 7320 7468 6520 6d69 ich sends the mi │ │ │ │ +0003b6f0: 6e69 6d75 6d0a 6e75 6d62 6572 206f 6620 nimum.number of │ │ │ │ +0003b700: 6d69 6e6f 7273 206e 6565 6465 6420 746f minors needed to │ │ │ │ +0003b710: 2076 6572 6966 7920 7468 6174 2073 6f6d verify that som │ │ │ │ +0003b720: 6574 6869 6e67 2069 7320 7265 6775 6c61 ething is regula │ │ │ │ +0003b730: 7220 696e 2063 6f64 696d 656e 7369 6f6e r in codimension │ │ │ │ +0003b740: 2024 6e24 0a28 7768 6963 6820 6973 2061 $n$.(which is a │ │ │ │ +0003b750: 6c77 6179 7320 246e 2b31 2429 2074 6f20 lways $n+1$) to │ │ │ │ +0003b760: 7468 6520 6e75 6d62 6572 206f 6620 6d69 the number of mi │ │ │ │ +0003b770: 6e6f 7273 2074 6f20 636f 6d70 7574 6520 nors to compute │ │ │ │ +0003b780: 6265 666f 7265 2063 6f6d 7075 7469 6e67 before computing │ │ │ │ +0003b790: 2074 6865 0a64 696d 656e 7369 6f6e 206f the.dimension o │ │ │ │ +0003b7a0: 6620 7468 6520 7061 7274 6961 6c20 6964 f the partial id │ │ │ │ +0003b7b0: 6561 6c20 6f66 206d 696e 6f72 7320 666f eal of minors fo │ │ │ │ +0003b7c0: 7220 7468 6520 6669 7273 7420 7469 6d65 r the first time │ │ │ │ +0003b7d0: 2e20 2020 596f 7520 6361 6e20 7365 6520 . You can see │ │ │ │ +0003b7e0: 7468 6174 0a74 6872 6565 206d 696e 6f72 that.three minor │ │ │ │ +0003b7f0: 7320 7765 7265 2063 6f6d 7075 7465 6420 s were computed │ │ │ │ +0003b800: 696e 2074 6865 2061 626f 7665 2065 7861 in the above exa │ │ │ │ +0003b810: 6d70 6c65 2062 6566 6f72 6520 7765 2061 mple before we a │ │ │ │ +0003b820: 7474 656d 7074 2074 6f20 636f 6d70 7574 ttempt to comput │ │ │ │ +0003b830: 650a 636f 6469 6d65 6e73 696f 6e2e 0a0a e.codimension... │ │ │ │ +0003b840: 436f 6469 6d43 6865 636b 4675 6e63 7469 CodimCheckFuncti │ │ │ │ +0003b850: 6f6e 2e20 5468 6520 6f70 7469 6f6e 2043 on. The option C │ │ │ │ +0003b860: 6f64 696d 4368 6563 6b46 756e 6374 696f odimCheckFunctio │ │ │ │ +0003b870: 6e20 636f 6e74 726f 6c73 2068 6f77 2066 n controls how f │ │ │ │ +0003b880: 7265 7175 656e 746c 7920 7468 650a 6469 requently the.di │ │ │ │ +0003b890: 6d65 6e73 696f 6e20 6f66 2074 6865 2070 mension of the p │ │ │ │ +0003b8a0: 6172 7469 616c 2069 6465 616c 206f 6620 artial ideal of │ │ │ │ +0003b8b0: 6d69 6e6f 7273 2069 7320 636f 6d70 7574 minors is comput │ │ │ │ +0003b8c0: 6564 2e20 2046 6f72 2069 6e73 7461 6e63 ed. For instanc │ │ │ │ +0003b8d0: 652c 2073 6574 7469 6e67 0a43 6f64 696d e, setting.Codim │ │ │ │ +0003b8e0: 4368 6563 6b46 756e 6374 696f 6e20 3d3e CheckFunction => │ │ │ │ +0003b8f0: 2074 202d 3e20 742f 3520 7769 6c6c 2073 t -> t/5 will s │ │ │ │ +0003b900: 6179 2069 7420 7368 6f75 6c64 2063 6f6d ay it should com │ │ │ │ +0003b910: 7075 7465 2064 696d 656e 7369 6f6e 2061 pute dimension a │ │ │ │ +0003b920: 6674 6572 2065 7665 7279 0a35 206d 696e fter every.5 min │ │ │ │ +0003b930: 6f72 7320 6172 6520 6578 616d 696e 6564 ors are examined │ │ │ │ +0003b940: 2e20 2049 6e20 6765 6e65 7261 6c2c 2061 . In general, a │ │ │ │ +0003b950: 6674 6572 2074 6865 206f 7574 7075 7420 fter the output │ │ │ │ +0003b960: 6f66 2074 6865 2043 6f64 696d 4368 6563 of the CodimChec │ │ │ │ +0003b970: 6b46 756e 6374 696f 6e0a 696e 6372 6561 kFunction.increa │ │ │ │ +0003b980: 7365 7320 6279 2061 6e20 696e 7465 6765 ses by an intege │ │ │ │ +0003b990: 7220 7765 2063 6f6d 7075 7465 2074 6865 r we compute the │ │ │ │ +0003b9a0: 2063 6f64 696d 656e 7369 6f6e 2061 6761 codimension aga │ │ │ │ +0003b9b0: 696e 2e20 2054 6865 2064 6566 6175 6c74 in. The default │ │ │ │ +0003b9c0: 2066 756e 6374 696f 6e0a 6861 7320 7468 function.has th │ │ │ │ +0003b9d0: 6520 7370 6163 6520 6265 7477 6565 6e20 e space between │ │ │ │ +0003b9e0: 636f 6d70 7574 6174 696f 6e73 2067 726f computations gro │ │ │ │ +0003b9f0: 7720 6578 706f 6e65 6e74 6961 6c6c 792e w exponentially. │ │ │ │ +0003ba00: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 0003ba10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ba20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ba30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ba40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ba50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 ---------+.|i10 │ │ │ │ -0003ba60: 3a20 7469 6d65 2072 6567 756c 6172 496e : time regularIn │ │ │ │ -0003ba70: 436f 6469 6d65 6e73 696f 6e28 312c 2053 Codimension(1, S │ │ │ │ -0003ba80: 2f4a 2c20 4d61 784d 696e 6f72 733d 3e32 /J, MaxMinors=>2 │ │ │ │ -0003ba90: 352c 2043 6f64 696d 4368 6563 6b46 756e 5, CodimCheckFun │ │ │ │ -0003baa0: 6374 696f 6e20 3d3e 207c 0a7c 202d 2d20 ction => |.| -- │ │ │ │ -0003bab0: 7573 6564 2030 2e36 3530 3732 3273 2028 used 0.650722s ( │ │ │ │ -0003bac0: 6370 7529 3b20 302e 3438 3834 3237 7320 cpu); 0.488427s │ │ │ │ -0003bad0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0003bae0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0003baf0: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003bb00: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003bb10: 3a20 7269 6e67 2064 696d 656e 7369 6f6e : ring dimension │ │ │ │ -0003bb20: 203d 342c 2074 6865 7265 2061 7265 2031 =4, there are 1 │ │ │ │ -0003bb30: 3436 3531 3238 2070 6f73 7369 626c 6520 465128 possible │ │ │ │ -0003bb40: 3520 6279 2035 206d 697c 0a7c 7265 6775 5 by 5 mi|.|regu │ │ │ │ -0003bb50: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003bb60: 3a20 4162 6f75 7420 746f 2065 6e74 6572 : About to enter │ │ │ │ -0003bb70: 206c 6f6f 7020 2020 2020 2020 2020 2020 loop │ │ │ │ +0003ba50: 2b0a 7c69 3130 203a 2074 696d 6520 7265 +.|i10 : time re │ │ │ │ +0003ba60: 6775 6c61 7249 6e43 6f64 696d 656e 7369 gularInCodimensi │ │ │ │ +0003ba70: 6f6e 2831 2c20 532f 4a2c 204d 6178 4d69 on(1, S/J, MaxMi │ │ │ │ +0003ba80: 6e6f 7273 3d3e 3235 2c20 436f 6469 6d43 nors=>25, CodimC │ │ │ │ +0003ba90: 6865 636b 4675 6e63 7469 6f6e 203d 3e20 heckFunction => │ │ │ │ +0003baa0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3930 |.| -- used 0.90 │ │ │ │ +0003bab0: 3534 3237 7320 2863 7075 293b 2030 2e35 5427s (cpu); 0.5 │ │ │ │ +0003bac0: 3936 3932 3973 2028 7468 7265 6164 293b 96929s (thread); │ │ │ │ +0003bad0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +0003bae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003baf0: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003bb00: 6d65 6e73 696f 6e3a 2072 696e 6720 6469 mension: ring di │ │ │ │ +0003bb10: 6d65 6e73 696f 6e20 3d34 2c20 7468 6572 mension =4, ther │ │ │ │ +0003bb20: 6520 6172 6520 3134 3635 3132 3820 706f e are 1465128 po │ │ │ │ +0003bb30: 7373 6962 6c65 2035 2062 7920 3520 6d69 ssible 5 by 5 mi │ │ │ │ +0003bb40: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003bb50: 6d65 6e73 696f 6e3a 2041 626f 7574 2074 mension: About t │ │ │ │ +0003bb60: 6f20 656e 7465 7220 6c6f 6f70 2020 2020 o enter loop │ │ │ │ +0003bb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003bb90: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003bba0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003bbb0: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003bbc0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020 xSmallestTerm │ │ │ │ +0003bb90: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003bba0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003bbb0: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003bbc0: 5465 726d 2020 2020 2020 2020 2020 2020 Term │ │ │ │ 0003bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003bbe0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003bbf0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003bc00: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003bc10: 7853 6d61 6c6c 6573 7454 6572 6d20 2020 xSmallestTerm │ │ │ │ +0003bbe0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003bbf0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003bc00: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003bc10: 5465 726d 2020 2020 2020 2020 2020 2020 Term │ │ │ │ 0003bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003bc30: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003bc40: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003bc50: 3a20 204c 6f6f 7020 7374 6570 2c20 6162 : Loop step, ab │ │ │ │ -0003bc60: 6f75 7420 746f 2063 6f6d 7075 7465 2064 out to compute d │ │ │ │ -0003bc70: 696d 656e 7369 6f6e 2e20 2053 7562 6d61 imension. Subma │ │ │ │ -0003bc80: 7472 6963 6573 2063 6f7c 0a7c 7265 6775 trices co|.|regu │ │ │ │ -0003bc90: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003bca0: 3a20 2069 7343 6f64 696d 4174 4c65 6173 : isCodimAtLeas │ │ │ │ -0003bcb0: 7420 6661 696c 6564 2c20 636f 6d70 7574 t failed, comput │ │ │ │ -0003bcc0: 696e 6720 636f 6469 6d2e 2020 2020 2020 ing codim. │ │ │ │ -0003bcd0: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003bce0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003bcf0: 3a20 2070 6172 7469 616c 2073 696e 6775 : partial singu │ │ │ │ -0003bd00: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73 lar locus dimens │ │ │ │ -0003bd10: 696f 6e20 636f 6d70 7574 6564 2c20 3d20 ion computed, = │ │ │ │ -0003bd20: 3420 2020 2020 2020 207c 0a7c 696e 7465 4 |.|inte │ │ │ │ -0003bd30: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003bd40: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003bd50: 7853 6d61 6c6c 6573 7420 2020 2020 2020 xSmallest │ │ │ │ +0003bc30: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003bc40: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073 mension: Loop s │ │ │ │ +0003bc50: 7465 702c 2061 626f 7574 2074 6f20 636f tep, about to co │ │ │ │ +0003bc60: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e mpute dimension. │ │ │ │ +0003bc70: 2020 5375 626d 6174 7269 6365 7320 636f Submatrices co │ │ │ │ +0003bc80: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003bc90: 6d65 6e73 696f 6e3a 2020 6973 436f 6469 mension: isCodi │ │ │ │ +0003bca0: 6d41 744c 6561 7374 2066 6169 6c65 642c mAtLeast failed, │ │ │ │ +0003bcb0: 2063 6f6d 7075 7469 6e67 2063 6f64 696d computing codim │ │ │ │ +0003bcc0: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ +0003bcd0: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003bce0: 6d65 6e73 696f 6e3a 2020 7061 7274 6961 mension: partia │ │ │ │ +0003bcf0: 6c20 7369 6e67 756c 6172 206c 6f63 7573 l singular locus │ │ │ │ +0003bd00: 2064 696d 656e 7369 6f6e 2063 6f6d 7075 dimension compu │ │ │ │ +0003bd10: 7465 642c 203d 2034 2020 2020 2020 2020 ted, = 4 │ │ │ │ +0003bd20: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003bd30: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003bd40: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003bd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003bd70: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003bd80: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003bd90: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003bda0: 7853 6d61 6c6c 6573 7420 2020 2020 2020 xSmallest │ │ │ │ +0003bd70: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003bd80: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003bd90: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003bda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003bdc0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003bdd0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003bde0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ +0003bdc0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003bdd0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003bde0: 2052 616e 646f 6d20 2020 2020 2020 2020 Random │ │ │ │ 0003bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003be00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003be10: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003be20: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003be30: 3a20 204c 6f6f 7020 7374 6570 2c20 6162 : Loop step, ab │ │ │ │ -0003be40: 6f75 7420 746f 2063 6f6d 7075 7465 2064 out to compute d │ │ │ │ -0003be50: 696d 656e 7369 6f6e 2e20 2053 7562 6d61 imension. Subma │ │ │ │ -0003be60: 7472 6963 6573 2063 6f7c 0a7c 7265 6775 trices co|.|regu │ │ │ │ -0003be70: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003be80: 3a20 2069 7343 6f64 696d 4174 4c65 6173 : isCodimAtLeas │ │ │ │ -0003be90: 7420 6661 696c 6564 2c20 636f 6d70 7574 t failed, comput │ │ │ │ -0003bea0: 696e 6720 636f 6469 6d2e 2020 2020 2020 ing codim. │ │ │ │ -0003beb0: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003bec0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003bed0: 3a20 2070 6172 7469 616c 2073 696e 6775 : partial singu │ │ │ │ -0003bee0: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73 lar locus dimens │ │ │ │ -0003bef0: 696f 6e20 636f 6d70 7574 6564 2c20 3d20 ion computed, = │ │ │ │ -0003bf00: 3420 2020 2020 2020 207c 0a7c 696e 7465 4 |.|inte │ │ │ │ -0003bf10: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003bf20: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003bf30: 7853 6d61 6c6c 6573 7454 6572 6d20 2020 xSmallestTerm │ │ │ │ +0003be10: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003be20: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073 mension: Loop s │ │ │ │ +0003be30: 7465 702c 2061 626f 7574 2074 6f20 636f tep, about to co │ │ │ │ +0003be40: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e mpute dimension. │ │ │ │ +0003be50: 2020 5375 626d 6174 7269 6365 7320 636f Submatrices co │ │ │ │ +0003be60: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003be70: 6d65 6e73 696f 6e3a 2020 6973 436f 6469 mension: isCodi │ │ │ │ +0003be80: 6d41 744c 6561 7374 2066 6169 6c65 642c mAtLeast failed, │ │ │ │ +0003be90: 2063 6f6d 7075 7469 6e67 2063 6f64 696d computing codim │ │ │ │ +0003bea0: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ +0003beb0: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003bec0: 6d65 6e73 696f 6e3a 2020 7061 7274 6961 mension: partia │ │ │ │ +0003bed0: 6c20 7369 6e67 756c 6172 206c 6f63 7573 l singular locus │ │ │ │ +0003bee0: 2064 696d 656e 7369 6f6e 2063 6f6d 7075 dimension compu │ │ │ │ +0003bef0: 7465 642c 203d 2034 2020 2020 2020 2020 ted, = 4 │ │ │ │ +0003bf00: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003bf10: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003bf20: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003bf30: 5465 726d 2020 2020 2020 2020 2020 2020 Term │ │ │ │ 0003bf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003bf50: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003bf60: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003bf70: 2043 686f 6f73 696e 6720 4c65 7853 6d61 Choosing LexSma │ │ │ │ -0003bf80: 6c6c 6573 7420 2020 2020 2020 2020 2020 llest │ │ │ │ +0003bf50: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003bf60: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003bf70: 204c 6578 536d 616c 6c65 7374 2020 2020 LexSmallest │ │ │ │ +0003bf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003bfa0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003bfb0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003bfc0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ -0003bfd0: 4e6f 6e5a 6572 6f20 2020 2020 2020 2020 NonZero │ │ │ │ +0003bfa0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003bfb0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003bfc0: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020 RandomNonZero │ │ │ │ +0003bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003bff0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c000: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c010: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003c020: 7853 6d61 6c6c 6573 7420 2020 2020 2020 xSmallest │ │ │ │ +0003bff0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c000: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c010: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c040: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c050: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c060: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003c070: 7853 6d61 6c6c 6573 7420 2020 2020 2020 xSmallest │ │ │ │ +0003c040: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c050: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c060: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c090: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003c0a0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c0b0: 3a20 204c 6f6f 7020 7374 6570 2c20 6162 : Loop step, ab │ │ │ │ -0003c0c0: 6f75 7420 746f 2063 6f6d 7075 7465 2064 out to compute d │ │ │ │ -0003c0d0: 696d 656e 7369 6f6e 2e20 2053 7562 6d61 imension. Subma │ │ │ │ -0003c0e0: 7472 6963 6573 2063 6f7c 0a7c 7265 6775 trices co|.|regu │ │ │ │ -0003c0f0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c100: 3a20 2069 7343 6f64 696d 4174 4c65 6173 : isCodimAtLeas │ │ │ │ -0003c110: 7420 6661 696c 6564 2c20 636f 6d70 7574 t failed, comput │ │ │ │ -0003c120: 696e 6720 636f 6469 6d2e 2020 2020 2020 ing codim. │ │ │ │ -0003c130: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003c140: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c150: 3a20 2070 6172 7469 616c 2073 696e 6775 : partial singu │ │ │ │ -0003c160: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73 lar locus dimens │ │ │ │ -0003c170: 696f 6e20 636f 6d70 7574 6564 2c20 3d20 ion computed, = │ │ │ │ -0003c180: 3420 2020 2020 2020 207c 0a7c 696e 7465 4 |.|inte │ │ │ │ -0003c190: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c1a0: 2043 686f 6f73 696e 6720 4c65 7853 6d61 Choosing LexSma │ │ │ │ -0003c1b0: 6c6c 6573 7454 6572 6d20 2020 2020 2020 llestTerm │ │ │ │ +0003c090: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c0a0: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073 mension: Loop s │ │ │ │ +0003c0b0: 7465 702c 2061 626f 7574 2074 6f20 636f tep, about to co │ │ │ │ +0003c0c0: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e mpute dimension. │ │ │ │ +0003c0d0: 2020 5375 626d 6174 7269 6365 7320 636f Submatrices co │ │ │ │ +0003c0e0: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c0f0: 6d65 6e73 696f 6e3a 2020 6973 436f 6469 mension: isCodi │ │ │ │ +0003c100: 6d41 744c 6561 7374 2066 6169 6c65 642c mAtLeast failed, │ │ │ │ +0003c110: 2063 6f6d 7075 7469 6e67 2063 6f64 696d computing codim │ │ │ │ +0003c120: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ +0003c130: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c140: 6d65 6e73 696f 6e3a 2020 7061 7274 6961 mension: partia │ │ │ │ +0003c150: 6c20 7369 6e67 756c 6172 206c 6f63 7573 l singular locus │ │ │ │ +0003c160: 2064 696d 656e 7369 6f6e 2063 6f6d 7075 dimension compu │ │ │ │ +0003c170: 7465 642c 203d 2034 2020 2020 2020 2020 ted, = 4 │ │ │ │ +0003c180: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c190: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c1a0: 204c 6578 536d 616c 6c65 7374 5465 726d LexSmallestTerm │ │ │ │ +0003c1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c1d0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c1e0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c1f0: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003c200: 7853 6d61 6c6c 6573 7454 6572 6d20 2020 xSmallestTerm │ │ │ │ +0003c1d0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c1e0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c1f0: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003c200: 5465 726d 2020 2020 2020 2020 2020 2020 Term │ │ │ │ 0003c210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c220: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c230: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c240: 2043 686f 6f73 696e 6720 4c65 7853 6d61 Choosing LexSma │ │ │ │ -0003c250: 6c6c 6573 7420 2020 2020 2020 2020 2020 llest │ │ │ │ +0003c220: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c230: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c240: 204c 6578 536d 616c 6c65 7374 2020 2020 LexSmallest │ │ │ │ +0003c250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c270: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c280: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c290: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003c2a0: 7853 6d61 6c6c 6573 7420 2020 2020 2020 xSmallest │ │ │ │ +0003c270: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c280: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c290: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003c2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c2c0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c2d0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c2e0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ -0003c2f0: 4e6f 6e5a 6572 6f20 2020 2020 2020 2020 NonZero │ │ │ │ +0003c2c0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c2d0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c2e0: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020 RandomNonZero │ │ │ │ +0003c2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c310: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003c320: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c330: 3a20 204c 6f6f 7020 7374 6570 2c20 6162 : Loop step, ab │ │ │ │ -0003c340: 6f75 7420 746f 2063 6f6d 7075 7465 2064 out to compute d │ │ │ │ -0003c350: 696d 656e 7369 6f6e 2e20 2053 7562 6d61 imension. Subma │ │ │ │ -0003c360: 7472 6963 6573 2063 6f7c 0a7c 7265 6775 trices co|.|regu │ │ │ │ -0003c370: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c380: 3a20 2069 7343 6f64 696d 4174 4c65 6173 : isCodimAtLeas │ │ │ │ -0003c390: 7420 6661 696c 6564 2c20 636f 6d70 7574 t failed, comput │ │ │ │ -0003c3a0: 696e 6720 636f 6469 6d2e 2020 2020 2020 ing codim. │ │ │ │ -0003c3b0: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003c3c0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c3d0: 3a20 2070 6172 7469 616c 2073 696e 6775 : partial singu │ │ │ │ -0003c3e0: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73 lar locus dimens │ │ │ │ -0003c3f0: 696f 6e20 636f 6d70 7574 6564 2c20 3d20 ion computed, = │ │ │ │ -0003c400: 3320 2020 2020 2020 207c 0a7c 696e 7465 3 |.|inte │ │ │ │ -0003c410: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c420: 2043 686f 6f73 696e 6720 4c65 7853 6d61 Choosing LexSma │ │ │ │ -0003c430: 6c6c 6573 7420 2020 2020 2020 2020 2020 llest │ │ │ │ +0003c310: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c320: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073 mension: Loop s │ │ │ │ +0003c330: 7465 702c 2061 626f 7574 2074 6f20 636f tep, about to co │ │ │ │ +0003c340: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e mpute dimension. │ │ │ │ +0003c350: 2020 5375 626d 6174 7269 6365 7320 636f Submatrices co │ │ │ │ +0003c360: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c370: 6d65 6e73 696f 6e3a 2020 6973 436f 6469 mension: isCodi │ │ │ │ +0003c380: 6d41 744c 6561 7374 2066 6169 6c65 642c mAtLeast failed, │ │ │ │ +0003c390: 2063 6f6d 7075 7469 6e67 2063 6f64 696d computing codim │ │ │ │ +0003c3a0: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ +0003c3b0: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c3c0: 6d65 6e73 696f 6e3a 2020 7061 7274 6961 mension: partia │ │ │ │ +0003c3d0: 6c20 7369 6e67 756c 6172 206c 6f63 7573 l singular locus │ │ │ │ +0003c3e0: 2064 696d 656e 7369 6f6e 2063 6f6d 7075 dimension compu │ │ │ │ +0003c3f0: 7465 642c 203d 2033 2020 2020 2020 2020 ted, = 3 │ │ │ │ +0003c400: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c410: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c420: 204c 6578 536d 616c 6c65 7374 2020 2020 LexSmallest │ │ │ │ +0003c430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c450: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c460: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c470: 2043 686f 6f73 696e 6720 4c65 7853 6d61 Choosing LexSma │ │ │ │ -0003c480: 6c6c 6573 7454 6572 6d20 2020 2020 2020 llestTerm │ │ │ │ +0003c450: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c460: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c470: 204c 6578 536d 616c 6c65 7374 5465 726d LexSmallestTerm │ │ │ │ +0003c480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c4a0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c4b0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c4c0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ -0003c4d0: 4e6f 6e5a 6572 6f20 2020 2020 2020 2020 NonZero │ │ │ │ +0003c4a0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c4b0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c4c0: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020 RandomNonZero │ │ │ │ +0003c4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c4f0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c500: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c510: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003c520: 7853 6d61 6c6c 6573 7454 6572 6d20 2020 xSmallestTerm │ │ │ │ +0003c4f0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c500: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c510: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003c520: 5465 726d 2020 2020 2020 2020 2020 2020 Term │ │ │ │ 0003c530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c540: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c550: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c560: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003c570: 7853 6d61 6c6c 6573 7454 6572 6d20 2020 xSmallestTerm │ │ │ │ +0003c540: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c550: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c560: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003c570: 5465 726d 2020 2020 2020 2020 2020 2020 Term │ │ │ │ 0003c580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c590: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003c5a0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c5b0: 3a20 204c 6f6f 7020 7374 6570 2c20 6162 : Loop step, ab │ │ │ │ -0003c5c0: 6f75 7420 746f 2063 6f6d 7075 7465 2064 out to compute d │ │ │ │ -0003c5d0: 696d 656e 7369 6f6e 2e20 2053 7562 6d61 imension. Subma │ │ │ │ -0003c5e0: 7472 6963 6573 2063 6f7c 0a7c 7265 6775 trices co|.|regu │ │ │ │ -0003c5f0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c600: 3a20 2069 7343 6f64 696d 4174 4c65 6173 : isCodimAtLeas │ │ │ │ -0003c610: 7420 6661 696c 6564 2c20 636f 6d70 7574 t failed, comput │ │ │ │ -0003c620: 696e 6720 636f 6469 6d2e 2020 2020 2020 ing codim. │ │ │ │ -0003c630: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003c640: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c650: 3a20 2070 6172 7469 616c 2073 696e 6775 : partial singu │ │ │ │ -0003c660: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73 lar locus dimens │ │ │ │ -0003c670: 696f 6e20 636f 6d70 7574 6564 2c20 3d20 ion computed, = │ │ │ │ -0003c680: 3320 2020 2020 2020 207c 0a7c 696e 7465 3 |.|inte │ │ │ │ -0003c690: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c6a0: 2043 686f 6f73 696e 6720 4c65 7853 6d61 Choosing LexSma │ │ │ │ -0003c6b0: 6c6c 6573 7420 2020 2020 2020 2020 2020 llest │ │ │ │ +0003c590: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c5a0: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073 mension: Loop s │ │ │ │ +0003c5b0: 7465 702c 2061 626f 7574 2074 6f20 636f tep, about to co │ │ │ │ +0003c5c0: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e mpute dimension. │ │ │ │ +0003c5d0: 2020 5375 626d 6174 7269 6365 7320 636f Submatrices co │ │ │ │ +0003c5e0: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c5f0: 6d65 6e73 696f 6e3a 2020 6973 436f 6469 mension: isCodi │ │ │ │ +0003c600: 6d41 744c 6561 7374 2066 6169 6c65 642c mAtLeast failed, │ │ │ │ +0003c610: 2063 6f6d 7075 7469 6e67 2063 6f64 696d computing codim │ │ │ │ +0003c620: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ +0003c630: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c640: 6d65 6e73 696f 6e3a 2020 7061 7274 6961 mension: partia │ │ │ │ +0003c650: 6c20 7369 6e67 756c 6172 206c 6f63 7573 l singular locus │ │ │ │ +0003c660: 2064 696d 656e 7369 6f6e 2063 6f6d 7075 dimension compu │ │ │ │ +0003c670: 7465 642c 203d 2033 2020 2020 2020 2020 ted, = 3 │ │ │ │ +0003c680: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c690: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c6a0: 204c 6578 536d 616c 6c65 7374 2020 2020 LexSmallest │ │ │ │ +0003c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c6d0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c6e0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c6f0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ -0003c700: 4e6f 6e5a 6572 6f20 2020 2020 2020 2020 NonZero │ │ │ │ +0003c6d0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c6e0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c6f0: 2052 616e 646f 6d4e 6f6e 5a65 726f 2020 RandomNonZero │ │ │ │ +0003c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c720: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c730: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c740: 2043 686f 6f73 696e 6720 4c65 7853 6d61 Choosing LexSma │ │ │ │ -0003c750: 6c6c 6573 7454 6572 6d20 2020 2020 2020 llestTerm │ │ │ │ +0003c720: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c730: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c740: 204c 6578 536d 616c 6c65 7374 5465 726d LexSmallestTerm │ │ │ │ +0003c750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c770: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c780: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c790: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003c7a0: 7853 6d61 6c6c 6573 7420 2020 2020 2020 xSmallest │ │ │ │ +0003c770: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c780: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c790: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003c7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003c7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c7c0: 2020 2020 2020 2020 207c 0a7c 696e 7465 |.|inte │ │ │ │ -0003c7d0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ -0003c7e0: 2043 686f 6f73 696e 6720 4752 6576 4c65 Choosing GRevLe │ │ │ │ -0003c7f0: 7853 6d61 6c6c 6573 7454 6572 6d20 2020 xSmallestTerm │ │ │ │ +0003c7c0: 7c0a 7c69 6e74 6572 6e61 6c43 686f 6f73 |.|internalChoos │ │ │ │ +0003c7d0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ +0003c7e0: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +0003c7f0: 5465 726d 2020 2020 2020 2020 2020 2020 Term │ │ │ │ 0003c800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c810: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003c820: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c830: 3a20 204c 6f6f 7020 7374 6570 2c20 6162 : Loop step, ab │ │ │ │ -0003c840: 6f75 7420 746f 2063 6f6d 7075 7465 2064 out to compute d │ │ │ │ -0003c850: 696d 656e 7369 6f6e 2e20 2053 7562 6d61 imension. Subma │ │ │ │ -0003c860: 7472 6963 6573 2063 6f7c 0a7c 7265 6775 trices co|.|regu │ │ │ │ -0003c870: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c880: 3a20 2069 7343 6f64 696d 4174 4c65 6173 : isCodimAtLeas │ │ │ │ -0003c890: 7420 6661 696c 6564 2c20 636f 6d70 7574 t failed, comput │ │ │ │ -0003c8a0: 696e 6720 636f 6469 6d2e 2020 2020 2020 ing codim. │ │ │ │ -0003c8b0: 2020 2020 2020 2020 207c 0a7c 7265 6775 |.|regu │ │ │ │ -0003c8c0: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c8d0: 3a20 2070 6172 7469 616c 2073 696e 6775 : partial singu │ │ │ │ -0003c8e0: 6c61 7220 6c6f 6375 7320 6469 6d65 6e73 lar locus dimens │ │ │ │ -0003c8f0: 696f 6e20 636f 6d70 7574 6564 2c20 3d20 ion computed, = │ │ │ │ -0003c900: 3320 2020 2020 2020 207c 0a7c 7265 6775 3 |.|regu │ │ │ │ -0003c910: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003c920: 3a20 204c 6f6f 7020 636f 6d70 6c65 7465 : Loop complete │ │ │ │ -0003c930: 642c 2073 7562 6d61 7472 6963 6573 2063 d, submatrices c │ │ │ │ -0003c940: 6f6e 7369 6465 7265 6420 3d20 3235 2c20 onsidered = 25, │ │ │ │ -0003c950: 616e 6420 636f 6d70 757c 0a7c 2d2d 2d2d and compu|.|---- │ │ │ │ +0003c810: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c820: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2073 mension: Loop s │ │ │ │ +0003c830: 7465 702c 2061 626f 7574 2074 6f20 636f tep, about to co │ │ │ │ +0003c840: 6d70 7574 6520 6469 6d65 6e73 696f 6e2e mpute dimension. │ │ │ │ +0003c850: 2020 5375 626d 6174 7269 6365 7320 636f Submatrices co │ │ │ │ +0003c860: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c870: 6d65 6e73 696f 6e3a 2020 6973 436f 6469 mension: isCodi │ │ │ │ +0003c880: 6d41 744c 6561 7374 2066 6169 6c65 642c mAtLeast failed, │ │ │ │ +0003c890: 2063 6f6d 7075 7469 6e67 2063 6f64 696d computing codim │ │ │ │ +0003c8a0: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ +0003c8b0: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c8c0: 6d65 6e73 696f 6e3a 2020 7061 7274 6961 mension: partia │ │ │ │ +0003c8d0: 6c20 7369 6e67 756c 6172 206c 6f63 7573 l singular locus │ │ │ │ +0003c8e0: 2064 696d 656e 7369 6f6e 2063 6f6d 7075 dimension compu │ │ │ │ +0003c8f0: 7465 642c 203d 2033 2020 2020 2020 2020 ted, = 3 │ │ │ │ +0003c900: 7c0a 7c72 6567 756c 6172 496e 436f 6469 |.|regularInCodi │ │ │ │ +0003c910: 6d65 6e73 696f 6e3a 2020 4c6f 6f70 2063 mension: Loop c │ │ │ │ +0003c920: 6f6d 706c 6574 6564 2c20 7375 626d 6174 ompleted, submat │ │ │ │ +0003c930: 7269 6365 7320 636f 6e73 6964 6572 6564 rices considered │ │ │ │ +0003c940: 203d 2032 352c 2061 6e64 2063 6f6d 7075 = 25, and compu │ │ │ │ +0003c950: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------- │ │ │ │ 0003c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c9a0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 742d 3e74 ---------|.|t->t │ │ │ │ -0003c9b0: 2f35 2c20 4d69 6e4d 696e 6f72 7346 756e /5, MinMinorsFun │ │ │ │ -0003c9c0: 6374 696f 6e20 3d3e 2074 2d3e 322c 2056 ction => t->2, V │ │ │ │ -0003c9d0: 6572 626f 7365 3d3e 7472 7565 2920 2020 erbose=>true) │ │ │ │ +0003c9a0: 7c0a 7c74 2d3e 742f 352c 204d 696e 4d69 |.|t->t/5, MinMi │ │ │ │ +0003c9b0: 6e6f 7273 4675 6e63 7469 6f6e 203d 3e20 norsFunction => │ │ │ │ +0003c9c0: 742d 3e32 2c20 5665 7262 6f73 653d 3e74 t->2, Verbose=>t │ │ │ │ +0003c9d0: 7275 6529 2020 2020 2020 2020 2020 2020 rue) │ │ │ │ 0003c9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c9f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003c9f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ca00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ca10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ca20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ca30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ca40: 2020 2020 2020 2020 207c 0a7c 6e6f 7273 |.|nors │ │ │ │ -0003ca50: 2c20 7765 2077 696c 6c20 636f 6d70 7574 , we will comput │ │ │ │ -0003ca60: 6520 7570 2074 6f20 3235 206f 6620 7468 e up to 25 of th │ │ │ │ -0003ca70: 656d 2e20 2020 2020 2020 2020 2020 2020 em. │ │ │ │ +0003ca40: 7c0a 7c6e 6f72 732c 2077 6520 7769 6c6c |.|nors, we will │ │ │ │ +0003ca50: 2063 6f6d 7075 7465 2075 7020 746f 2032 compute up to 2 │ │ │ │ +0003ca60: 3520 6f66 2074 6865 6d2e 2020 2020 2020 5 of them. │ │ │ │ +0003ca70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ca80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ca90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003ca90: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003caa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cae0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cae0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003caf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cb30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cb30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003cb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cb80: 2020 2020 2020 2020 207c 0a7c 6e73 6964 |.|nsid │ │ │ │ -0003cb90: 6572 6564 3a20 322c 2061 6e64 2063 6f6d ered: 2, and com │ │ │ │ -0003cba0: 7075 7465 6420 3d20 3220 2020 2020 2020 puted = 2 │ │ │ │ +0003cb80: 7c0a 7c6e 7369 6465 7265 643a 2032 2c20 |.|nsidered: 2, │ │ │ │ +0003cb90: 616e 6420 636f 6d70 7574 6564 203d 2032 and computed = 2 │ │ │ │ +0003cba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cbd0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cbd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003cbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cc20: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cc20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003cc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cc70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cc70: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003cc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ccc0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003ccc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ccd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ccf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cd10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cd10: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cd60: 2020 2020 2020 2020 207c 0a7c 6e73 6964 |.|nsid │ │ │ │ -0003cd70: 6572 6564 3a20 352c 2061 6e64 2063 6f6d ered: 5, and com │ │ │ │ -0003cd80: 7075 7465 6420 3d20 3520 2020 2020 2020 puted = 5 │ │ │ │ +0003cd60: 7c0a 7c6e 7369 6465 7265 643a 2035 2c20 |.|nsidered: 5, │ │ │ │ +0003cd70: 616e 6420 636f 6d70 7574 6564 203d 2035 and computed = 5 │ │ │ │ +0003cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cdb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cdb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ce00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003ce00: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ce10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ce20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ce30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ce40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ce50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003ce50: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ce60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ce70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ce80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ce90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cea0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cea0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ceb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ced0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cef0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cef0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003cf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cf40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cf40: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003cf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cf90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003cf90: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003cfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003cfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003cfe0: 2020 2020 2020 2020 207c 0a7c 6e73 6964 |.|nsid │ │ │ │ -0003cff0: 6572 6564 3a20 3130 2c20 616e 6420 636f ered: 10, and co │ │ │ │ -0003d000: 6d70 7574 6564 203d 2031 3020 2020 2020 mputed = 10 │ │ │ │ +0003cfe0: 7c0a 7c6e 7369 6465 7265 643a 2031 302c |.|nsidered: 10, │ │ │ │ +0003cff0: 2061 6e64 2063 6f6d 7075 7465 6420 3d20 and computed = │ │ │ │ +0003d000: 3130 2020 2020 2020 2020 2020 2020 2020 10 │ │ │ │ 0003d010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d030: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d030: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d080: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d080: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d0d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d0d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d120: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d120: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d170: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d170: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d1c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d1c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d210: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d210: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d260: 2020 2020 2020 2020 207c 0a7c 6e73 6964 |.|nsid │ │ │ │ -0003d270: 6572 6564 3a20 3135 2c20 616e 6420 636f ered: 15, and co │ │ │ │ -0003d280: 6d70 7574 6564 203d 2031 3520 2020 2020 mputed = 15 │ │ │ │ +0003d260: 7c0a 7c6e 7369 6465 7265 643a 2031 352c |.|nsidered: 15, │ │ │ │ +0003d270: 2061 6e64 2063 6f6d 7075 7465 6420 3d20 and computed = │ │ │ │ +0003d280: 3135 2020 2020 2020 2020 2020 2020 2020 15 │ │ │ │ 0003d290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d2b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d2b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d300: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d300: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d350: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d350: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d3a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d3a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d3f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d3f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d440: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d440: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d490: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d490: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d4e0: 2020 2020 2020 2020 207c 0a7c 6e73 6964 |.|nsid │ │ │ │ -0003d4f0: 6572 6564 3a20 3230 2c20 616e 6420 636f ered: 20, and co │ │ │ │ -0003d500: 6d70 7574 6564 203d 2031 3820 2020 2020 mputed = 18 │ │ │ │ +0003d4e0: 7c0a 7c6e 7369 6465 7265 643a 2032 302c |.|nsidered: 20, │ │ │ │ +0003d4f0: 2061 6e64 2063 6f6d 7075 7465 6420 3d20 and computed = │ │ │ │ +0003d500: 3138 2020 2020 2020 2020 2020 2020 2020 18 │ │ │ │ 0003d510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d530: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d530: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d580: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d580: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d5d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d5d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d620: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d620: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d670: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d670: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d690: 2020 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 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d6c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d710: 7c0a 7c20 2020 2020 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 2020 2020 2020 207c 0a7c 6e73 6964 |.|nsid │ │ │ │ -0003d770: 6572 6564 3a20 3235 2c20 616e 6420 636f ered: 25, and co │ │ │ │ -0003d780: 6d70 7574 6564 203d 2032 3320 2020 2020 mputed = 23 │ │ │ │ +0003d760: 7c0a 7c6e 7369 6465 7265 643a 2032 352c |.|nsidered: 25, │ │ │ │ +0003d770: 2061 6e64 2063 6f6d 7075 7465 6420 3d20 and computed = │ │ │ │ +0003d780: 3233 2020 2020 2020 2020 2020 2020 2020 23 │ │ │ │ 0003d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d7b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d800: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003d800: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d820: 2020 2020 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 207c 0a7c 7465 6420 |.|ted │ │ │ │ -0003d860: 3d20 3233 2e20 2073 696e 6775 6c61 7220 = 23. singular │ │ │ │ -0003d870: 6c6f 6375 7320 6469 6d65 6e73 696f 6e20 locus dimension │ │ │ │ -0003d880: 6170 7065 6172 7320 746f 2062 6520 3d20 appears to be = │ │ │ │ -0003d890: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -0003d8a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003d850: 7c0a 7c74 6564 203d 2032 332e 2020 7369 |.|ted = 23. si │ │ │ │ +0003d860: 6e67 756c 6172 206c 6f63 7573 2064 696d ngular locus dim │ │ │ │ +0003d870: 656e 7369 6f6e 2061 7070 6561 7273 2074 ension appears t │ │ │ │ +0003d880: 6f20 6265 203d 2033 2020 2020 2020 2020 o be = 3 │ │ │ │ +0003d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003d8a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0003d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d8f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 6973 436f ---------+..isCo │ │ │ │ -0003d900: 6469 6d41 744c 6561 7374 2061 6e64 2064 dimAtLeast and d │ │ │ │ -0003d910: 696d 2e20 2057 6520 7365 6520 7468 6520 im. We see the │ │ │ │ -0003d920: 6c69 6e65 7320 6162 6f75 7420 7468 6520 lines about the │ │ │ │ -0003d930: 6060 6973 436f 6469 6d41 744c 6561 7374 ``isCodimAtLeast │ │ │ │ -0003d940: 2066 6169 6c65 6427 272e 0a54 6869 7320 failed''..This │ │ │ │ -0003d950: 6d65 616e 7320 7468 6174 2069 7343 6f64 means that isCod │ │ │ │ -0003d960: 696d 4174 4c65 6173 7420 7761 7320 6e6f imAtLeast was no │ │ │ │ -0003d970: 7420 656e 6f75 6768 206f 6e20 6974 7320 t enough on its │ │ │ │ -0003d980: 6f77 6e20 746f 2076 6572 6966 7920 7468 own to verify th │ │ │ │ -0003d990: 6174 206f 7572 0a72 696e 6720 6973 2072 at our.ring is r │ │ │ │ -0003d9a0: 6567 756c 6172 2069 6e20 636f 6469 6d65 egular in codime │ │ │ │ -0003d9b0: 6e73 696f 6e20 312e 2020 4166 7465 7220 nsion 1. After │ │ │ │ -0003d9c0: 7468 6973 2c20 6060 7061 7274 6961 6c20 this, ``partial │ │ │ │ -0003d9d0: 7369 6e67 756c 6172 206c 6f63 7573 0a64 singular locus.d │ │ │ │ -0003d9e0: 696d 656e 7369 6f6e 2063 6f6d 7075 7465 imension compute │ │ │ │ -0003d9f0: 6427 2720 696e 6469 6361 7465 7320 7765 d'' indicates we │ │ │ │ -0003da00: 2064 6964 2061 2063 6f6d 706c 6574 6520 did a complete │ │ │ │ -0003da10: 6469 6d65 6e73 696f 6e20 636f 6d70 7574 dimension comput │ │ │ │ -0003da20: 6174 696f 6e20 6f66 2074 6865 0a70 6172 ation of the.par │ │ │ │ -0003da30: 7469 616c 2069 6465 616c 2064 6566 696e tial ideal defin │ │ │ │ -0003da40: 696e 6720 7468 6520 7369 6e67 756c 6172 ing the singular │ │ │ │ -0003da50: 206c 6f63 7573 2e20 2048 6f77 2069 7343 locus. How isC │ │ │ │ -0003da60: 6f64 696d 4174 4c65 6173 7420 6973 2063 odimAtLeast is c │ │ │ │ -0003da70: 616c 6c65 6420 6361 6e20 6265 0a63 6f6e alled can be.con │ │ │ │ -0003da80: 7472 6f6c 6c65 6420 7669 6120 7468 6520 trolled via the │ │ │ │ -0003da90: 6f70 7469 6f6e 7320 5350 6169 7273 4675 options SPairsFu │ │ │ │ -0003daa0: 6e63 7469 6f6e 2061 6e64 2050 6169 724c nction and PairL │ │ │ │ -0003dab0: 696d 6974 2c20 7768 6963 6820 6172 6520 imit, which are │ │ │ │ -0003dac0: 7369 6d70 6c79 0a70 6173 7365 6420 746f simply.passed to │ │ │ │ -0003dad0: 202a 6e6f 7465 2069 7343 6f64 696d 4174 *note isCodimAt │ │ │ │ -0003dae0: 4c65 6173 743a 2069 7343 6f64 696d 4174 Least: isCodimAt │ │ │ │ -0003daf0: 4c65 6173 742c 2e20 2059 6f75 2063 616e Least,. You can │ │ │ │ -0003db00: 2066 6f72 6365 2074 6865 2066 756e 6374 force the funct │ │ │ │ -0003db10: 696f 6e20 746f 0a6f 6e6c 7920 7573 6520 ion to.only use │ │ │ │ -0003db20: 6973 436f 6469 6d41 744c 6561 7374 2061 isCodimAtLeast a │ │ │ │ -0003db30: 6e64 206e 6f74 2063 616c 6c20 6469 6d65 nd not call dime │ │ │ │ -0003db40: 6e73 696f 6e20 6279 2073 6574 7469 6e67 nsion by setting │ │ │ │ -0003db50: 2055 7365 4f6e 6c79 4661 7374 436f 6469 UseOnlyFastCodi │ │ │ │ -0003db60: 6d20 3d3e 0a74 7275 652e 0a0a 2b2d 2d2d m =>.true...+--- │ │ │ │ +0003d8f0: 2b0a 0a69 7343 6f64 696d 4174 4c65 6173 +..isCodimAtLeas │ │ │ │ +0003d900: 7420 616e 6420 6469 6d2e 2020 5765 2073 t and dim. We s │ │ │ │ +0003d910: 6565 2074 6865 206c 696e 6573 2061 626f ee the lines abo │ │ │ │ +0003d920: 7574 2074 6865 2060 6069 7343 6f64 696d ut the ``isCodim │ │ │ │ +0003d930: 4174 4c65 6173 7420 6661 696c 6564 2727 AtLeast failed'' │ │ │ │ +0003d940: 2e0a 5468 6973 206d 6561 6e73 2074 6861 ..This means tha │ │ │ │ +0003d950: 7420 6973 436f 6469 6d41 744c 6561 7374 t isCodimAtLeast │ │ │ │ +0003d960: 2077 6173 206e 6f74 2065 6e6f 7567 6820 was not enough │ │ │ │ +0003d970: 6f6e 2069 7473 206f 776e 2074 6f20 7665 on its own to ve │ │ │ │ +0003d980: 7269 6679 2074 6861 7420 6f75 720a 7269 rify that our.ri │ │ │ │ +0003d990: 6e67 2069 7320 7265 6775 6c61 7220 696e ng is regular in │ │ │ │ +0003d9a0: 2063 6f64 696d 656e 7369 6f6e 2031 2e20 codimension 1. │ │ │ │ +0003d9b0: 2041 6674 6572 2074 6869 732c 2060 6070 After this, ``p │ │ │ │ +0003d9c0: 6172 7469 616c 2073 696e 6775 6c61 7220 artial singular │ │ │ │ +0003d9d0: 6c6f 6375 730a 6469 6d65 6e73 696f 6e20 locus.dimension │ │ │ │ +0003d9e0: 636f 6d70 7574 6564 2727 2069 6e64 6963 computed'' indic │ │ │ │ +0003d9f0: 6174 6573 2077 6520 6469 6420 6120 636f ates we did a co │ │ │ │ +0003da00: 6d70 6c65 7465 2064 696d 656e 7369 6f6e mplete dimension │ │ │ │ +0003da10: 2063 6f6d 7075 7461 7469 6f6e 206f 6620 computation of │ │ │ │ +0003da20: 7468 650a 7061 7274 6961 6c20 6964 6561 the.partial idea │ │ │ │ +0003da30: 6c20 6465 6669 6e69 6e67 2074 6865 2073 l defining the s │ │ │ │ +0003da40: 696e 6775 6c61 7220 6c6f 6375 732e 2020 ingular locus. │ │ │ │ +0003da50: 486f 7720 6973 436f 6469 6d41 744c 6561 How isCodimAtLea │ │ │ │ +0003da60: 7374 2069 7320 6361 6c6c 6564 2063 616e st is called can │ │ │ │ +0003da70: 2062 650a 636f 6e74 726f 6c6c 6564 2076 be.controlled v │ │ │ │ +0003da80: 6961 2074 6865 206f 7074 696f 6e73 2053 ia the options S │ │ │ │ +0003da90: 5061 6972 7346 756e 6374 696f 6e20 616e PairsFunction an │ │ │ │ +0003daa0: 6420 5061 6972 4c69 6d69 742c 2077 6869 d PairLimit, whi │ │ │ │ +0003dab0: 6368 2061 7265 2073 696d 706c 790a 7061 ch are simply.pa │ │ │ │ +0003dac0: 7373 6564 2074 6f20 2a6e 6f74 6520 6973 ssed to *note is │ │ │ │ +0003dad0: 436f 6469 6d41 744c 6561 7374 3a20 6973 CodimAtLeast: is │ │ │ │ +0003dae0: 436f 6469 6d41 744c 6561 7374 2c2e 2020 CodimAtLeast,. │ │ │ │ +0003daf0: 596f 7520 6361 6e20 666f 7263 6520 7468 You can force th │ │ │ │ +0003db00: 6520 6675 6e63 7469 6f6e 2074 6f0a 6f6e e function to.on │ │ │ │ +0003db10: 6c79 2075 7365 2069 7343 6f64 696d 4174 ly use isCodimAt │ │ │ │ +0003db20: 4c65 6173 7420 616e 6420 6e6f 7420 6361 Least and not ca │ │ │ │ +0003db30: 6c6c 2064 696d 656e 7369 6f6e 2062 7920 ll dimension by │ │ │ │ +0003db40: 7365 7474 696e 6720 5573 654f 6e6c 7946 setting UseOnlyF │ │ │ │ +0003db50: 6173 7443 6f64 696d 203d 3e0a 7472 7565 astCodim =>.true │ │ │ │ +0003db60: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 0003db70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003db90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003dba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003dbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 ----------+.|i11 │ │ │ │ -0003dbc0: 203a 2074 696d 6520 7265 6775 6c61 7249 : time regularI │ │ │ │ -0003dbd0: 6e43 6f64 696d 656e 7369 6f6e 2831 2c20 nCodimension(1, │ │ │ │ -0003dbe0: 532f 4a2c 204d 6178 4d69 6e6f 7273 3d3e S/J, MaxMinors=> │ │ │ │ -0003dbf0: 3235 2c20 5573 654f 6e6c 7946 6173 7443 25, UseOnlyFastC │ │ │ │ -0003dc00: 6f64 696d 203d 3e20 7472 7c0a 7c20 2d2d odim => tr|.| -- │ │ │ │ -0003dc10: 2075 7365 6420 302e 3434 3138 3339 7320 used 0.441839s │ │ │ │ -0003dc20: 2863 7075 293b 2030 2e33 3139 3639 3273 (cpu); 0.319692s │ │ │ │ -0003dc30: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -0003dc40: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ -0003dc50: 2020 2020 2020 2020 2020 7c0a 7c72 6567 |.|reg │ │ │ │ -0003dc60: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ -0003dc70: 6e3a 2072 696e 6720 6469 6d65 6e73 696f n: ring dimensio │ │ │ │ -0003dc80: 6e20 3d34 2c20 7468 6572 6520 6172 6520 n =4, there are │ │ │ │ -0003dc90: 3134 3635 3132 3820 706f 7373 6962 6c65 1465128 possible │ │ │ │ -0003dca0: 2035 2062 7920 3520 6d69 7c0a 7c72 6567 5 by 5 mi|.|reg │ │ │ │ -0003dcb0: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ -0003dcc0: 6e3a 2041 626f 7574 2074 6f20 656e 7465 n: About to ente │ │ │ │ -0003dcd0: 7220 6c6f 6f70 2020 2020 2020 2020 2020 r loop │ │ │ │ +0003dbb0: 2d2b 0a7c 6931 3120 3a20 7469 6d65 2072 -+.|i11 : time r │ │ │ │ +0003dbc0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +0003dbd0: 696f 6e28 312c 2053 2f4a 2c20 4d61 784d ion(1, S/J, MaxM │ │ │ │ +0003dbe0: 696e 6f72 733d 3e32 352c 2055 7365 4f6e inors=>25, UseOn │ │ │ │ +0003dbf0: 6c79 4661 7374 436f 6469 6d20 3d3e 2074 lyFastCodim => t │ │ │ │ +0003dc00: 727c 0a7c 202d 2d20 7573 6564 2030 2e35 r|.| -- used 0.5 │ │ │ │ +0003dc10: 3833 3539 3373 2028 6370 7529 3b20 302e 83593s (cpu); 0. │ │ │ │ +0003dc20: 3336 3136 3938 7320 2874 6872 6561 6429 361698s (thread) │ │ │ │ +0003dc30: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0003dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dc50: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ +0003dc60: 696d 656e 7369 6f6e 3a20 7269 6e67 2064 imension: ring d │ │ │ │ +0003dc70: 696d 656e 7369 6f6e 203d 342c 2074 6865 imension =4, the │ │ │ │ +0003dc80: 7265 2061 7265 2031 3436 3531 3238 2070 re are 1465128 p │ │ │ │ +0003dc90: 6f73 7369 626c 6520 3520 6279 2035 206d ossible 5 by 5 m │ │ │ │ +0003dca0: 697c 0a7c 7265 6775 6c61 7249 6e43 6f64 i|.|regularInCod │ │ │ │ +0003dcb0: 696d 656e 7369 6f6e 3a20 4162 6f75 7420 imension: About │ │ │ │ +0003dcc0: 746f 2065 6e74 6572 206c 6f6f 7020 2020 to enter loop │ │ │ │ +0003dcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dcf0: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003dd00: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003dd10: 3a20 4368 6f6f 7369 6e67 2047 5265 764c : Choosing GRevL │ │ │ │ -0003dd20: 6578 536d 616c 6c65 7374 2020 2020 2020 exSmallest │ │ │ │ +0003dcf0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003dd00: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003dd10: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ +0003dd20: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 0003dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dd40: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003dd50: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003dd60: 3a20 4368 6f6f 7369 6e67 204c 6578 536d : Choosing LexSm │ │ │ │ -0003dd70: 616c 6c65 7374 2020 2020 2020 2020 2020 allest │ │ │ │ +0003dd40: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003dd50: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003dd60: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +0003dd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dd90: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003dda0: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003ddb0: 3a20 4368 6f6f 7369 6e67 2052 616e 646f : Choosing Rando │ │ │ │ -0003ddc0: 6d4e 6f6e 5a65 726f 2020 2020 2020 2020 mNonZero │ │ │ │ +0003dd90: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003dda0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003ddb0: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ +0003ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dde0: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003ddf0: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003de00: 3a20 4368 6f6f 7369 6e67 2052 616e 646f : Choosing Rando │ │ │ │ -0003de10: 6d4e 6f6e 5a65 726f 2020 2020 2020 2020 mNonZero │ │ │ │ +0003dde0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003ddf0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003de00: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ +0003de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003de20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de30: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003de40: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003de50: 3a20 4368 6f6f 7369 6e67 2047 5265 764c : Choosing GRevL │ │ │ │ -0003de60: 6578 536d 616c 6c65 7374 5465 726d 2020 exSmallestTerm │ │ │ │ +0003de30: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003de40: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003de50: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ +0003de60: 7454 6572 6d20 2020 2020 2020 2020 2020 tTerm │ │ │ │ 0003de70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de80: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003de90: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003dea0: 3a20 4368 6f6f 7369 6e67 204c 6578 536d : Choosing LexSm │ │ │ │ -0003deb0: 616c 6c65 7374 2020 2020 2020 2020 2020 allest │ │ │ │ +0003de80: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003de90: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003dea0: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +0003deb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ded0: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003dee0: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003def0: 3a20 4368 6f6f 7369 6e67 2052 616e 646f : Choosing Rando │ │ │ │ -0003df00: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ +0003ded0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003dee0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003def0: 6720 5261 6e64 6f6d 2020 2020 2020 2020 g Random │ │ │ │ +0003df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003df20: 2020 2020 2020 2020 2020 7c0a 7c72 6567 |.|reg │ │ │ │ -0003df30: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ -0003df40: 6e3a 2020 4c6f 6f70 2073 7465 702c 2061 n: Loop step, a │ │ │ │ -0003df50: 626f 7574 2074 6f20 636f 6d70 7574 6520 bout to compute │ │ │ │ -0003df60: 6469 6d65 6e73 696f 6e2e 2020 5375 626d dimension. Subm │ │ │ │ -0003df70: 6174 7269 6365 7320 636f 7c0a 7c72 6567 atrices co|.|reg │ │ │ │ -0003df80: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ -0003df90: 6e3a 2020 7061 7274 6961 6c20 7369 6e67 n: partial sing │ │ │ │ -0003dfa0: 756c 6172 206c 6f63 7573 2064 696d 656e ular locus dimen │ │ │ │ -0003dfb0: 7369 6f6e 2063 6f6d 7075 7465 642c 203d sion computed, = │ │ │ │ -0003dfc0: 2034 2020 2020 2020 2020 7c0a 7c69 6e74 4 |.|int │ │ │ │ -0003dfd0: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003dfe0: 3a20 4368 6f6f 7369 6e67 204c 6578 536d : Choosing LexSm │ │ │ │ -0003dff0: 616c 6c65 7374 2020 2020 2020 2020 2020 allest │ │ │ │ +0003df20: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ +0003df30: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ +0003df40: 7374 6570 2c20 6162 6f75 7420 746f 2063 step, about to c │ │ │ │ +0003df50: 6f6d 7075 7465 2064 696d 656e 7369 6f6e ompute dimension │ │ │ │ +0003df60: 2e20 2053 7562 6d61 7472 6963 6573 2063 . Submatrices c │ │ │ │ +0003df70: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64 o|.|regularInCod │ │ │ │ +0003df80: 696d 656e 7369 6f6e 3a20 2070 6172 7469 imension: parti │ │ │ │ +0003df90: 616c 2073 696e 6775 6c61 7220 6c6f 6375 al singular locu │ │ │ │ +0003dfa0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70 s dimension comp │ │ │ │ +0003dfb0: 7574 6564 2c20 3d20 3420 2020 2020 2020 uted, = 4 │ │ │ │ +0003dfc0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003dfd0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003dfe0: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +0003dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e010: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003e020: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003e030: 3a20 4368 6f6f 7369 6e67 2047 5265 764c : Choosing GRevL │ │ │ │ -0003e040: 6578 536d 616c 6c65 7374 5465 726d 2020 exSmallestTerm │ │ │ │ +0003e010: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003e020: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003e030: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ +0003e040: 7454 6572 6d20 2020 2020 2020 2020 2020 tTerm │ │ │ │ 0003e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e060: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003e070: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003e080: 3a20 4368 6f6f 7369 6e67 2047 5265 764c : Choosing GRevL │ │ │ │ -0003e090: 6578 536d 616c 6c65 7374 2020 2020 2020 exSmallest │ │ │ │ +0003e060: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003e070: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003e080: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ +0003e090: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 0003e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e0b0: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003e0c0: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003e0d0: 3a20 4368 6f6f 7369 6e67 2047 5265 764c : Choosing GRevL │ │ │ │ -0003e0e0: 6578 536d 616c 6c65 7374 2020 2020 2020 exSmallest │ │ │ │ +0003e0b0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003e0c0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003e0d0: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ +0003e0e0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 0003e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e100: 2020 2020 2020 2020 2020 7c0a 7c72 6567 |.|reg │ │ │ │ -0003e110: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ -0003e120: 6e3a 2020 4c6f 6f70 2073 7465 702c 2061 n: Loop step, a │ │ │ │ -0003e130: 626f 7574 2074 6f20 636f 6d70 7574 6520 bout to compute │ │ │ │ -0003e140: 6469 6d65 6e73 696f 6e2e 2020 5375 626d dimension. Subm │ │ │ │ -0003e150: 6174 7269 6365 7320 636f 7c0a 7c72 6567 atrices co|.|reg │ │ │ │ -0003e160: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ -0003e170: 6e3a 2020 7061 7274 6961 6c20 7369 6e67 n: partial sing │ │ │ │ -0003e180: 756c 6172 206c 6f63 7573 2064 696d 656e ular locus dimen │ │ │ │ -0003e190: 7369 6f6e 2063 6f6d 7075 7465 642c 203d sion computed, = │ │ │ │ -0003e1a0: 2034 2020 2020 2020 2020 7c0a 7c69 6e74 4 |.|int │ │ │ │ -0003e1b0: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003e1c0: 3a20 4368 6f6f 7369 6e67 204c 6578 536d : Choosing LexSm │ │ │ │ -0003e1d0: 616c 6c65 7374 5465 726d 2020 2020 2020 allestTerm │ │ │ │ +0003e100: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ +0003e110: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ +0003e120: 7374 6570 2c20 6162 6f75 7420 746f 2063 step, about to c │ │ │ │ +0003e130: 6f6d 7075 7465 2064 696d 656e 7369 6f6e ompute dimension │ │ │ │ +0003e140: 2e20 2053 7562 6d61 7472 6963 6573 2063 . Submatrices c │ │ │ │ +0003e150: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64 o|.|regularInCod │ │ │ │ +0003e160: 696d 656e 7369 6f6e 3a20 2070 6172 7469 imension: parti │ │ │ │ +0003e170: 616c 2073 696e 6775 6c61 7220 6c6f 6375 al singular locu │ │ │ │ +0003e180: 7320 6469 6d65 6e73 696f 6e20 636f 6d70 s dimension comp │ │ │ │ +0003e190: 7574 6564 2c20 3d20 3420 2020 2020 2020 uted, = 4 │ │ │ │ +0003e1a0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003e1b0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003e1c0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ +0003e1d0: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ 0003e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e1f0: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003e200: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003e210: 3a20 4368 6f6f 7369 6e67 204c 6578 536d : Choosing LexSm │ │ │ │ -0003e220: 616c 6c65 7374 2020 2020 2020 2020 2020 allest │ │ │ │ +0003e1f0: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003e200: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003e210: 6720 4c65 7853 6d61 6c6c 6573 7420 2020 g LexSmallest │ │ │ │ +0003e220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e240: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003e250: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003e260: 3a20 4368 6f6f 7369 6e67 2047 5265 764c : Choosing GRevL │ │ │ │ -0003e270: 6578 536d 616c 6c65 7374 5465 726d 2020 exSmallestTerm │ │ │ │ +0003e240: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003e250: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003e260: 6720 4752 6576 4c65 7853 6d61 6c6c 6573 g GRevLexSmalles │ │ │ │ +0003e270: 7454 6572 6d20 2020 2020 2020 2020 2020 tTerm │ │ │ │ 0003e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e290: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ -0003e2a0: 6572 6e61 6c43 686f 6f73 654d 696e 6f72 ernalChooseMinor │ │ │ │ -0003e2b0: 3a20 4368 6f6f 7369 6e67 204c 6578 536d : Choosing LexSm │ │ │ │ -0003e2c0: 616c 6c65 7374 5465 726d 2020 2020 2020 allestTerm │ │ │ │ +0003e290: 207c 0a7c 696e 7465 726e 616c 4368 6f6f |.|internalChoo │ │ │ │ +0003e2a0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ +0003e2b0: 6720 4c65 7853 6d61 6c6c 6573 7454 6572 g LexSmallestTer │ │ │ │ +0003e2c0: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ 0003e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e2e0: 2020 2020 2020 2020 2020 7c0a 7c72 6567 |.|reg │ │ │ │ -0003e2f0: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ -0003e300: 6e3a 2020 4c6f 6f70 2073 7465 702c 2061 n: Loop step, a │ │ │ │ -0003e310: 626f 7574 2074 6f20 636f 6d70 7574 6520 bout to compute │ │ │ │ -0003e320: 6469 6d65 6e73 696f 6e2e 2020 5375 626d dimension. Subm │ │ │ │ -0003e330: 6174 7269 6365 7320 636f 7c0a 7c72 6567 atrices co|.|reg │ │ │ │ -0003e340: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ -0003e350: 6e3a 2020 7369 6e67 756c 6172 4c6f 6375 n: singularLocu │ │ │ │ -0003e360: 7320 6469 6d65 6e73 696f 6e20 7665 7269 s dimension veri │ │ │ │ -0003e370: 6669 6564 2062 7920 6973 436f 6469 6d41 fied by isCodimA │ │ │ │ -0003e380: 744c 6561 7374 2020 2020 7c0a 7c72 6567 tLeast |.|reg │ │ │ │ -0003e390: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ -0003e3a0: 6e3a 2020 7061 7274 6961 6c20 7369 6e67 n: partial sing │ │ │ │ -0003e3b0: 756c 6172 206c 6f63 7573 2064 696d 656e ular locus dimen │ │ │ │ -0003e3c0: 7369 6f6e 2063 6f6d 7075 7465 642c 203d sion computed, = │ │ │ │ -0003e3d0: 2032 2020 2020 2020 2020 7c0a 7c72 6567 2 |.|reg │ │ │ │ -0003e3e0: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ -0003e3f0: 6e3a 2020 4c6f 6f70 2063 6f6d 706c 6574 n: Loop complet │ │ │ │ -0003e400: 6564 2c20 7375 626d 6174 7269 6365 7320 ed, submatrices │ │ │ │ -0003e410: 636f 6e73 6964 6572 6564 203d 2031 352c considered = 15, │ │ │ │ -0003e420: 2061 6e64 2063 6f6d 7075 7c0a 7c20 2020 and compu|.| │ │ │ │ +0003e2e0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ +0003e2f0: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ +0003e300: 7374 6570 2c20 6162 6f75 7420 746f 2063 step, about to c │ │ │ │ +0003e310: 6f6d 7075 7465 2064 696d 656e 7369 6f6e ompute dimension │ │ │ │ +0003e320: 2e20 2053 7562 6d61 7472 6963 6573 2063 . Submatrices c │ │ │ │ +0003e330: 6f7c 0a7c 7265 6775 6c61 7249 6e43 6f64 o|.|regularInCod │ │ │ │ +0003e340: 696d 656e 7369 6f6e 3a20 2073 696e 6775 imension: singu │ │ │ │ +0003e350: 6c61 724c 6f63 7573 2064 696d 656e 7369 larLocus dimensi │ │ │ │ +0003e360: 6f6e 2076 6572 6966 6965 6420 6279 2069 on verified by i │ │ │ │ +0003e370: 7343 6f64 696d 4174 4c65 6173 7420 2020 sCodimAtLeast │ │ │ │ +0003e380: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ +0003e390: 696d 656e 7369 6f6e 3a20 2070 6172 7469 imension: parti │ │ │ │ +0003e3a0: 616c 2073 696e 6775 6c61 7220 6c6f 6375 al singular locu │ │ │ │ +0003e3b0: 7320 6469 6d65 6e73 696f 6e20 636f 6d70 s dimension comp │ │ │ │ +0003e3c0: 7574 6564 2c20 3d20 3220 2020 2020 2020 uted, = 2 │ │ │ │ +0003e3d0: 207c 0a7c 7265 6775 6c61 7249 6e43 6f64 |.|regularInCod │ │ │ │ +0003e3e0: 696d 656e 7369 6f6e 3a20 204c 6f6f 7020 imension: Loop │ │ │ │ +0003e3f0: 636f 6d70 6c65 7465 642c 2073 7562 6d61 completed, subma │ │ │ │ +0003e400: 7472 6963 6573 2063 6f6e 7369 6465 7265 trices considere │ │ │ │ +0003e410: 6420 3d20 3135 2c20 616e 6420 636f 6d70 d = 15, and comp │ │ │ │ +0003e420: 757c 0a7c 2020 2020 2020 2020 2020 2020 u|.| │ │ │ │ 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 2020 7c0a 7c6f 3131 |.|o11 │ │ │ │ -0003e480: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ +0003e470: 207c 0a7c 6f31 3120 3d20 7472 7565 2020 |.|o11 = true │ │ │ │ +0003e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 2020 2020 │ │ │ │ -0003e4c0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d |.|--- │ │ │ │ +0003e4c0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ 0003e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e510: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c75 652c ----------|.|ue, │ │ │ │ -0003e520: 2056 6572 626f 7365 3d3e 7472 7565 2920 Verbose=>true) │ │ │ │ +0003e510: 2d7c 0a7c 7565 2c20 5665 7262 6f73 653d -|.|ue, Verbose= │ │ │ │ +0003e520: 3e74 7275 6529 2020 2020 2020 2020 2020 >true) │ │ │ │ 0003e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e560: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e560: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e5b0: 2020 2020 2020 2020 2020 7c0a 7c6e 6f72 |.|nor │ │ │ │ -0003e5c0: 732c 2077 6520 7769 6c6c 2063 6f6d 7075 s, we will compu │ │ │ │ -0003e5d0: 7465 2075 7020 746f 2032 3520 6f66 2074 te up to 25 of t │ │ │ │ -0003e5e0: 6865 6d2e 2020 2020 2020 2020 2020 2020 hem. │ │ │ │ +0003e5b0: 207c 0a7c 6e6f 7273 2c20 7765 2077 696c |.|nors, we wil │ │ │ │ +0003e5c0: 6c20 636f 6d70 7574 6520 7570 2074 6f20 l compute up to │ │ │ │ +0003e5d0: 3235 206f 6620 7468 656d 2e20 2020 2020 25 of them. │ │ │ │ +0003e5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e600: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e600: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e650: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e650: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e6a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e6c0: 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020 |.| │ │ │ │ +0003e6f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e740: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e740: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e790: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e790: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e7e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e7e0: 207c 0a7c 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 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e830: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e830: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e880: 2020 2020 2020 2020 2020 7c0a 7c6e 7369 |.|nsi │ │ │ │ -0003e890: 6465 7265 643a 2037 2c20 616e 6420 636f dered: 7, and co │ │ │ │ -0003e8a0: 6d70 7574 6564 203d 2037 2020 2020 2020 mputed = 7 │ │ │ │ +0003e880: 207c 0a7c 6e73 6964 6572 6564 3a20 372c |.|nsidered: 7, │ │ │ │ +0003e890: 2061 6e64 2063 6f6d 7075 7465 6420 3d20 and computed = │ │ │ │ +0003e8a0: 3720 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ 0003e8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e8d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e8d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e920: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e920: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e970: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e970: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e9c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e9c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ea00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ea10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003ea10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ea20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ea30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ea40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ea60: 2020 2020 2020 2020 2020 7c0a 7c6e 7369 |.|nsi │ │ │ │ -0003ea70: 6465 7265 643a 2031 312c 2061 6e64 2063 dered: 11, and c │ │ │ │ -0003ea80: 6f6d 7075 7465 6420 3d20 3130 2020 2020 omputed = 10 │ │ │ │ +0003ea60: 207c 0a7c 6e73 6964 6572 6564 3a20 3131 |.|nsidered: 11 │ │ │ │ +0003ea70: 2c20 616e 6420 636f 6d70 7574 6564 203d , and computed = │ │ │ │ +0003ea80: 2031 3020 2020 2020 2020 2020 2020 2020 10 │ │ │ │ 0003ea90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003eaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eab0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003eab0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003eac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003eae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eb00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003eb00: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eb50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003eb50: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003eb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003eb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eba0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003eba0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ebd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ebe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ebf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003ebf0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ec20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ec30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ec40: 2020 2020 2020 2020 2020 7c0a 7c6e 7369 |.|nsi │ │ │ │ -0003ec50: 6465 7265 643a 2031 352c 2061 6e64 2063 dered: 15, and c │ │ │ │ -0003ec60: 6f6d 7075 7465 6420 3d20 3133 2020 2020 omputed = 13 │ │ │ │ +0003ec40: 207c 0a7c 6e73 6964 6572 6564 3a20 3135 |.|nsidered: 15 │ │ │ │ +0003ec50: 2c20 616e 6420 636f 6d70 7574 6564 203d , and computed = │ │ │ │ +0003ec60: 2031 3320 2020 2020 2020 2020 2020 2020 13 │ │ │ │ 0003ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ec80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ec90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003ec90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003eca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ecb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ecc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ecd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ece0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003ece0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ed00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ed10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ed20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ed30: 2020 2020 2020 2020 2020 7c0a 7c74 6564 |.|ted │ │ │ │ -0003ed40: 203d 2031 332e 2020 7369 6e67 756c 6172 = 13. singular │ │ │ │ -0003ed50: 206c 6f63 7573 2064 696d 656e 7369 6f6e locus dimension │ │ │ │ -0003ed60: 2061 7070 6561 7273 2074 6f20 6265 203d appears to be = │ │ │ │ -0003ed70: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0003ed80: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0003ed30: 207c 0a7c 7465 6420 3d20 3133 2e20 2073 |.|ted = 13. s │ │ │ │ +0003ed40: 696e 6775 6c61 7220 6c6f 6375 7320 6469 ingular locus di │ │ │ │ +0003ed50: 6d65 6e73 696f 6e20 6170 7065 6172 7320 mension appears │ │ │ │ +0003ed60: 746f 2062 6520 3d20 3220 2020 2020 2020 to be = 2 │ │ │ │ +0003ed70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ed80: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0003ed90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003eda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6869 ----------+..Thi │ │ │ │ -0003ede0: 7320 6361 6e20 6265 2075 7365 6675 6c20 s can be useful │ │ │ │ -0003edf0: 6966 2074 6865 2066 756e 6374 696f 6e20 if the function │ │ │ │ -0003ee00: 6973 2068 616e 6769 6e67 2077 6865 6e20 is hanging when │ │ │ │ -0003ee10: 7472 7969 6e67 2074 6f20 636f 6d70 7574 trying to comput │ │ │ │ -0003ee20: 6520 7468 650a 6469 6d65 6e73 696f 6e2c e the.dimension, │ │ │ │ -0003ee30: 2062 7574 2079 6f75 206d 6179 2077 6973 but you may wis │ │ │ │ -0003ee40: 6820 696e 6372 6561 7365 2050 6169 724c h increase PairL │ │ │ │ -0003ee50: 696d 6974 2e0a 0a53 756d 6d61 7279 2e20 imit...Summary. │ │ │ │ -0003ee60: 2049 6620 796f 7520 6578 7065 6374 2074 If you expect t │ │ │ │ -0003ee70: 6861 7420 6669 6e64 696e 6720 6120 7375 hat finding a su │ │ │ │ -0003ee80: 626d 6174 7269 7820 6f72 2063 6f6d 7075 bmatrix or compu │ │ │ │ -0003ee90: 7469 6e67 2061 206d 696e 6f72 2069 730a ting a minor is. │ │ │ │ -0003eea0: 7265 6c61 7469 7665 6c79 2063 6f73 746c relatively costl │ │ │ │ -0003eeb0: 7920 6672 6f6d 2061 2074 696d 6520 7065 y from a time pe │ │ │ │ -0003eec0: 7273 7065 6374 6976 652c 2074 6865 6e20 rspective, then │ │ │ │ -0003eed0: 6974 206d 616b 6573 2073 656e 7365 2074 it makes sense t │ │ │ │ -0003eee0: 6f20 636f 6d70 7574 6520 7468 650a 636f o compute the.co │ │ │ │ -0003eef0: 6469 6d65 6e73 696f 6e20 6d6f 7265 2066 dimension more f │ │ │ │ -0003ef00: 7265 7175 656e 746c 792e 2020 4966 2063 requently. If c │ │ │ │ -0003ef10: 6f6d 7075 7469 6e67 2074 6865 2063 6f64 omputing the cod │ │ │ │ -0003ef20: 696d 656e 7369 6f6e 2069 7320 7265 6c61 imension is rela │ │ │ │ -0003ef30: 7469 7665 6c79 2063 6f73 746c 790a 7765 tively costly.we │ │ │ │ -0003ef40: 2072 6563 6f6d 6d65 6e64 2063 6f6d 7075 recommend compu │ │ │ │ -0003ef50: 7469 6e67 2074 6865 2063 6f64 696d 656e ting the codimen │ │ │ │ -0003ef60: 7369 6f6e 206c 6573 7320 6672 6571 7565 sion less freque │ │ │ │ -0003ef70: 6e74 6c79 2c20 6f72 2075 7369 6e67 2074 ntly, or using t │ │ │ │ -0003ef80: 6865 0a55 7365 4f6e 6c79 4661 7374 436f he.UseOnlyFastCo │ │ │ │ -0003ef90: 6469 6d20 3d3e 2074 7275 6520 7769 7468 dim => true with │ │ │ │ -0003efa0: 2061 2068 6967 6820 5061 6972 4c69 6d69 a high PairLimi │ │ │ │ -0003efb0: 742e 2020 466f 7220 6578 616d 706c 652c t. For example, │ │ │ │ -0003efc0: 2069 6620 7573 696e 670a 5374 7261 7465 if using.Strate │ │ │ │ -0003efd0: 6779 506f 696e 7473 2c20 7468 656e 2063 gyPoints, then c │ │ │ │ -0003efe0: 686f 6f73 696e 6720 6120 7375 626d 6174 hoosing a submat │ │ │ │ -0003eff0: 7269 7820 6361 6e20 6265 2071 7569 7465 rix can be quite │ │ │ │ -0003f000: 2073 6c6f 772c 2068 6f77 6576 6572 2065 slow, however e │ │ │ │ -0003f010: 6163 680a 7375 626d 6174 7269 7820 6973 ach.submatrix is │ │ │ │ -0003f020: 2076 6572 7920 6060 7661 6c75 6162 6c65 very ``valuable │ │ │ │ -0003f030: 2727 2c20 696e 2074 6861 7420 6164 6469 '', in that addi │ │ │ │ -0003f040: 6e67 2069 7420 746f 2074 6865 2069 6465 ng it to the ide │ │ │ │ -0003f050: 616c 206f 6620 6d69 6e6f 7273 2073 6f20 al of minors so │ │ │ │ -0003f060: 6661 720a 6973 2071 7569 7465 206c 696b far.is quite lik │ │ │ │ -0003f070: 656c 7920 746f 2072 6564 7563 6520 7468 ely to reduce th │ │ │ │ -0003f080: 6520 6469 6d65 6e73 696f 6e20 6f66 2074 e dimension of t │ │ │ │ -0003f090: 6865 2073 696e 6775 6c61 7220 6c6f 6375 he singular locu │ │ │ │ -0003f0a0: 732e 0a0a 4f6e 6520 6d61 7920 616c 736f s...One may also │ │ │ │ -0003f0b0: 2063 6861 6e67 6520 686f 7720 6d69 6e6f change how mino │ │ │ │ -0003f0c0: 7273 2028 6465 7465 726d 696e 616e 7473 rs (determinants │ │ │ │ -0003f0d0: 206f 6620 7468 6520 4a61 636f 6269 616e of the Jacobian │ │ │ │ -0003f0e0: 2073 7562 6d61 7472 6978 2920 6172 650a submatrix) are. │ │ │ │ -0003f0f0: 636f 6d70 7574 6564 2062 7920 7573 696e computed by usin │ │ │ │ -0003f100: 6720 7468 6520 2a6e 6f74 6520 4465 7453 g the *note DetS │ │ │ │ -0003f110: 7472 6174 6567 793a 2044 6574 5374 7261 trategy: DetStra │ │ │ │ -0003f120: 7465 6779 2c20 6f70 7469 6f6e 2e0a 0a53 tegy, option...S │ │ │ │ -0003f130: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ -0003f140: 0a0a 2020 2a20 2a6e 6f74 6520 7265 6775 .. * *note regu │ │ │ │ -0003f150: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003f160: 3a20 7265 6775 6c61 7249 6e43 6f64 696d : regularInCodim │ │ │ │ -0003f170: 656e 7369 6f6e 2c20 2d2d 2061 7474 656d ension, -- attem │ │ │ │ -0003f180: 7074 7320 746f 2073 686f 7720 7468 6174 pts to show that │ │ │ │ -0003f190: 0a20 2020 2074 6865 2072 696e 6720 6973 . the ring is │ │ │ │ -0003f1a0: 2072 6567 756c 6172 2069 6e20 636f 6469 regular in codi │ │ │ │ -0003f1b0: 6d65 6e73 696f 6e20 6e0a 2020 2a20 2a6e mension n. * *n │ │ │ │ -0003f1c0: 6f74 6520 4661 7374 4d69 6e6f 7273 5374 ote FastMinorsSt │ │ │ │ -0003f1d0: 7261 7465 6779 5475 746f 7269 616c 3a20 rategyTutorial: │ │ │ │ -0003f1e0: 4661 7374 4d69 6e6f 7273 5374 7261 7465 FastMinorsStrate │ │ │ │ -0003f1f0: 6779 5475 746f 7269 616c 2c20 2d2d 2048 gyTutorial, -- H │ │ │ │ -0003f200: 6f77 2074 6f20 7573 650a 2020 2020 616e ow to use. an │ │ │ │ -0003f210: 6420 636f 6e73 7472 7563 7420 7374 7261 d construct stra │ │ │ │ -0003f220: 7465 6769 6573 2066 6f72 2073 656c 6563 tegies for selec │ │ │ │ -0003f230: 7469 6e67 2073 7562 6d61 7472 6963 6573 ting submatrices │ │ │ │ -0003f240: 2069 6e20 7661 7269 6f75 7320 6675 6e63 in various func │ │ │ │ -0003f250: 7469 6f6e 730a 2020 2a20 2a6e 6f74 6520 tions. * *note │ │ │ │ -0003f260: 4465 7453 7472 6174 6567 793a 2044 6574 DetStrategy: Det │ │ │ │ -0003f270: 5374 7261 7465 6779 2c20 2d2d 2044 6574 Strategy, -- Det │ │ │ │ -0003f280: 5374 7261 7465 6779 2069 7320 6120 7374 Strategy is a st │ │ │ │ -0003f290: 7261 7465 6779 2066 6f72 2061 6c6c 6f77 rategy for allow │ │ │ │ -0003f2a0: 696e 670a 2020 2020 7468 6520 7573 6572 ing. the user │ │ │ │ -0003f2b0: 2074 6f20 6368 6f6f 7365 2068 6f77 2064 to choose how d │ │ │ │ -0003f2c0: 6574 6572 6d69 6e61 6e74 7320 286f 7220 eterminants (or │ │ │ │ -0003f2d0: 7261 6e6b 292c 2069 7320 636f 6d70 7574 rank), is comput │ │ │ │ -0003f2e0: 6564 0a0a 466f 7220 7468 6520 7072 6f67 ed..For the prog │ │ │ │ -0003f2f0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -0003f300: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -0003f310: 626a 6563 7420 2a6e 6f74 6520 5265 6775 bject *note Regu │ │ │ │ -0003f320: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -0003f330: 5475 746f 7269 616c 3a20 5265 6775 6c61 Tutorial: Regula │ │ │ │ -0003f340: 7249 6e43 6f64 696d 656e 7369 6f6e 5475 rInCodimensionTu │ │ │ │ -0003f350: 746f 7269 616c 2c20 6973 0a61 202a 6e6f torial, is.a *no │ │ │ │ -0003f360: 7465 2073 796d 626f 6c3a 2028 4d61 6361 te symbol: (Maca │ │ │ │ -0003f370: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ -0003f380: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0003edd0: 2d2b 0a0a 5468 6973 2063 616e 2062 6520 -+..This can be │ │ │ │ +0003ede0: 7573 6566 756c 2069 6620 7468 6520 6675 useful if the fu │ │ │ │ +0003edf0: 6e63 7469 6f6e 2069 7320 6861 6e67 696e nction is hangin │ │ │ │ +0003ee00: 6720 7768 656e 2074 7279 696e 6720 746f g when trying to │ │ │ │ +0003ee10: 2063 6f6d 7075 7465 2074 6865 0a64 696d compute the.dim │ │ │ │ +0003ee20: 656e 7369 6f6e 2c20 6275 7420 796f 7520 ension, but you │ │ │ │ +0003ee30: 6d61 7920 7769 7368 2069 6e63 7265 6173 may wish increas │ │ │ │ +0003ee40: 6520 5061 6972 4c69 6d69 742e 0a0a 5375 e PairLimit...Su │ │ │ │ +0003ee50: 6d6d 6172 792e 2020 4966 2079 6f75 2065 mmary. If you e │ │ │ │ +0003ee60: 7870 6563 7420 7468 6174 2066 696e 6469 xpect that findi │ │ │ │ +0003ee70: 6e67 2061 2073 7562 6d61 7472 6978 206f ng a submatrix o │ │ │ │ +0003ee80: 7220 636f 6d70 7574 696e 6720 6120 6d69 r computing a mi │ │ │ │ +0003ee90: 6e6f 7220 6973 0a72 656c 6174 6976 656c nor is.relativel │ │ │ │ +0003eea0: 7920 636f 7374 6c79 2066 726f 6d20 6120 y costly from a │ │ │ │ +0003eeb0: 7469 6d65 2070 6572 7370 6563 7469 7665 time perspective │ │ │ │ +0003eec0: 2c20 7468 656e 2069 7420 6d61 6b65 7320 , then it makes │ │ │ │ +0003eed0: 7365 6e73 6520 746f 2063 6f6d 7075 7465 sense to compute │ │ │ │ +0003eee0: 2074 6865 0a63 6f64 696d 656e 7369 6f6e the.codimension │ │ │ │ +0003eef0: 206d 6f72 6520 6672 6571 7565 6e74 6c79 more frequently │ │ │ │ +0003ef00: 2e20 2049 6620 636f 6d70 7574 696e 6720 . If computing │ │ │ │ +0003ef10: 7468 6520 636f 6469 6d65 6e73 696f 6e20 the codimension │ │ │ │ +0003ef20: 6973 2072 656c 6174 6976 656c 7920 636f is relatively co │ │ │ │ +0003ef30: 7374 6c79 0a77 6520 7265 636f 6d6d 656e stly.we recommen │ │ │ │ +0003ef40: 6420 636f 6d70 7574 696e 6720 7468 6520 d computing the │ │ │ │ +0003ef50: 636f 6469 6d65 6e73 696f 6e20 6c65 7373 codimension less │ │ │ │ +0003ef60: 2066 7265 7175 656e 746c 792c 206f 7220 frequently, or │ │ │ │ +0003ef70: 7573 696e 6720 7468 650a 5573 654f 6e6c using the.UseOnl │ │ │ │ +0003ef80: 7946 6173 7443 6f64 696d 203d 3e20 7472 yFastCodim => tr │ │ │ │ +0003ef90: 7565 2077 6974 6820 6120 6869 6768 2050 ue with a high P │ │ │ │ +0003efa0: 6169 724c 696d 6974 2e20 2046 6f72 2065 airLimit. For e │ │ │ │ +0003efb0: 7861 6d70 6c65 2c20 6966 2075 7369 6e67 xample, if using │ │ │ │ +0003efc0: 0a53 7472 6174 6567 7950 6f69 6e74 732c .StrategyPoints, │ │ │ │ +0003efd0: 2074 6865 6e20 6368 6f6f 7369 6e67 2061 then choosing a │ │ │ │ +0003efe0: 2073 7562 6d61 7472 6978 2063 616e 2062 submatrix can b │ │ │ │ +0003eff0: 6520 7175 6974 6520 736c 6f77 2c20 686f e quite slow, ho │ │ │ │ +0003f000: 7765 7665 7220 6561 6368 0a73 7562 6d61 wever each.subma │ │ │ │ +0003f010: 7472 6978 2069 7320 7665 7279 2060 6076 trix is very ``v │ │ │ │ +0003f020: 616c 7561 626c 6527 272c 2069 6e20 7468 aluable'', in th │ │ │ │ +0003f030: 6174 2061 6464 696e 6720 6974 2074 6f20 at adding it to │ │ │ │ +0003f040: 7468 6520 6964 6561 6c20 6f66 206d 696e the ideal of min │ │ │ │ +0003f050: 6f72 7320 736f 2066 6172 0a69 7320 7175 ors so far.is qu │ │ │ │ +0003f060: 6974 6520 6c69 6b65 6c79 2074 6f20 7265 ite likely to re │ │ │ │ +0003f070: 6475 6365 2074 6865 2064 696d 656e 7369 duce the dimensi │ │ │ │ +0003f080: 6f6e 206f 6620 7468 6520 7369 6e67 756c on of the singul │ │ │ │ +0003f090: 6172 206c 6f63 7573 2e0a 0a4f 6e65 206d ar locus...One m │ │ │ │ +0003f0a0: 6179 2061 6c73 6f20 6368 616e 6765 2068 ay also change h │ │ │ │ +0003f0b0: 6f77 206d 696e 6f72 7320 2864 6574 6572 ow minors (deter │ │ │ │ +0003f0c0: 6d69 6e61 6e74 7320 6f66 2074 6865 204a minants of the J │ │ │ │ +0003f0d0: 6163 6f62 6961 6e20 7375 626d 6174 7269 acobian submatri │ │ │ │ +0003f0e0: 7829 2061 7265 0a63 6f6d 7075 7465 6420 x) are.computed │ │ │ │ +0003f0f0: 6279 2075 7369 6e67 2074 6865 202a 6e6f by using the *no │ │ │ │ +0003f100: 7465 2044 6574 5374 7261 7465 6779 3a20 te DetStrategy: │ │ │ │ +0003f110: 4465 7453 7472 6174 6567 792c 206f 7074 DetStrategy, opt │ │ │ │ +0003f120: 696f 6e2e 0a0a 5365 6520 616c 736f 0a3d ion...See also.= │ │ │ │ +0003f130: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +0003f140: 7465 2072 6567 756c 6172 496e 436f 6469 te regularInCodi │ │ │ │ +0003f150: 6d65 6e73 696f 6e3a 2072 6567 756c 6172 mension: regular │ │ │ │ +0003f160: 496e 436f 6469 6d65 6e73 696f 6e2c 202d InCodimension, - │ │ │ │ +0003f170: 2d20 6174 7465 6d70 7473 2074 6f20 7368 - attempts to sh │ │ │ │ +0003f180: 6f77 2074 6861 740a 2020 2020 7468 6520 ow that. the │ │ │ │ +0003f190: 7269 6e67 2069 7320 7265 6775 6c61 7220 ring is regular │ │ │ │ +0003f1a0: 696e 2063 6f64 696d 656e 7369 6f6e 206e in codimension n │ │ │ │ +0003f1b0: 0a20 202a 202a 6e6f 7465 2046 6173 744d . * *note FastM │ │ │ │ +0003f1c0: 696e 6f72 7353 7472 6174 6567 7954 7574 inorsStrategyTut │ │ │ │ +0003f1d0: 6f72 6961 6c3a 2046 6173 744d 696e 6f72 orial: FastMinor │ │ │ │ +0003f1e0: 7353 7472 6174 6567 7954 7574 6f72 6961 sStrategyTutoria │ │ │ │ +0003f1f0: 6c2c 202d 2d20 486f 7720 746f 2075 7365 l, -- How to use │ │ │ │ +0003f200: 0a20 2020 2061 6e64 2063 6f6e 7374 7275 . and constru │ │ │ │ +0003f210: 6374 2073 7472 6174 6567 6965 7320 666f ct strategies fo │ │ │ │ +0003f220: 7220 7365 6c65 6374 696e 6720 7375 626d r selecting subm │ │ │ │ +0003f230: 6174 7269 6365 7320 696e 2076 6172 696f atrices in vario │ │ │ │ +0003f240: 7573 2066 756e 6374 696f 6e73 0a20 202a us functions. * │ │ │ │ +0003f250: 202a 6e6f 7465 2044 6574 5374 7261 7465 *note DetStrate │ │ │ │ +0003f260: 6779 3a20 4465 7453 7472 6174 6567 792c gy: DetStrategy, │ │ │ │ +0003f270: 202d 2d20 4465 7453 7472 6174 6567 7920 -- DetStrategy │ │ │ │ +0003f280: 6973 2061 2073 7472 6174 6567 7920 666f is a strategy fo │ │ │ │ +0003f290: 7220 616c 6c6f 7769 6e67 0a20 2020 2074 r allowing. t │ │ │ │ +0003f2a0: 6865 2075 7365 7220 746f 2063 686f 6f73 he user to choos │ │ │ │ +0003f2b0: 6520 686f 7720 6465 7465 726d 696e 616e e how determinan │ │ │ │ +0003f2c0: 7473 2028 6f72 2072 616e 6b29 2c20 6973 ts (or rank), is │ │ │ │ +0003f2d0: 2063 6f6d 7075 7465 640a 0a46 6f72 2074 computed..For t │ │ │ │ +0003f2e0: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +0003f2f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0003f300: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ +0003f310: 7465 2052 6567 756c 6172 496e 436f 6469 te RegularInCodi │ │ │ │ +0003f320: 6d65 6e73 696f 6e54 7574 6f72 6961 6c3a mensionTutorial: │ │ │ │ +0003f330: 2052 6567 756c 6172 496e 436f 6469 6d65 RegularInCodime │ │ │ │ +0003f340: 6e73 696f 6e54 7574 6f72 6961 6c2c 2069 nsionTutorial, i │ │ │ │ +0003f350: 730a 6120 2a6e 6f74 6520 7379 6d62 6f6c s.a *note symbol │ │ │ │ +0003f360: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0003f370: 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d Symbol,...------ │ │ │ │ +0003f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003f3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003f3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f3d0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0003f3e0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0003f3f0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0003f400: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0003f410: 6163 6175 6c61 7932 2d31 2e32 352e 3131 acaulay2-1.25.11 │ │ │ │ -0003f420: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0003f430: 2f70 6163 6b61 6765 732f 4661 7374 4d69 /packages/FastMi │ │ │ │ -0003f440: 6e6f 7273 2e0a 6d32 3a31 3538 383a 302e nors..m2:1588:0. │ │ │ │ -0003f450: 0a1f 0a46 696c 653a 2046 6173 744d 696e ...File: FastMin │ │ │ │ -0003f460: 6f72 732e 696e 666f 2c20 4e6f 6465 3a20 ors.info, Node: │ │ │ │ -0003f470: 7265 6f72 6465 7250 6f6c 796e 6f6d 6961 reorderPolynomia │ │ │ │ -0003f480: 6c52 696e 672c 204e 6578 743a 2053 7472 lRing, Next: Str │ │ │ │ -0003f490: 6174 6567 7944 6566 6175 6c74 2c20 5072 ategyDefault, Pr │ │ │ │ -0003f4a0: 6576 3a20 5265 6775 6c61 7249 6e43 6f64 ev: RegularInCod │ │ │ │ -0003f4b0: 696d 656e 7369 6f6e 5475 746f 7269 616c imensionTutorial │ │ │ │ -0003f4c0: 2c20 5570 3a20 546f 700a 0a72 656f 7264 , Up: Top..reord │ │ │ │ -0003f4d0: 6572 506f 6c79 6e6f 6d69 616c 5269 6e67 erPolynomialRing │ │ │ │ -0003f4e0: 202d 2d20 7072 6f64 7563 6573 2061 6e20 -- produces an │ │ │ │ -0003f4f0: 6973 6f6d 6f72 7068 6963 2070 6f6c 796e isomorphic polyn │ │ │ │ -0003f500: 6f6d 6961 6c20 7269 6e67 2077 6974 6820 omial ring with │ │ │ │ -0003f510: 6120 6469 6666 6572 656e 742c 2072 616e a different, ran │ │ │ │ -0003f520: 646f 6d69 7a65 642c 206d 6f6e 6f6d 6961 domized, monomia │ │ │ │ -0003f530: 6c20 6f72 6465 720a 2a2a 2a2a 2a2a 2a2a l order.******** │ │ │ │ +0003f3c0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +0003f3d0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +0003f3e0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +0003f3f0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +0003f400: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +0003f410: 312e 3235 2e31 312b 6473 2f4d 322f 4d61 1.25.11+ds/M2/Ma │ │ │ │ +0003f420: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +0003f430: 2f46 6173 744d 696e 6f72 732e 0a6d 323a /FastMinors..m2: │ │ │ │ +0003f440: 3135 3838 3a30 2e0a 1f0a 4669 6c65 3a20 1588:0....File: │ │ │ │ +0003f450: 4661 7374 4d69 6e6f 7273 2e69 6e66 6f2c FastMinors.info, │ │ │ │ +0003f460: 204e 6f64 653a 2072 656f 7264 6572 506f Node: reorderPo │ │ │ │ +0003f470: 6c79 6e6f 6d69 616c 5269 6e67 2c20 4e65 lynomialRing, Ne │ │ │ │ +0003f480: 7874 3a20 5374 7261 7465 6779 4465 6661 xt: StrategyDefa │ │ │ │ +0003f490: 756c 742c 2050 7265 763a 2052 6567 756c ult, Prev: Regul │ │ │ │ +0003f4a0: 6172 496e 436f 6469 6d65 6e73 696f 6e54 arInCodimensionT │ │ │ │ +0003f4b0: 7574 6f72 6961 6c2c 2055 703a 2054 6f70 utorial, Up: Top │ │ │ │ +0003f4c0: 0a0a 7265 6f72 6465 7250 6f6c 796e 6f6d ..reorderPolynom │ │ │ │ +0003f4d0: 6961 6c52 696e 6720 2d2d 2070 726f 6475 ialRing -- produ │ │ │ │ +0003f4e0: 6365 7320 616e 2069 736f 6d6f 7270 6869 ces an isomorphi │ │ │ │ +0003f4f0: 6320 706f 6c79 6e6f 6d69 616c 2072 696e c polynomial rin │ │ │ │ +0003f500: 6720 7769 7468 2061 2064 6966 6665 7265 g with a differe │ │ │ │ +0003f510: 6e74 2c20 7261 6e64 6f6d 697a 6564 2c20 nt, randomized, │ │ │ │ +0003f520: 6d6f 6e6f 6d69 616c 206f 7264 6572 0a2a monomial order.* │ │ │ │ +0003f530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003f540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003f550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003f560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003f570: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003f580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003f590: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003f5a0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -0003f5b0: 200a 2020 2020 2020 2020 5231 203d 2072 . R1 = r │ │ │ │ -0003f5c0: 656f 7264 6572 506f 6c79 6e6f 6d69 616c eorderPolynomial │ │ │ │ -0003f5d0: 5269 6e67 286f 7264 6572 5479 7065 2c20 Ring(orderType, │ │ │ │ -0003f5e0: 5229 0a20 202a 2049 6e70 7574 733a 0a20 R). * Inputs:. │ │ │ │ -0003f5f0: 2020 2020 202a 2052 2c20 6120 2a6e 6f74 * R, a *not │ │ │ │ -0003f600: 6520 7269 6e67 3a20 284d 6163 6175 6c61 e ring: (Macaula │ │ │ │ -0003f610: 7932 446f 6329 5269 6e67 2c2c 2061 2070 y2Doc)Ring,, a p │ │ │ │ -0003f620: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a20 olynomial ring. │ │ │ │ -0003f630: 2020 2020 202a 206f 7264 6572 5479 7065 * orderType │ │ │ │ -0003f640: 2c20 6120 2a6e 6f74 6520 7379 6d62 6f6c , a *note symbol │ │ │ │ -0003f650: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0003f660: 5379 6d62 6f6c 2c2c 2061 2076 616c 6964 Symbol,, a valid │ │ │ │ -0003f670: 206d 6f6e 6f6d 6961 6c0a 2020 2020 2020 monomial. │ │ │ │ -0003f680: 2020 6f72 6465 722c 2073 7563 6820 6173 order, such as │ │ │ │ -0003f690: 2047 5265 764c 6578 0a20 202a 204f 7574 GRevLex. * Out │ │ │ │ -0003f6a0: 7075 7473 3a0a 2020 2020 2020 2a20 532c puts:. * S, │ │ │ │ -0003f6b0: 2061 202a 6e6f 7465 2072 696e 673a 2028 a *note ring: ( │ │ │ │ -0003f6c0: 4d61 6361 756c 6179 3244 6f63 2952 696e Macaulay2Doc)Rin │ │ │ │ -0003f6d0: 672c 2c20 6120 706f 6c79 6e6f 6d69 616c g,, a polynomial │ │ │ │ -0003f6e0: 2072 696e 6720 7769 7468 2061 206e 6577 ring with a new │ │ │ │ -0003f6f0: 0a20 2020 2020 2020 2072 616e 646f 6d20 . random │ │ │ │ -0003f700: 6d6f 6e6f 6d69 616c 206f 7264 6572 0a0a monomial order.. │ │ │ │ -0003f710: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -0003f720: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6675 =======..This fu │ │ │ │ -0003f730: 6e63 7469 6f6e 2074 616b 6573 2061 2070 nction takes a p │ │ │ │ -0003f740: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2061 olynomial ring a │ │ │ │ -0003f750: 6e64 2070 726f 6475 6365 7320 6120 6e65 nd produces a ne │ │ │ │ -0003f760: 7720 706f 6c79 6e6f 6d69 616c 2072 696e w polynomial rin │ │ │ │ -0003f770: 6720 7769 7468 0a4d 6f6e 6f6d 6961 6c4f g with.MonomialO │ │ │ │ -0003f780: 7264 6572 206f 6620 7479 7065 206f 7264 rder of type ord │ │ │ │ -0003f790: 6572 5479 7065 2e20 5468 6520 6f72 6465 erType. The orde │ │ │ │ -0003f7a0: 7220 6f66 2074 6865 2076 6172 6961 626c r of the variabl │ │ │ │ -0003f7b0: 6573 2069 7320 7261 6e64 6f6d 697a 6564 es is randomized │ │ │ │ -0003f7c0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ +0003f590: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0003f5a0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0003f5b0: 2052 3120 3d20 7265 6f72 6465 7250 6f6c R1 = reorderPol │ │ │ │ +0003f5c0: 796e 6f6d 6961 6c52 696e 6728 6f72 6465 ynomialRing(orde │ │ │ │ +0003f5d0: 7254 7970 652c 2052 290a 2020 2a20 496e rType, R). * In │ │ │ │ +0003f5e0: 7075 7473 3a0a 2020 2020 2020 2a20 522c puts:. * R, │ │ │ │ +0003f5f0: 2061 202a 6e6f 7465 2072 696e 673a 2028 a *note ring: ( │ │ │ │ +0003f600: 4d61 6361 756c 6179 3244 6f63 2952 696e Macaulay2Doc)Rin │ │ │ │ +0003f610: 672c 2c20 6120 706f 6c79 6e6f 6d69 616c g,, a polynomial │ │ │ │ +0003f620: 2072 696e 670a 2020 2020 2020 2a20 6f72 ring. * or │ │ │ │ +0003f630: 6465 7254 7970 652c 2061 202a 6e6f 7465 derType, a *note │ │ │ │ +0003f640: 2073 796d 626f 6c3a 2028 4d61 6361 756c symbol: (Macaul │ │ │ │ +0003f650: 6179 3244 6f63 2953 796d 626f 6c2c 2c20 ay2Doc)Symbol,, │ │ │ │ +0003f660: 6120 7661 6c69 6420 6d6f 6e6f 6d69 616c a valid monomial │ │ │ │ +0003f670: 0a20 2020 2020 2020 206f 7264 6572 2c20 . order, │ │ │ │ +0003f680: 7375 6368 2061 7320 4752 6576 4c65 780a such as GRevLex. │ │ │ │ +0003f690: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +0003f6a0: 2020 202a 2053 2c20 6120 2a6e 6f74 6520 * S, a *note │ │ │ │ +0003f6b0: 7269 6e67 3a20 284d 6163 6175 6c61 7932 ring: (Macaulay2 │ │ │ │ +0003f6c0: 446f 6329 5269 6e67 2c2c 2061 2070 6f6c Doc)Ring,, a pol │ │ │ │ +0003f6d0: 796e 6f6d 6961 6c20 7269 6e67 2077 6974 ynomial ring wit │ │ │ │ +0003f6e0: 6820 6120 6e65 770a 2020 2020 2020 2020 h a new. │ │ │ │ +0003f6f0: 7261 6e64 6f6d 206d 6f6e 6f6d 6961 6c20 random monomial │ │ │ │ +0003f700: 6f72 6465 720a 0a44 6573 6372 6970 7469 order..Descripti │ │ │ │ +0003f710: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0003f720: 5468 6973 2066 756e 6374 696f 6e20 7461 This function ta │ │ │ │ +0003f730: 6b65 7320 6120 706f 6c79 6e6f 6d69 616c kes a polynomial │ │ │ │ +0003f740: 2072 696e 6720 616e 6420 7072 6f64 7563 ring and produc │ │ │ │ +0003f750: 6573 2061 206e 6577 2070 6f6c 796e 6f6d es a new polynom │ │ │ │ +0003f760: 6961 6c20 7269 6e67 2077 6974 680a 4d6f ial ring with.Mo │ │ │ │ +0003f770: 6e6f 6d69 616c 4f72 6465 7220 6f66 2074 nomialOrder of t │ │ │ │ +0003f780: 7970 6520 6f72 6465 7254 7970 652e 2054 ype orderType. T │ │ │ │ +0003f790: 6865 206f 7264 6572 206f 6620 7468 6520 he order of the │ │ │ │ +0003f7a0: 7661 7269 6162 6c65 7320 6973 2072 616e variables is ran │ │ │ │ +0003f7b0: 646f 6d69 7a65 642e 0a0a 2b2d 2d2d 2d2d domized...+----- │ │ │ │ +0003f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0003f7f0: 7c69 3120 3a20 5220 3d20 5151 5b78 2c79 |i1 : R = QQ[x,y │ │ │ │ -0003f800: 2c7a 2c77 5d3b 2020 2020 2020 2020 2020 ,z,w]; │ │ │ │ -0003f810: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0003f7e0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d -----+.|i1 : R = │ │ │ │ +0003f7f0: 2051 515b 782c 792c 7a2c 775d 3b20 2020 QQ[x,y,z,w]; │ │ │ │ +0003f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f810: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0003f820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f840: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -0003f850: 7820 3e20 7920 616e 6420 7920 3e20 7a20 x > y and y > z │ │ │ │ -0003f860: 616e 6420 7a20 3e20 7720 2020 2020 2020 and z > w │ │ │ │ -0003f870: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003f840: 0a7c 6932 203a 2078 203e 2079 2061 6e64 .|i2 : x > y and │ │ │ │ +0003f850: 2079 203e 207a 2061 6e64 207a 203e 2077 y > z and z > w │ │ │ │ +0003f860: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f8a0: 2020 7c0a 7c6f 3220 3d20 7472 7565 2020 |.|o2 = true │ │ │ │ +0003f890: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ +0003f8a0: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ 0003f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f8c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003f8d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003f8c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0003f8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003f900: 3320 3a20 7573 6520 7265 6f72 6465 7250 3 : use reorderP │ │ │ │ -0003f910: 6f6c 796e 6f6d 6961 6c52 696e 6728 4752 olynomialRing(GR │ │ │ │ -0003f920: 6576 4c65 782c 2052 297c 0a7c 2020 2020 evLex, R)|.| │ │ │ │ +0003f8f0: 2d2d 2d2b 0a7c 6933 203a 2075 7365 2072 ---+.|i3 : use r │ │ │ │ +0003f900: 656f 7264 6572 506f 6c79 6e6f 6d69 616c eorderPolynomial │ │ │ │ +0003f910: 5269 6e67 2847 5265 764c 6578 2c20 5229 Ring(GRevLex, R) │ │ │ │ +0003f920: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003f930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f950: 2020 2020 2020 7c0a 7c6f 3320 3d20 5151 |.|o3 = QQ │ │ │ │ -0003f960: 5b7a 2c20 772e 2e79 5d20 2020 2020 2020 [z, w..y] │ │ │ │ -0003f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f980: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003f940: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003f950: 6f33 203d 2051 515b 7a2c 2077 2e2e 795d o3 = QQ[z, w..y] │ │ │ │ +0003f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f970: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f9b0: 7c0a 7c6f 3320 3a20 506f 6c79 6e6f 6d69 |.|o3 : Polynomi │ │ │ │ -0003f9c0: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ -0003f9d0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0003f9a0: 2020 2020 2020 207c 0a7c 6f33 203a 2050 |.|o3 : P │ │ │ │ +0003f9b0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +0003f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f9d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0003f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ -0003fa10: 3a20 7820 3e20 7920 2020 2020 2020 2020 : x > y │ │ │ │ -0003fa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003fa30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0003fa00: 2d2b 0a7c 6934 203a 2078 203e 2079 2020 -+.|i4 : x > y │ │ │ │ +0003fa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fa20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0003fa30: 7c20 2020 2020 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---------------- │ │ │ │ -0003fb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003fb70: 2d2d 2b0a 7c69 3620 3a20 7a20 3e20 7720 --+.|i6 : z > w │ │ │ │ +0003fb60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ +0003fb70: 207a 203e 2077 2020 2020 2020 2020 2020 z > w │ │ │ │ 0003fb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003fb90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003fba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003fb90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0003fba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003fbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003fbd0: 3620 3d20 7472 7565 2020 2020 2020 2020 6 = true │ │ │ │ +0003fbc0: 2020 207c 0a7c 6f36 203d 2074 7275 6520 |.|o6 = true │ │ │ │ +0003fbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2272 656f 7264 6572 506f 6c79 * "reorderPoly │ │ │ │ -0003fc80: 6e6f 6d69 616c 5269 6e67 2853 796d 626f nomialRing(Symbo │ │ │ │ -0003fc90: 6c2c 5269 6e67 2922 0a0a 466f 7220 7468 l,Ring)"..For th │ │ │ │ -0003fca0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0003fcb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0003fcc0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0003fcd0: 6520 7265 6f72 6465 7250 6f6c 796e 6f6d e reorderPolynom │ │ │ │ -0003fce0: 6961 6c52 696e 673a 2072 656f 7264 6572 ialRing: reorder │ │ │ │ -0003fcf0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20 PolynomialRing, │ │ │ │ -0003fd00: 6973 2061 202a 6e6f 7465 0a6d 6574 686f is a *note.metho │ │ │ │ -0003fd10: 6420 6675 6e63 7469 6f6e 2077 6974 6820 d function with │ │ │ │ -0003fd20: 6f70 7469 6f6e 733a 2028 4d61 6361 756c options: (Macaul │ │ │ │ -0003fd30: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ -0003fd40: 6374 696f 6e57 6974 684f 7074 696f 6e73 ctionWithOptions │ │ │ │ -0003fd50: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +0003fc60: 3d3d 3d3d 3d0a 0a20 202a 2022 7265 6f72 =====.. * "reor │ │ │ │ +0003fc70: 6465 7250 6f6c 796e 6f6d 6961 6c52 696e derPolynomialRin │ │ │ │ +0003fc80: 6728 5379 6d62 6f6c 2c52 696e 6729 220a g(Symbol,Ring)". │ │ │ │ +0003fc90: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0003fca0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0003fcb0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0003fcc0: 6374 202a 6e6f 7465 2072 656f 7264 6572 ct *note reorder │ │ │ │ +0003fcd0: 506f 6c79 6e6f 6d69 616c 5269 6e67 3a20 PolynomialRing: │ │ │ │ +0003fce0: 7265 6f72 6465 7250 6f6c 796e 6f6d 6961 reorderPolynomia │ │ │ │ +0003fcf0: 6c52 696e 672c 2069 7320 6120 2a6e 6f74 lRing, is a *not │ │ │ │ +0003fd00: 650a 6d65 7468 6f64 2066 756e 6374 696f e.method functio │ │ │ │ +0003fd10: 6e20 7769 7468 206f 7074 696f 6e73 3a20 n with options: │ │ │ │ +0003fd20: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +0003fd30: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ +0003fd40: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +0003fd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003fd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003fd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003fd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003fd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003fda0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0003fdb0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0003fdc0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0003fdd0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0003fde0: 6d61 6361 756c 6179 322d 312e 3235 2e31 macaulay2-1.25.1 │ │ │ │ -0003fdf0: 312b 6473 2f4d 322f 4d61 6361 756c 6179 1+ds/M2/Macaulay │ │ │ │ -0003fe00: 322f 7061 636b 6167 6573 2f46 6173 744d 2/packages/FastM │ │ │ │ -0003fe10: 696e 6f72 732e 0a6d 323a 3231 3638 3a30 inors..m2:2168:0 │ │ │ │ -0003fe20: 2e0a 1f0a 4669 6c65 3a20 4661 7374 4d69 ....File: FastMi │ │ │ │ -0003fe30: 6e6f 7273 2e69 6e66 6f2c 204e 6f64 653a nors.info, Node: │ │ │ │ -0003fe40: 2053 7472 6174 6567 7944 6566 6175 6c74 StrategyDefault │ │ │ │ -0003fe50: 2c20 5072 6576 3a20 7265 6f72 6465 7250 , Prev: reorderP │ │ │ │ -0003fe60: 6f6c 796e 6f6d 6961 6c52 696e 672c 2055 olynomialRing, U │ │ │ │ -0003fe70: 703a 2054 6f70 0a0a 5374 7261 7465 6779 p: Top..Strategy │ │ │ │ -0003fe80: 4465 6661 756c 7420 2d2d 2073 7472 6174 Default -- strat │ │ │ │ -0003fe90: 6567 6965 7320 666f 7220 6368 6f6f 7369 egies for choosi │ │ │ │ -0003fea0: 6e67 2073 7562 6d61 7472 6963 6573 0a2a ng submatrices.* │ │ │ │ +0003fd90: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ 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7472 6174 6567 7944 6566 6175 6c74 202d trategyDefault - │ │ │ │ +0003fe80: 2d20 7374 7261 7465 6769 6573 2066 6f72 - strategies for │ │ │ │ +0003fe90: 2063 686f 6f73 696e 6720 7375 626d 6174 choosing submat │ │ │ │ +0003fea0: 7269 6365 730a 2a2a 2a2a 2a2a 2a2a 2a2a rices.********** │ │ │ │ 0003feb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003fec0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003fed0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003fee0: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ -0003fef0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -0003ff00: 4d61 6e79 206f 6620 7468 6520 636f 7265 Many of the core │ │ │ │ -0003ff10: 2066 756e 6374 696f 6e73 206f 6620 7468 functions of th │ │ │ │ -0003ff20: 6973 2070 6163 6b61 6765 2061 6c6c 6f77 is package allow │ │ │ │ -0003ff30: 2074 6865 2075 7365 7220 746f 2066 696e the user to fin │ │ │ │ -0003ff40: 6520 7475 6e65 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Different s │ │ │ │ -0003ff80: 7472 6174 6567 6965 7320 7969 656c 6420 trategies yield │ │ │ │ -0003ff90: 6d61 726b 6564 6c79 0a64 6966 6665 7265 markedly.differe │ │ │ │ -0003ffa0: 6e74 2070 6572 666f 726d 616e 6365 206f nt performance o │ │ │ │ -0003ffb0: 7220 7265 7375 6c74 7320 6f6e 2076 6172 r results on var │ │ │ │ -0003ffc0: 696f 7573 2065 7861 6d70 6c65 732e 2054 ious examples. T │ │ │ │ -0003ffd0: 6865 7365 2061 7265 2063 6f6e 7472 6f6c hese are control │ │ │ │ -0003ffe0: 6c65 6420 6279 0a73 7065 6369 6679 696e led by.specifyin │ │ │ │ -0003fff0: 6720 6120 2053 7472 6174 6567 7920 3d3e g a Strategy => │ │ │ │ -00040000: 2020 6f70 7469 6f6e 2c20 706f 696e 7469 option, pointi │ │ │ │ -00040010: 6e67 2074 6f20 6120 2048 6173 6854 6162 ng to a HashTab │ │ │ │ -00040020: 6c65 7768 6963 6820 7370 6563 6966 6965 lewhich specifie │ │ │ │ -00040030: 730a 7365 7665 7261 6c20 7374 7261 7465 s.several strate │ │ │ │ -00040040: 6769 6573 2073 686f 756c 6420 6265 2075 gies should be u │ │ │ │ -00040050: 7365 6420 7369 6d75 6c74 616e 656f 7573 sed simultaneous │ │ │ │ -00040060: 6c79 2c20 6f72 2074 6f20 6120 7379 6d62 ly, or to a symb │ │ │ │ -00040070: 6f6c 2073 6179 696e 6720 7765 0a73 686f ol saying we.sho │ │ │ │ -00040080: 756c 6420 7573 6520 6f6e 6c79 2061 2073 uld use only a s │ │ │ │ -00040090: 696e 676c 6520 7374 7261 7465 6779 2e20 ingle strategy. │ │ │ │ -000400a0: 2046 6f72 2061 206d 6f72 6520 6465 7461 For a more deta │ │ │ │ -000400b0: 696c 6564 206c 6f6f 6b20 6174 2074 6869 iled look at thi │ │ │ │ -000400c0: 7320 696e 2061 6e0a 6578 616d 706c 6520 s in an.example │ │ │ │ -000400d0: 706c 6561 7365 2073 6565 202a 6e6f 7465 please see *note │ │ │ │ -000400e0: 2046 6173 744d 696e 6f72 7353 7472 6174 FastMinorsStrat │ │ │ │ -000400f0: 6567 7954 7574 6f72 6961 6c3a 0a46 6173 egyTutorial:.Fas │ │ │ │ -00040100: 744d 696e 6f72 7353 7472 6174 6567 7954 tMinorsStrategyT │ │ │ │ -00040110: 7574 6f72 6961 6c2c 4265 666f 7265 2064 utorial,Before d │ │ │ │ -00040120: 6573 6372 6962 696e 6720 7468 6520 6176 escribing the av │ │ │ │ -00040130: 6169 6c61 626c 6520 7374 7261 7465 6769 ailable strategi │ │ │ │ -00040140: 6573 2c20 7765 2062 6567 696e 0a62 7920 es, we begin.by │ │ │ │ -00040150: 726f 7567 686c 7920 6f75 746c 696e 696e roughly outlinin │ │ │ │ -00040160: 6720 7468 6520 6469 6666 6572 656e 7420 g the different │ │ │ │ -00040170: 6170 7072 6f61 6368 6573 2e0a 2020 2a20 approaches.. * │ │ │ │ -00040180: 4865 7572 6973 7469 6320 7375 626d 6174 Heuristic submat │ │ │ │ -00040190: 7269 7820 7365 6c65 6374 696f 6e3a 2049 rix selection: I │ │ │ │ -000401a0: 6e20 7468 6973 2063 6173 652c 2061 2073 n this case, a s │ │ │ │ -000401b0: 7562 6d61 7472 6978 2069 7320 6368 6f73 ubmatrix is chos │ │ │ │ -000401c0: 656e 2076 6961 2061 0a20 2020 2067 7265 en via a. gre │ │ │ │ -000401d0: 6564 7920 616c 676f 7269 7468 6d2c 206c edy algorithm, l │ │ │ │ -000401e0: 6f6f 6b69 6e67 2066 6f72 2061 2073 7562 ooking for a sub │ │ │ │ -000401f0: 6d61 7472 6978 2077 6974 6820 736d 616c matrix with smal │ │ │ │ -00040200: 6c65 7374 2028 6f72 206c 6172 6765 7374 lest (or largest │ │ │ │ -00040210: 2920 6465 6772 6565 0a20 2020 2077 6974 ) degree. wit │ │ │ │ -00040220: 6820 7265 7370 6563 7420 746f 2061 2072 h respect to a r │ │ │ │ -00040230: 616e 646f 6d20 6d6f 6e6f 6d69 616c 206f andom monomial o │ │ │ │ -00040240: 7264 6572 2e0a 2020 2a20 5375 626d 6174 rder.. * Submat │ │ │ │ -00040250: 7269 7820 7365 6c65 6374 696f 6e20 7669 rix selection vi │ │ │ │ -00040260: 6120 7261 7469 6f6e 616c 2061 6e64 2067 a rational and g │ │ │ │ -00040270: 656f 6d65 7472 6963 2070 6f69 6e74 733a eometric points: │ │ │ │ -00040280: 2048 6572 6520 6120 7261 7469 6f6e 616c Here a rational │ │ │ │ -00040290: 206f 720a 2020 2020 6765 6f6d 6574 7269 or. geometri │ │ │ │ -000402a0: 6320 706f 696e 7420 6973 2066 6f75 6e64 c point is found │ │ │ │ -000402b0: 2077 6865 7265 2061 2067 6976 656e 2069 where a given i │ │ │ │ -000402c0: 6465 616c 2076 616e 6973 6865 732e 2020 deal vanishes. │ │ │ │ -000402d0: 5468 6174 2070 6f69 6e74 2069 730a 2020 That point is. │ │ │ │ -000402e0: 2020 706c 7567 6765 6420 696e 746f 2074 plugged into t │ │ │ │ -000402f0: 6865 206d 6174 7269 7820 616e 6420 6120 he matrix and a │ │ │ │ -00040300: 7375 626d 6174 7269 7820 6f66 2066 756c submatrix of ful │ │ │ │ -00040310: 6c20 7261 6e6b 2069 7320 6964 656e 7469 l rank is identi │ │ │ │ -00040320: 6669 6564 2e20 2020 5468 6973 0a20 2020 fied. This. │ │ │ │ -00040330: 2061 7070 726f 6163 6820 6375 7272 656e approach curren │ │ │ │ -00040340: 746c 7920 6f6e 6c79 2077 6f72 6b73 206f tly only works o │ │ │ │ -00040350: 7665 7220 6120 6669 6e69 7465 2066 6965 ver a finite fie │ │ │ │ -00040360: 6c64 2061 6e64 2069 7320 6163 636f 6d70 ld and is accomp │ │ │ │ -00040370: 6c69 7368 6564 2077 6974 680a 2020 2020 lished with. │ │ │ │ -00040380: 7468 6520 6865 6c70 206f 6620 7468 6520 the help of the │ │ │ │ -00040390: 7061 636b 6167 6520 2a6e 6f74 6520 5261 package *note Ra │ │ │ │ -000403a0: 6e64 6f6d 506f 696e 7473 3a20 2852 616e ndomPoints: (Ran │ │ │ │ -000403b0: 646f 6d50 6f69 6e74 7329 546f 702c 2e0a domPoints)Top,.. │ │ │ │ -000403c0: 2020 2a20 5261 6e64 6f6d 2073 7562 6d61 * Random subma │ │ │ │ -000403d0: 7472 6978 2073 656c 6563 7469 6f6e 3a20 trix selection: │ │ │ │ -000403e0: 5468 6973 2065 6974 6865 7220 6368 6f6f This either choo │ │ │ │ -000403f0: 7365 7320 6120 636f 6d70 6c65 7465 6c79 ses a completely │ │ │ │ -00040400: 2072 616e 646f 6d0a 2020 2020 7375 626d random. subm │ │ │ │ -00040410: 6174 7269 782c 206f 7220 6120 7375 626d atrix, or a subm │ │ │ │ -00040420: 6174 7269 7820 7768 6963 6820 6861 7320 atrix which has │ │ │ │ -00040430: 6e6f 207a 6572 6f20 636f 6c75 6d6e 7320 no zero columns │ │ │ │ -00040440: 6f72 2072 6f77 732e 0a54 6865 7265 2077 or rows..There w │ │ │ │ -00040450: 6520 6869 6768 6c69 6768 7420 6669 7665 e highlight five │ │ │ │ -00040460: 2070 7265 2d70 726f 6772 616d 6d65 6420 pre-programmed │ │ │ │ -00040470: 7374 7261 7465 6769 6573 2070 726f 7669 strategies provi │ │ │ │ -00040480: 6465 6420 746f 2074 6865 2075 7365 722e ded to the user. │ │ │ │ -00040490: 0a20 202a 2053 7472 6174 6567 7944 6566 . * StrategyDef │ │ │ │ -000404a0: 6175 6c74 3a20 7468 6973 2075 7365 7320 ault: this uses │ │ │ │ -000404b0: 6120 6d69 7820 6f66 2068 6575 7269 7374 a mix of heurist │ │ │ │ -000404c0: 6963 7320 616e 6420 7261 6e64 6f6d 2073 ics and random s │ │ │ │ -000404d0: 7562 6d61 7472 6963 6573 2e0a 2020 2a20 ubmatrices.. * │ │ │ │ -000404e0: 5374 7261 7465 6779 5261 6e64 6f6d 3a20 StrategyRandom: │ │ │ │ -000404f0: 7468 6973 2075 7365 7320 7075 7265 6c79 this uses purely │ │ │ │ -00040500: 2072 616e 646f 6d20 7375 626d 6174 7269 random submatri │ │ │ │ -00040510: 6365 732e 0a20 202a 2053 7472 6174 6567 ces.. * Strateg │ │ │ │ -00040520: 7944 6566 6175 6c74 4e6f 6e52 616e 646f yDefaultNonRando │ │ │ │ -00040530: 6d3a 2074 6869 7320 7573 6573 2061 206d m: this uses a m │ │ │ │ -00040540: 6978 206f 6620 6865 7572 6973 7469 6373 ix of heuristics │ │ │ │ -00040550: 2062 7574 206e 6f20 7261 6e64 6f6d 0a20 but no random. │ │ │ │ -00040560: 2020 2073 7562 6d61 7472 6963 6573 2e0a submatrices.. │ │ │ │ -00040570: 2020 2a20 5374 7261 7465 6779 506f 696e * StrategyPoin │ │ │ │ -00040580: 7473 3a20 7468 6973 206f 6e6c 7920 7573 ts: this only us │ │ │ │ -00040590: 6573 2072 6174 696f 6e61 6c20 2f20 6765 es rational / ge │ │ │ │ -000405a0: 6f6d 6574 7269 6320 706f 696e 7473 2074 ometric points t │ │ │ │ -000405b0: 6f20 6669 6e64 0a20 2020 2073 7562 6d61 o find. subma │ │ │ │ -000405c0: 7472 6963 6573 2e0a 2020 2a20 5374 7261 trices.. * Stra │ │ │ │ -000405d0: 7465 6779 4465 6661 756c 7457 6974 6850 tegyDefaultWithP │ │ │ │ -000405e0: 6f69 6e74 733a 2074 6869 7320 7573 6573 oints: this uses │ │ │ │ -000405f0: 2061 206d 6978 206f 6620 6865 7572 6973 a mix of heuris │ │ │ │ -00040600: 7469 6373 2061 6e64 2073 7562 6d61 7472 tics and submatr │ │ │ │ -00040610: 6963 6573 0a20 2020 2063 686f 7365 6e20 ices. chosen │ │ │ │ -00040620: 7769 7468 2072 6174 696f 6e61 6c20 616e with rational an │ │ │ │ -00040630: 6420 6765 6f6d 6574 7269 6320 706f 696e d geometric poin │ │ │ │ -00040640: 7473 2e0a 4265 6c6f 7720 7468 6520 6465 ts..Below the de │ │ │ │ -00040650: 7461 696c 7320 6f66 2068 6f77 2074 6865 tails of how the │ │ │ │ -00040660: 7365 2073 7472 6174 6567 6965 7320 6172 se strategies ar │ │ │ │ -00040670: 6520 636f 6e73 7472 7563 7465 6420 7769 e constructed wi │ │ │ │ -00040680: 6c6c 2062 6520 6465 7461 696c 6564 0a62 ll be detailed.b │ │ │ │ -00040690: 656c 6f77 2e20 2042 7574 2066 6972 7374 elow. But first │ │ │ │ -000406a0: 2c20 7765 2070 726f 7669 6465 2061 6e20 , we provide an │ │ │ │ -000406b0: 6578 616d 706c 6520 7368 6f77 696e 6720 example showing │ │ │ │ -000406c0: 7468 6174 2074 6865 7365 2073 7472 6174 that these strat │ │ │ │ -000406d0: 6567 6965 7320 6361 6e0a 7065 7266 6f72 egies can.perfor │ │ │ │ -000406e0: 6d20 7175 6974 6520 6469 6666 6572 656e m quite differen │ │ │ │ -000406f0: 746c 792e 2020 5468 6520 666f 6c6c 6f77 tly. The follow │ │ │ │ -00040700: 696e 6720 6973 2074 6865 2063 6f6e 6520 ing is the cone │ │ │ │ -00040710: 6f76 6572 2074 6865 2070 726f 6475 6374 over the product │ │ │ │ -00040720: 206f 6620 7477 6f0a 656c 6c69 7074 6963 of two.elliptic │ │ │ │ -00040730: 2063 7572 7665 732e 2020 5765 2076 6572 curves. We ver │ │ │ │ -00040740: 6966 7920 7468 6174 2074 6869 7320 7269 ify that this ri │ │ │ │ -00040750: 6e67 2069 7320 7265 6775 6c61 7220 696e ng is regular in │ │ │ │ -00040760: 2063 6f64 696d 656e 7369 6f6e 2031 2075 codimension 1 u │ │ │ │ -00040770: 7369 6e67 0a64 6966 6665 7265 6e74 2073 sing.different s │ │ │ │ -00040780: 7472 6174 6567 6965 732e 2020 4573 7365 trategies. Esse │ │ │ │ -00040790: 6e74 6961 6c6c 792c 206d 696e 6f72 7320 ntially, minors │ │ │ │ -000407a0: 6172 6520 636f 6d70 7574 6564 2075 6e74 are computed unt │ │ │ │ -000407b0: 696c 2069 7420 6973 2076 6572 6966 6965 il it is verifie │ │ │ │ -000407c0: 640a 7468 6174 2074 6865 2072 696e 6720 d.that the ring │ │ │ │ -000407d0: 6973 2072 6567 756c 6172 2069 6e20 636f is regular in co │ │ │ │ -000407e0: 6469 6d65 6e73 696f 6e20 312e 0a2b 2d2d dimension 1..+-- │ │ │ │ +0003fed0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 ************..De │ │ │ │ +0003fee0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +0003fef0: 3d3d 3d3d 3d0a 0a4d 616e 7920 6f66 2074 =====..Many of t │ │ │ │ +0003ff00: 6865 2063 6f72 6520 6675 6e63 7469 6f6e he core function │ │ │ │ +0003ff10: 7320 6f66 2074 6869 7320 7061 636b 6167 s of this packag │ │ │ │ +0003ff20: 6520 616c 6c6f 7720 7468 6520 7573 6572 e allow the user │ │ │ │ +0003ff30: 2074 6f20 6669 6e65 2074 756e 6520 7468 to fine tune th │ │ │ │ +0003ff40: 650a 7374 7261 7465 6779 2075 7365 6420 e.strategy used │ │ │ │ +0003ff50: 666f 7220 7365 6c65 6374 696e 6720 7375 for selecting su │ │ │ │ +0003ff60: 626d 6174 7269 6365 732e 2020 4469 6666 bmatrices. Diff │ │ │ │ +0003ff70: 6572 656e 7420 7374 7261 7465 6769 6573 erent strategies │ │ │ │ +0003ff80: 2079 6965 6c64 206d 6172 6b65 646c 790a yield markedly. │ │ │ │ +0003ff90: 6469 6666 6572 656e 7420 7065 7266 6f72 different perfor │ │ │ │ +0003ffa0: 6d61 6e63 6520 6f72 2072 6573 756c 7473 mance or results │ │ │ │ +0003ffb0: 206f 6e20 7661 7269 6f75 7320 6578 616d on various exam │ │ │ │ +0003ffc0: 706c 6573 2e20 5468 6573 6520 6172 6520 ples. These are │ │ │ │ +0003ffd0: 636f 6e74 726f 6c6c 6564 2062 790a 7370 controlled by.sp │ │ │ │ +0003ffe0: 6563 6966 7969 6e67 2061 2020 5374 7261 ecifying a Stra │ │ │ │ +0003fff0: 7465 6779 203d 3e20 206f 7074 696f 6e2c tegy => option, │ │ │ │ +00040000: 2070 6f69 6e74 696e 6720 746f 2061 2020 pointing to a │ │ │ │ +00040010: 4861 7368 5461 626c 6577 6869 6368 2073 HashTablewhich s │ │ │ │ +00040020: 7065 6369 6669 6573 0a73 6576 6572 616c pecifies.several │ │ │ │ +00040030: 2073 7472 6174 6567 6965 7320 7368 6f75 strategies shou │ │ │ │ +00040040: 6c64 2062 6520 7573 6564 2073 696d 756c ld be used simul │ │ │ │ +00040050: 7461 6e65 6f75 736c 792c 206f 7220 746f taneously, or to │ │ │ │ +00040060: 2061 2073 796d 626f 6c20 7361 7969 6e67 a symbol saying │ │ │ │ +00040070: 2077 650a 7368 6f75 6c64 2075 7365 206f we.should use o │ │ │ │ +00040080: 6e6c 7920 6120 7369 6e67 6c65 2073 7472 nly a single str │ │ │ │ +00040090: 6174 6567 792e 2020 466f 7220 6120 6d6f ategy. For a mo │ │ │ │ +000400a0: 7265 2064 6574 6169 6c65 6420 6c6f 6f6b re detailed look │ │ │ │ +000400b0: 2061 7420 7468 6973 2069 6e20 616e 0a65 at this in an.e │ │ │ │ +000400c0: 7861 6d70 6c65 2070 6c65 6173 6520 7365 xample please se │ │ │ │ +000400d0: 6520 2a6e 6f74 6520 4661 7374 4d69 6e6f e *note FastMino │ │ │ │ +000400e0: 7273 5374 7261 7465 6779 5475 746f 7269 rsStrategyTutori │ │ │ │ +000400f0: 616c 3a0a 4661 7374 4d69 6e6f 7273 5374 al:.FastMinorsSt │ │ │ │ +00040100: 7261 7465 6779 5475 746f 7269 616c 2c42 rategyTutorial,B │ │ │ │ +00040110: 6566 6f72 6520 6465 7363 7269 6269 6e67 efore describing │ │ │ │ +00040120: 2074 6865 2061 7661 696c 6162 6c65 2073 the available s │ │ │ │ +00040130: 7472 6174 6567 6965 732c 2077 6520 6265 trategies, we be │ │ │ │ +00040140: 6769 6e0a 6279 2072 6f75 6768 6c79 206f gin.by roughly o │ │ │ │ +00040150: 7574 6c69 6e69 6e67 2074 6865 2064 6966 utlining the dif │ │ │ │ +00040160: 6665 7265 6e74 2061 7070 726f 6163 6865 ferent approache │ │ │ │ +00040170: 732e 0a20 202a 2048 6575 7269 7374 6963 s.. * Heuristic │ │ │ │ +00040180: 2073 7562 6d61 7472 6978 2073 656c 6563 submatrix selec │ │ │ │ +00040190: 7469 6f6e 3a20 496e 2074 6869 7320 6361 tion: In this ca │ │ │ │ +000401a0: 7365 2c20 6120 7375 626d 6174 7269 7820 se, a submatrix │ │ │ │ +000401b0: 6973 2063 686f 7365 6e20 7669 6120 610a is chosen via a. │ │ │ │ +000401c0: 2020 2020 6772 6565 6479 2061 6c67 6f72 greedy algor │ │ │ │ +000401d0: 6974 686d 2c20 6c6f 6f6b 696e 6720 666f ithm, looking fo │ │ │ │ +000401e0: 7220 6120 7375 626d 6174 7269 7820 7769 r a submatrix wi │ │ │ │ +000401f0: 7468 2073 6d61 6c6c 6573 7420 286f 7220 th smallest (or │ │ │ │ +00040200: 6c61 7267 6573 7429 2064 6567 7265 650a largest) degree. │ │ │ │ +00040210: 2020 2020 7769 7468 2072 6573 7065 6374 with respect │ │ │ │ +00040220: 2074 6f20 6120 7261 6e64 6f6d 206d 6f6e to a random mon │ │ │ │ +00040230: 6f6d 6961 6c20 6f72 6465 722e 0a20 202a omial order.. * │ │ │ │ +00040240: 2053 7562 6d61 7472 6978 2073 656c 6563 Submatrix selec │ │ │ │ +00040250: 7469 6f6e 2076 6961 2072 6174 696f 6e61 tion via rationa │ │ │ │ +00040260: 6c20 616e 6420 6765 6f6d 6574 7269 6320 l and geometric │ │ │ │ +00040270: 706f 696e 7473 3a20 4865 7265 2061 2072 points: Here a r │ │ │ │ +00040280: 6174 696f 6e61 6c20 6f72 0a20 2020 2067 ational or. g │ │ │ │ +00040290: 656f 6d65 7472 6963 2070 6f69 6e74 2069 eometric point i │ │ │ │ +000402a0: 7320 666f 756e 6420 7768 6572 6520 6120 s found where a │ │ │ │ +000402b0: 6769 7665 6e20 6964 6561 6c20 7661 6e69 given ideal vani │ │ │ │ +000402c0: 7368 6573 2e20 2054 6861 7420 706f 696e shes. That poin │ │ │ │ +000402d0: 7420 6973 0a20 2020 2070 6c75 6767 6564 t is. plugged │ │ │ │ +000402e0: 2069 6e74 6f20 7468 6520 6d61 7472 6978 into the matrix │ │ │ │ +000402f0: 2061 6e64 2061 2073 7562 6d61 7472 6978 and a submatrix │ │ │ │ +00040300: 206f 6620 6675 6c6c 2072 616e 6b20 6973 of full rank is │ │ │ │ +00040310: 2069 6465 6e74 6966 6965 642e 2020 2054 identified. T │ │ │ │ +00040320: 6869 730a 2020 2020 6170 7072 6f61 6368 his. approach │ │ │ │ +00040330: 2063 7572 7265 6e74 6c79 206f 6e6c 7920 currently only │ │ │ │ +00040340: 776f 726b 7320 6f76 6572 2061 2066 696e works over a fin │ │ │ │ +00040350: 6974 6520 6669 656c 6420 616e 6420 6973 ite field and is │ │ │ │ +00040360: 2061 6363 6f6d 706c 6973 6865 6420 7769 accomplished wi │ │ │ │ +00040370: 7468 0a20 2020 2074 6865 2068 656c 7020 th. the help │ │ │ │ +00040380: 6f66 2074 6865 2070 6163 6b61 6765 202a of the package * │ │ │ │ +00040390: 6e6f 7465 2052 616e 646f 6d50 6f69 6e74 note RandomPoint │ │ │ │ +000403a0: 733a 2028 5261 6e64 6f6d 506f 696e 7473 s: (RandomPoints │ │ │ │ +000403b0: 2954 6f70 2c2e 0a20 202a 2052 616e 646f )Top,.. * Rando │ │ │ │ +000403c0: 6d20 7375 626d 6174 7269 7820 7365 6c65 m submatrix sele │ │ │ │ +000403d0: 6374 696f 6e3a 2054 6869 7320 6569 7468 ction: This eith │ │ │ │ +000403e0: 6572 2063 686f 6f73 6573 2061 2063 6f6d er chooses a com │ │ │ │ +000403f0: 706c 6574 656c 7920 7261 6e64 6f6d 0a20 pletely random. │ │ │ │ +00040400: 2020 2073 7562 6d61 7472 6978 2c20 6f72 submatrix, or │ │ │ │ +00040410: 2061 2073 7562 6d61 7472 6978 2077 6869 a submatrix whi │ │ │ │ +00040420: 6368 2068 6173 206e 6f20 7a65 726f 2063 ch has no zero c │ │ │ │ +00040430: 6f6c 756d 6e73 206f 7220 726f 7773 2e0a olumns or rows.. │ │ │ │ +00040440: 5468 6572 6520 7765 2068 6967 686c 6967 There we highlig │ │ │ │ +00040450: 6874 2066 6976 6520 7072 652d 7072 6f67 ht five pre-prog │ │ │ │ +00040460: 7261 6d6d 6564 2073 7472 6174 6567 6965 rammed strategie │ │ │ │ +00040470: 7320 7072 6f76 6964 6564 2074 6f20 7468 s provided to th │ │ │ │ +00040480: 6520 7573 6572 2e0a 2020 2a20 5374 7261 e user.. * Stra │ │ │ │ +00040490: 7465 6779 4465 6661 756c 743a 2074 6869 tegyDefault: thi │ │ │ │ +000404a0: 7320 7573 6573 2061 206d 6978 206f 6620 s uses a mix of │ │ │ │ +000404b0: 6865 7572 6973 7469 6373 2061 6e64 2072 heuristics and r │ │ │ │ +000404c0: 616e 646f 6d20 7375 626d 6174 7269 6365 andom submatrice │ │ │ │ +000404d0: 732e 0a20 202a 2053 7472 6174 6567 7952 s.. * StrategyR │ │ │ │ +000404e0: 616e 646f 6d3a 2074 6869 7320 7573 6573 andom: this uses │ │ │ │ +000404f0: 2070 7572 656c 7920 7261 6e64 6f6d 2073 purely random s │ │ │ │ +00040500: 7562 6d61 7472 6963 6573 2e0a 2020 2a20 ubmatrices.. * │ │ │ │ +00040510: 5374 7261 7465 6779 4465 6661 756c 744e StrategyDefaultN │ │ │ │ +00040520: 6f6e 5261 6e64 6f6d 3a20 7468 6973 2075 onRandom: this u │ │ │ │ +00040530: 7365 7320 6120 6d69 7820 6f66 2068 6575 ses a mix of heu │ │ │ │ +00040540: 7269 7374 6963 7320 6275 7420 6e6f 2072 ristics but no r │ │ │ │ +00040550: 616e 646f 6d0a 2020 2020 7375 626d 6174 andom. submat │ │ │ │ +00040560: 7269 6365 732e 0a20 202a 2053 7472 6174 rices.. * Strat │ │ │ │ +00040570: 6567 7950 6f69 6e74 733a 2074 6869 7320 egyPoints: this │ │ │ │ +00040580: 6f6e 6c79 2075 7365 7320 7261 7469 6f6e only uses ration │ │ │ │ +00040590: 616c 202f 2067 656f 6d65 7472 6963 2070 al / geometric p │ │ │ │ +000405a0: 6f69 6e74 7320 746f 2066 696e 640a 2020 oints to find. │ │ │ │ +000405b0: 2020 7375 626d 6174 7269 6365 732e 0a20 submatrices.. │ │ │ │ +000405c0: 202a 2053 7472 6174 6567 7944 6566 6175 * StrategyDefau │ │ │ │ +000405d0: 6c74 5769 7468 506f 696e 7473 3a20 7468 ltWithPoints: th │ │ │ │ +000405e0: 6973 2075 7365 7320 6120 6d69 7820 6f66 is uses a mix of │ │ │ │ +000405f0: 2068 6575 7269 7374 6963 7320 616e 6420 heuristics and │ │ │ │ +00040600: 7375 626d 6174 7269 6365 730a 2020 2020 submatrices. │ │ │ │ +00040610: 6368 6f73 656e 2077 6974 6820 7261 7469 chosen with rati │ │ │ │ +00040620: 6f6e 616c 2061 6e64 2067 656f 6d65 7472 onal and geometr │ │ │ │ +00040630: 6963 2070 6f69 6e74 732e 0a42 656c 6f77 ic points..Below │ │ │ │ +00040640: 2074 6865 2064 6574 6169 6c73 206f 6620 the details of │ │ │ │ +00040650: 686f 7720 7468 6573 6520 7374 7261 7465 how these strate │ │ │ │ +00040660: 6769 6573 2061 7265 2063 6f6e 7374 7275 gies are constru │ │ │ │ +00040670: 6374 6564 2077 696c 6c20 6265 2064 6574 cted will be det │ │ │ │ +00040680: 6169 6c65 640a 6265 6c6f 772e 2020 4275 ailed.below. Bu │ │ │ │ +00040690: 7420 6669 7273 742c 2077 6520 7072 6f76 t first, we prov │ │ │ │ +000406a0: 6964 6520 616e 2065 7861 6d70 6c65 2073 ide an example s │ │ │ │ +000406b0: 686f 7769 6e67 2074 6861 7420 7468 6573 howing that thes │ │ │ │ +000406c0: 6520 7374 7261 7465 6769 6573 2063 616e e strategies can │ │ │ │ +000406d0: 0a70 6572 666f 726d 2071 7569 7465 2064 .perform quite d │ │ │ │ +000406e0: 6966 6665 7265 6e74 6c79 2e20 2054 6865 ifferently. The │ │ │ │ +000406f0: 2066 6f6c 6c6f 7769 6e67 2069 7320 7468 following is th │ │ │ │ +00040700: 6520 636f 6e65 206f 7665 7220 7468 6520 e cone over the │ │ │ │ +00040710: 7072 6f64 7563 7420 6f66 2074 776f 0a65 product of two.e │ │ │ │ +00040720: 6c6c 6970 7469 6320 6375 7276 6573 2e20 lliptic curves. │ │ │ │ +00040730: 2057 6520 7665 7269 6679 2074 6861 7420 We verify that │ │ │ │ +00040740: 7468 6973 2072 696e 6720 6973 2072 6567 this ring is reg │ │ │ │ +00040750: 756c 6172 2069 6e20 636f 6469 6d65 6e73 ular in codimens │ │ │ │ +00040760: 696f 6e20 3120 7573 696e 670a 6469 6666 ion 1 using.diff │ │ │ │ +00040770: 6572 656e 7420 7374 7261 7465 6769 6573 erent strategies │ │ │ │ +00040780: 2e20 2045 7373 656e 7469 616c 6c79 2c20 . Essentially, │ │ │ │ +00040790: 6d69 6e6f 7273 2061 7265 2063 6f6d 7075 minors are compu │ │ │ │ +000407a0: 7465 6420 756e 7469 6c20 6974 2069 7320 ted until it is │ │ │ │ +000407b0: 7665 7269 6669 6564 0a74 6861 7420 7468 verified.that th │ │ │ │ +000407c0: 6520 7269 6e67 2069 7320 7265 6775 6c61 e ring is regula │ │ │ │ +000407d0: 7220 696e 2063 6f64 696d 656e 7369 6f6e r in codimension │ │ │ │ +000407e0: 2031 2e0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 1..+----------- │ │ │ │ 000407f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00040840: 203a 2054 3d5a 5a2f 375b 612e 2e69 5d2f : T=ZZ/7[a..i]/ │ │ │ │ -00040850: 6964 6561 6c28 662a 682d 652a 692c 632a ideal(f*h-e*i,c* │ │ │ │ -00040860: 682d 622a 692c 662a 672d 642a 692c 652a h-b*i,f*g-d*i,e* │ │ │ │ -00040870: 672d 642a 682c 632a 672d 612a 692c 622a g-d*h,c*g-a*i,b* │ │ │ │ -00040880: 672d 612a 682c 632a 652d 627c 0a7c 2d2d g-a*h,c*e-b|.|-- │ │ │ │ +00040830: 2d2d 2b0a 7c69 3120 3a20 543d 5a5a 2f37 --+.|i1 : T=ZZ/7 │ │ │ │ +00040840: 5b61 2e2e 695d 2f69 6465 616c 2866 2a68 [a..i]/ideal(f*h │ │ │ │ +00040850: 2d65 2a69 2c63 2a68 2d62 2a69 2c66 2a67 -e*i,c*h-b*i,f*g │ │ │ │ +00040860: 2d64 2a69 2c65 2a67 2d64 2a68 2c63 2a67 -d*i,e*g-d*h,c*g │ │ │ │ +00040870: 2d61 2a69 2c62 2a67 2d61 2a68 2c63 2a65 -a*i,b*g-a*h,c*e │ │ │ │ +00040880: 2d62 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d -b|.|----------- │ │ │ │ 00040890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000408a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000408b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000408c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000408d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a66 -----------|.|*f │ │ │ │ -000408e0: 2c63 2a64 2d61 2a66 2c62 2a64 2d61 2a65 ,c*d-a*f,b*d-a*e │ │ │ │ -000408f0: 2c67 5e33 2d68 5e32 2a69 2d67 2a69 5e32 ,g^3-h^2*i-g*i^2 │ │ │ │ -00040900: 2c64 2a67 5e32 2d65 2a68 2a69 2d64 2a69 ,d*g^2-e*h*i-d*i │ │ │ │ -00040910: 5e32 2c61 2a67 5e32 2d62 2a68 2a69 2d61 ^2,a*g^2-b*h*i-a │ │ │ │ -00040920: 2a69 5e32 2c64 5e32 2a67 2d7c 0a7c 2d2d *i^2,d^2*g-|.|-- │ │ │ │ +000408d0: 2d2d 7c0a 7c2a 662c 632a 642d 612a 662c --|.|*f,c*d-a*f, │ │ │ │ +000408e0: 622a 642d 612a 652c 675e 332d 685e 322a b*d-a*e,g^3-h^2* │ │ │ │ +000408f0: 692d 672a 695e 322c 642a 675e 322d 652a i-g*i^2,d*g^2-e* │ │ │ │ +00040900: 682a 692d 642a 695e 322c 612a 675e 322d h*i-d*i^2,a*g^2- │ │ │ │ +00040910: 622a 682a 692d 612a 695e 322c 645e 322a b*h*i-a*i^2,d^2* │ │ │ │ +00040920: 672d 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d g-|.|----------- │ │ │ │ 00040930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 655e -----------|.|e^ │ │ │ │ -00040980: 322a 692d 642a 662a 692c 612a 642a 672d 2*i-d*f*i,a*d*g- │ │ │ │ -00040990: 622a 652a 692d 612a 662a 692c 615e 322a b*e*i-a*f*i,a^2* │ │ │ │ -000409a0: 672d 625e 322a 692d 612a 632a 692c 645e g-b^2*i-a*c*i,d^ │ │ │ │ -000409b0: 332d 655e 322a 662d 642a 665e 322c 612a 3-e^2*f-d*f^2,a* │ │ │ │ -000409c0: 645e 322d 622a 652a 662d 617c 0a7c 2d2d d^2-b*e*f-a|.|-- │ │ │ │ +00040970: 2d2d 7c0a 7c65 5e32 2a69 2d64 2a66 2a69 --|.|e^2*i-d*f*i │ │ │ │ +00040980: 2c61 2a64 2a67 2d62 2a65 2a69 2d61 2a66 ,a*d*g-b*e*i-a*f │ │ │ │ +00040990: 2a69 2c61 5e32 2a67 2d62 5e32 2a69 2d61 *i,a^2*g-b^2*i-a │ │ │ │ +000409a0: 2a63 2a69 2c64 5e33 2d65 5e32 2a66 2d64 *c*i,d^3-e^2*f-d │ │ │ │ +000409b0: 2a66 5e32 2c61 2a64 5e32 2d62 2a65 2a66 *f^2,a*d^2-b*e*f │ │ │ │ +000409c0: 2d61 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d -a|.|----------- │ │ │ │ 000409d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000409e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000409f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2a66 -----------|.|*f │ │ │ │ -00040a20: 5e32 2c61 5e32 2a64 2d62 5e32 2a66 2d61 ^2,a^2*d-b^2*f-a │ │ │ │ -00040a30: 2a63 2a66 2c63 5e33 2b66 5e33 2d69 5e33 *c*f,c^3+f^3-i^3 │ │ │ │ -00040a40: 2c62 2a63 5e32 2b65 2a66 5e32 2d68 2a69 ,b*c^2+e*f^2-h*i │ │ │ │ -00040a50: 5e32 2c61 2a63 5e32 2b64 2a66 5e32 2d67 ^2,a*c^2+d*f^2-g │ │ │ │ -00040a60: 2a69 5e32 2c62 5e32 2a63 2b7c 0a7c 2d2d *i^2,b^2*c+|.|-- │ │ │ │ +00040a10: 2d2d 7c0a 7c2a 665e 322c 615e 322a 642d --|.|*f^2,a^2*d- │ │ │ │ +00040a20: 625e 322a 662d 612a 632a 662c 635e 332b b^2*f-a*c*f,c^3+ │ │ │ │ +00040a30: 665e 332d 695e 332c 622a 635e 322b 652a f^3-i^3,b*c^2+e* │ │ │ │ +00040a40: 665e 322d 682a 695e 322c 612a 635e 322b f^2-h*i^2,a*c^2+ │ │ │ │ +00040a50: 642a 665e 322d 672a 695e 322c 625e 322a d*f^2-g*i^2,b^2* │ │ │ │ +00040a60: 632b 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d c+|.|----------- │ │ │ │ 00040a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 655e -----------|.|e^ │ │ │ │ -00040ac0: 322a 662d 685e 322a 692c 612a 622a 632b 2*f-h^2*i,a*b*c+ │ │ │ │ -00040ad0: 642a 652a 662d 672a 682a 692c 615e 322a d*e*f-g*h*i,a^2* │ │ │ │ -00040ae0: 632b 645e 322a 662d 675e 322a 692c 625e c+d^2*f-g^2*i,b^ │ │ │ │ -00040af0: 332b 655e 332d 685e 332c 612a 625e 322b 3+e^3-h^3,a*b^2+ │ │ │ │ -00040b00: 642a 655e 322d 672a 685e 327c 0a7c 2d2d d*e^2-g*h^2|.|-- │ │ │ │ +00040ab0: 2d2d 7c0a 7c65 5e32 2a66 2d68 5e32 2a69 --|.|e^2*f-h^2*i │ │ │ │ +00040ac0: 2c61 2a62 2a63 2b64 2a65 2a66 2d67 2a68 ,a*b*c+d*e*f-g*h │ │ │ │ +00040ad0: 2a69 2c61 5e32 2a63 2b64 5e32 2a66 2d67 *i,a^2*c+d^2*f-g │ │ │ │ +00040ae0: 5e32 2a69 2c62 5e33 2b65 5e33 2d68 5e33 ^2*i,b^3+e^3-h^3 │ │ │ │ +00040af0: 2c61 2a62 5e32 2b64 2a65 5e32 2d67 2a68 ,a*b^2+d*e^2-g*h │ │ │ │ +00040b00: 5e32 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d ^2|.|----------- │ │ │ │ 00040b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2c61 -----------|.|,a │ │ │ │ -00040b60: 5e32 2a62 2b64 5e32 2a65 2d67 5e32 2a68 ^2*b+d^2*e-g^2*h │ │ │ │ -00040b70: 2c61 5e33 2b65 5e32 2a66 2b64 2a66 5e32 ,a^3+e^2*f+d*f^2 │ │ │ │ -00040b80: 2d68 5e32 2a69 2d67 2a69 5e32 293b 2020 -h^2*i-g*i^2); │ │ │ │ +00040b50: 2d2d 7c0a 7c2c 615e 322a 622b 645e 322a --|.|,a^2*b+d^2* │ │ │ │ +00040b60: 652d 675e 322a 682c 615e 332b 655e 322a e-g^2*h,a^3+e^2* │ │ │ │ +00040b70: 662b 642a 665e 322d 685e 322a 692d 672a f+d*f^2-h^2*i-g* │ │ │ │ +00040b80: 695e 3229 3b20 2020 2020 2020 2020 2020 i^2); │ │ │ │ 00040b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040ba0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00040ba0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00040bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00040c00: 203a 2065 6c61 7073 6564 5469 6d65 2072 : elapsedTime r │ │ │ │ -00040c10: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ -00040c20: 696f 6e28 312c 2054 2c20 5374 7261 7465 ion(1, T, Strate │ │ │ │ -00040c30: 6779 3d3e 5374 7261 7465 6779 4465 6661 gy=>StrategyDefa │ │ │ │ -00040c40: 756c 7429 2020 2020 2020 207c 0a7c 202d ult) |.| - │ │ │ │ -00040c50: 2d20 312e 3539 3232 3873 2065 6c61 7073 - 1.59228s elaps │ │ │ │ -00040c60: 6564 2020 2020 2020 2020 2020 2020 2020 ed │ │ │ │ +00040bf0: 2d2d 2b0a 7c69 3220 3a20 656c 6170 7365 --+.|i2 : elapse │ │ │ │ +00040c00: 6454 696d 6520 7265 6775 6c61 7249 6e43 dTime regularInC │ │ │ │ +00040c10: 6f64 696d 656e 7369 6f6e 2831 2c20 542c odimension(1, T, │ │ │ │ +00040c20: 2053 7472 6174 6567 793d 3e53 7472 6174 Strategy=>Strat │ │ │ │ +00040c30: 6567 7944 6566 6175 6c74 2920 2020 2020 egyDefault) │ │ │ │ +00040c40: 2020 7c0a 7c20 2d2d 2031 2e35 3632 3835 |.| -- 1.56285 │ │ │ │ +00040c50: 7320 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2020 2020 2020 │ │ │ │ -00040d30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00040d30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00040d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00040d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a49 6e20 -----------+.In │ │ │ │ -00040d90: 7468 6973 2070 6172 7469 6375 6c61 7220 this particular │ │ │ │ -00040da0: 6578 616d 706c 652c 206f 6e20 6f6e 6520 example, on one │ │ │ │ -00040db0: 6d61 6368 696e 652c 2077 6520 6c69 7374 machine, we list │ │ │ │ -00040dc0: 2061 7665 7261 6765 2074 696d 6520 746f average time to │ │ │ │ -00040dd0: 2063 6f6d 706c 6574 696f 6e0a 6f66 2065 completion.of e │ │ │ │ -00040de0: 6163 6820 6f66 2074 6865 2061 626f 7665 ach of the above │ │ │ │ -00040df0: 2073 7472 6174 6567 6965 7320 6166 7465 strategies afte │ │ │ │ -00040e00: 7220 3130 3020 7275 6e73 2e0a 2020 2a20 r 100 runs.. * │ │ │ │ -00040e10: 5374 7261 7465 6779 4465 6661 756c 743a StrategyDefault: │ │ │ │ -00040e20: 2031 2e36 3520 7365 636f 6e64 730a 2020 1.65 seconds. │ │ │ │ -00040e30: 2a20 5374 7261 7465 6779 5261 6e64 6f6d * StrategyRandom │ │ │ │ -00040e40: 3a20 382e 3332 2073 6563 6f6e 6473 0a20 : 8.32 seconds. │ │ │ │ -00040e50: 202a 2053 7472 6174 6567 7944 6566 6175 * StrategyDefau │ │ │ │ -00040e60: 6c74 4e6f 6e52 616e 646f 6d3a 2030 2e39 ltNonRandom: 0.9 │ │ │ │ -00040e70: 3920 7365 636f 6e64 730a 2020 2a20 5374 9 seconds. * St │ │ │ │ -00040e80: 7261 7465 6779 506f 696e 7473 3a20 332e rategyPoints: 3. │ │ │ │ -00040e90: 3237 2073 6563 6f6e 6473 0a20 202a 2053 27 seconds. * S │ │ │ │ -00040ea0: 7472 6174 6567 7944 6566 6175 6c74 5769 trategyDefaultWi │ │ │ │ -00040eb0: 7468 506f 696e 7473 3a20 332e 3337 0a52 thPoints: 3.37.R │ │ │ │ -00040ec0: 6f75 6768 6c79 2073 7065 616b 696e 672c oughly speaking, │ │ │ │ -00040ed0: 2068 6575 7269 7374 6963 7320 7465 6e64 heuristics tend │ │ │ │ -00040ee0: 2074 6f20 7072 6f76 6964 6520 6d6f 7265 to provide more │ │ │ │ -00040ef0: 2069 6e66 6f72 6d61 7469 6f6e 2074 6861 information tha │ │ │ │ -00040f00: 6e20 7261 6e64 6f6d 0a73 7562 6d61 7472 n random.submatr │ │ │ │ -00040f10: 6963 6573 2061 6e64 2073 6f20 7468 6579 ices and so they │ │ │ │ -00040f20: 2077 6f72 6b20 6d75 6368 2066 6173 7465 work much faste │ │ │ │ -00040f30: 7220 7369 6e63 6520 7468 6579 2063 6f6e r since they con │ │ │ │ -00040f40: 7369 6465 7220 6661 7220 6665 7765 720a sider far fewer. │ │ │ │ -00040f50: 7375 626d 6174 7269 6365 732e 2020 4672 submatrices. 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In. │ │ │ │ -00041170: 7061 7274 6963 756c 6172 2c20 7468 6520 particular, the │ │ │ │ -00041180: 7573 6572 2063 616e 2065 7665 6e20 6375 user can even cu │ │ │ │ -00041190: 7374 6f6d 697a 6520 7468 6520 6865 7572 stomize the heur │ │ │ │ -000411a0: 6973 7469 6373 2075 7365 642e 2020 5365 istics used. Se │ │ │ │ -000411b0: 6520 6265 6c6f 7720 666f 7220 686f 770a e below for how. │ │ │ │ -000411c0: 746f 2065 6173 696c 7920 7573 6520 6f6e to easily use on │ │ │ │ -000411d0: 6c79 2061 2073 696e 676c 6520 6865 7572 ly a single heur │ │ │ │ -000411e0: 6973 7469 632e 2054 6f20 6375 7374 6f6d istic. 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-000412c0: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ -000412d0: 3a20 7472 7920 746f 2066 696e 6420 7375 : try to find su │ │ │ │ -000412e0: 626d 6174 7269 6365 7320 7768 6572 6520 bmatrices where │ │ │ │ -000412f0: 6561 6368 2072 6f77 2061 6e64 2063 6f6c each row and col │ │ │ │ -00041300: 756d 6e20 6861 7320 610a 2020 2020 736d umn has a. sm │ │ │ │ -00041310: 616c 6c20 656e 7472 7920 7769 7468 2072 all entry with r │ │ │ │ -00041320: 6573 7065 6374 2074 6f20 6120 7261 6e64 espect to a rand │ │ │ │ -00041330: 6f6d 2047 5265 764c 6578 6f72 6465 722e om GRevLexorder. │ │ │ │ -00041340: 0a20 202a 2047 5265 764c 6578 536d 616c . * GRevLexSmal │ │ │ │ -00041350: 6c65 7374 5465 726d 3a20 6669 6e64 2073 lestTerm: find s │ │ │ │ -00041360: 7562 6d61 7472 6963 6573 2077 6865 7265 ubmatrices where │ │ │ │ -00041370: 2065 6163 6820 726f 7720 616e 6420 636f each row and co │ │ │ │ -00041380: 6c75 6d6e 2068 6173 2061 6e0a 2020 2020 lumn has an. │ │ │ │ -00041390: 656e 7472 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7472 6963 o find submatric │ │ │ │ -00041470: 6573 2077 6865 7265 2065 6163 6820 726f es where each ro │ │ │ │ -00041480: 7720 616e 6420 636f 6c75 6d6e 2068 6173 w and column has │ │ │ │ -00041490: 2061 2073 6d61 6c6c 0a20 2020 2065 6e74 a small. ent │ │ │ │ -000414a0: 7279 2077 6974 6820 7265 7370 6563 7420 ry with respect │ │ │ │ -000414b0: 746f 2061 2072 616e 646f 6d20 4c65 786f to a random Lexo │ │ │ │ -000414c0: 7264 6572 2e0a 2020 2a20 4c65 7853 6d61 rder.. * LexSma │ │ │ │ -000414d0: 6c6c 6573 7454 6572 6d3a 2066 696e 6420 llestTerm: find │ │ │ │ -000414e0: 7375 626d 6174 7269 6365 7320 7768 6572 submatrices wher │ │ │ │ -000414f0: 6520 6561 6368 2072 6f77 2061 6e64 2063 e each row and c │ │ │ │ -00041500: 6f6c 756d 6e20 6861 7320 616e 2065 6e74 olumn has an ent │ │ │ │ -00041510: 7279 0a20 2020 2077 6974 6820 6120 736d ry. with a sm │ │ │ │ -00041520: 616c 6c20 7465 726d 2077 6974 6820 7265 all term with re │ │ │ │ -00041530: 7370 6563 7420 746f 2061 2072 616e 646f spect to a rando │ │ │ │ -00041540: 6d20 4c65 786f 7264 6572 2e0a 2020 2a20 m Lexorder.. * │ │ │ │ -00041550: 5261 6e64 6f6d 3a20 6669 6e64 2072 616e Random: find ran │ │ │ │ -00041560: 646f 6d20 7375 626d 6174 7269 6365 7320 dom submatrices │ │ │ │ -00041570: 0a20 202a 2052 616e 646f 6d4e 6f6e 7a65 . * RandomNonze │ │ │ │ -00041580: 726f 3a20 6669 6e64 2072 616e 646f 6d20 ro: find random │ │ │ │ -00041590: 7375 626d 6174 7269 6365 7320 7468 6174 submatrices that │ │ │ │ -000415a0: 2068 6176 6520 6e6f 6e7a 6572 6f20 726f have nonzero ro │ │ │ │ -000415b0: 7773 2061 6e64 2063 6f6c 756d 6e73 0a20 ws and columns. │ │ │ │ -000415c0: 202a 2050 6f69 6e74 733a 2066 696e 6420 * Points: find │ │ │ │ -000415d0: 7375 626d 6174 7269 6365 7320 7468 6174 submatrices that │ │ │ │ -000415e0: 2061 7265 206e 6f74 2073 696e 6775 6c61 are not singula │ │ │ │ -000415f0: 7220 6174 2074 6865 2067 6976 656e 2069 r at the given i │ │ │ │ -00041600: 6465 616c 2062 790a 2020 2020 6669 6e64 deal by. find │ │ │ │ -00041610: 696e 6720 6120 706f 696e 7420 7768 6572 ing a point wher │ │ │ │ -00041620: 6520 7468 6174 2069 6465 616c 2076 616e e that ideal van │ │ │ │ -00041630: 6973 6865 732c 2061 6e64 2065 7661 6c75 ishes, and evalu │ │ │ │ -00041640: 6174 696e 6720 7468 6520 6d61 7472 6978 ating the matrix │ │ │ │ -00041650: 2061 740a 2020 2020 7468 6174 2070 6f69 at. that poi │ │ │ │ -00041660: 6e74 2028 7669 6120 7468 6520 7061 636b nt (via the pack │ │ │ │ -00041670: 6167 6520 2a6e 6f74 6520 5261 6e64 6f6d age *note Random │ │ │ │ -00041680: 506f 696e 7473 3a20 2852 616e 646f 6d50 Points: (RandomP │ │ │ │ -00041690: 6f69 6e74 7329 546f 702c 292e 2020 4966 oints)Top,). 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P │ │ │ │ -00041730: 6f69 6e74 4f70 7469 6f6e 733a 2050 6f69 ointOptions: Poi │ │ │ │ -00041740: 6e74 4f70 7469 6f6e 732c 206f 7074 696f ntOptions, optio │ │ │ │ -00041750: 6e2e 0a46 6f72 2065 7861 6d70 6c65 3a0a n..For example:. │ │ │ │ -00041760: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00040d80: 2d2d 2b0a 496e 2074 6869 7320 7061 7274 --+.In this part │ │ │ │ +00040d90: 6963 756c 6172 2065 7861 6d70 6c65 2c20 icular example, │ │ │ │ +00040da0: 6f6e 206f 6e65 206d 6163 6869 6e65 2c20 on one machine, │ │ │ │ +00040db0: 7765 206c 6973 7420 6176 6572 6167 6520 we list average │ │ │ │ +00040dc0: 7469 6d65 2074 6f20 636f 6d70 6c65 7469 time to completi │ │ │ │ +00040dd0: 6f6e 0a6f 6620 6561 6368 206f 6620 7468 on.of each of th │ │ │ │ +00040de0: 6520 6162 6f76 6520 7374 7261 7465 6769 e above strategi │ │ │ │ +00040df0: 6573 2061 6674 6572 2031 3030 2072 756e es after 100 run │ │ │ │ +00040e00: 732e 0a20 202a 2053 7472 6174 6567 7944 s.. * StrategyD │ │ │ │ +00040e10: 6566 6175 6c74 3a20 312e 3635 2073 6563 efault: 1.65 sec │ │ │ │ +00040e20: 6f6e 6473 0a20 202a 2053 7472 6174 6567 onds. * Strateg │ │ │ │ +00040e30: 7952 616e 646f 6d3a 2038 2e33 3220 7365 yRandom: 8.32 se │ │ │ │ +00040e40: 636f 6e64 730a 2020 2a20 5374 7261 7465 conds. * Strate │ │ │ │ +00040e50: 6779 4465 6661 756c 744e 6f6e 5261 6e64 gyDefaultNonRand │ │ │ │ +00040e60: 6f6d 3a20 302e 3939 2073 6563 6f6e 6473 om: 0.99 seconds │ │ │ │ +00040e70: 0a20 202a 2053 7472 6174 6567 7950 6f69 . * StrategyPoi │ │ │ │ +00040e80: 6e74 733a 2033 2e32 3720 7365 636f 6e64 nts: 3.27 second │ │ │ │ +00040e90: 730a 2020 2a20 5374 7261 7465 6779 4465 s. * StrategyDe │ │ │ │ +00040ea0: 6661 756c 7457 6974 6850 6f69 6e74 733a faultWithPoints: │ │ │ │ +00040eb0: 2033 2e33 370a 526f 7567 686c 7920 7370 3.37.Roughly sp │ │ │ │ +00040ec0: 6561 6b69 6e67 2c20 6865 7572 6973 7469 eaking, heuristi │ │ │ │ +00040ed0: 6373 2074 656e 6420 746f 2070 726f 7669 cs tend to provi │ │ │ │ +00040ee0: 6465 206d 6f72 6520 696e 666f 726d 6174 de more informat │ │ │ │ +00040ef0: 696f 6e20 7468 616e 2072 616e 646f 6d0a ion than random. │ │ │ │ +00040f00: 7375 626d 6174 7269 6365 7320 616e 6420 submatrices and │ │ │ │ +00040f10: 736f 2074 6865 7920 776f 726b 206d 7563 so they work muc │ │ │ │ +00040f20: 6820 6661 7374 6572 2073 696e 6365 2074 h faster since t │ │ │ │ +00040f30: 6865 7920 636f 6e73 6964 6572 2066 6172 hey consider far │ │ │ │ +00040f40: 2066 6577 6572 0a73 7562 6d61 7472 6963 fewer.submatric │ │ │ │ +00040f50: 6573 2e20 2046 7265 7175 656e 746c 7920 es. Frequently │ │ │ │ +00040f60: 616c 736f 2c20 636f 6d70 7574 696e 6720 also, computing │ │ │ │ +00040f70: 7261 6e64 6f6d 206f 7220 7261 7469 6f6e random or ration │ │ │ │ +00040f80: 616c 2070 6f69 6e74 7320 646f 6573 2068 al points does h │ │ │ │ +00040f90: 6176 650a 6164 7661 6e74 6167 6573 2061 ave.advantages a │ │ │ │ +00040fa0: 7320 7479 7069 6361 6c6c 7920 6665 7765 s typically fewe │ │ │ │ +00040fb0: 7220 7374 696c 6c20 6d69 6e6f 7273 2061 r still minors a │ │ │ │ +00040fc0: 7265 206e 6565 6465 6420 2868 656e 6365 re needed (hence │ │ │ │ +00040fd0: 2069 6620 636f 6d70 7574 696e 670a 6d69 if computing.mi │ │ │ │ +00040fe0: 6e6f 7273 2069 7320 736c 6f77 2053 7472 nors is slow Str │ │ │ │ +00040ff0: 6174 6567 7950 6f69 6e74 7320 6973 2061 ategyPoints is a │ │ │ │ +00041000: 2067 6f6f 6420 6368 6f69 6365 292e 2020 good choice). │ │ │ │ +00041010: 486f 7765 7665 722c 2073 6f6d 6574 696d However, sometim │ │ │ │ +00041020: 6573 2074 6861 740a 6e6f 6e2d 7472 6976 es that.non-triv │ │ │ │ +00041030: 6961 6c20 706f 696e 7420 636f 6d70 7574 ial point comput │ │ │ │ +00041040: 6174 696f 6e20 7769 6c6c 2062 6563 6f6d ation will becom │ │ │ │ +00041050: 6520 7374 7563 6b20 2869 6e20 7468 6520 e stuck (in the │ │ │ │ +00041060: 6162 6f76 6520 6578 616d 706c 652c 2074 above example, t │ │ │ │ +00041070: 6865 0a6d 6564 6961 6e20 7469 6d65 2066 he.median time f │ │ │ │ +00041080: 6f72 2053 7472 6174 6567 7950 6f69 6e74 or StrategyPoint │ │ │ │ +00041090: 7320 616e 6420 5374 7261 7465 6779 4465 s and StrategyDe │ │ │ │ +000410a0: 6661 756c 7457 6974 6850 6f69 6e74 7320 faultWithPoints │ │ │ │ +000410b0: 7761 7320 636c 6f73 6520 746f 2031 2e35 was close to 1.5 │ │ │ │ +000410c0: 0a73 6563 6f6e 6473 2c20 6275 7420 6120 .seconds, but a │ │ │ │ +000410d0: 636f 7570 6c65 2072 756e 7320 696e 2065 couple runs in e │ │ │ │ +000410e0: 6163 6820 6361 7365 2077 6572 6520 6f72 ach case were or │ │ │ │ +000410f0: 6465 7273 206f 6620 6d61 676e 6974 7564 ders of magnitud │ │ │ │ +00041100: 6520 736c 6f77 6572 292e 0a0a 4375 7374 e slower)...Cust │ │ │ │ +00041110: 6f6d 2053 7472 6174 6567 6965 730a 5468 om Strategies.Th │ │ │ │ +00041120: 6520 7573 6572 2063 616e 2063 7265 6174 e user can creat │ │ │ │ +00041130: 6520 7468 6569 7220 6f77 6e20 7374 7261 e their own stra │ │ │ │ +00041140: 7465 6769 6573 2061 7320 7765 6c6c 2c20 tegies as well, │ │ │ │ +00041150: 6173 2077 6520 6e6f 7720 6578 706c 6169 as we now explai │ │ │ │ +00041160: 6e2e 2020 496e 0a70 6172 7469 6375 6c61 n. In.particula │ │ │ │ +00041170: 722c 2074 6865 2075 7365 7220 6361 6e20 r, the user can │ │ │ │ +00041180: 6576 656e 2063 7573 746f 6d69 7a65 2074 even customize t │ │ │ │ +00041190: 6865 2068 6575 7269 7374 6963 7320 7573 he heuristics us │ │ │ │ +000411a0: 6564 2e20 2053 6565 2062 656c 6f77 2066 ed. See below f │ │ │ │ +000411b0: 6f72 2068 6f77 0a74 6f20 6561 7369 6c79 or how.to easily │ │ │ │ +000411c0: 2075 7365 206f 6e6c 7920 6120 7369 6e67 use only a sing │ │ │ │ +000411d0: 6c65 2068 6575 7269 7374 6963 2e20 546f le heuristic. To │ │ │ │ +000411e0: 2063 7573 746f 6d20 7374 7261 7465 6779 custom strategy │ │ │ │ +000411f0: 2069 7320 7370 6563 6966 6965 6420 6279 is specified by │ │ │ │ +00041200: 2061 0a48 6173 6854 6162 6c65 2077 6869 a.HashTable whi │ │ │ │ +00041210: 6368 206d 7573 7420 6861 7665 2074 6865 ch must have the │ │ │ │ +00041220: 2066 6f6c 6c6f 7769 6e67 206b 6579 732e following keys. │ │ │ │ +00041230: 0a20 202a 2047 5265 764c 6578 4c61 7267 . * GRevLexLarg │ │ │ │ +00041240: 6573 743a 2074 7279 2074 6f20 6669 6e64 est: try to find │ │ │ │ +00041250: 2073 7562 6d61 7472 6963 6573 2077 6865 submatrices whe │ │ │ │ +00041260: 7265 2065 6163 6820 726f 7720 616e 6420 re each row and │ │ │ │ +00041270: 636f 6c75 6d6e 2068 6173 2061 0a20 2020 column has a. │ │ │ │ +00041280: 206c 6172 6765 2065 6e74 7279 2077 6974 large entry wit │ │ │ │ +00041290: 6820 7265 7370 6563 7420 746f 2061 2072 h respect to a r │ │ │ │ +000412a0: 616e 646f 6d20 4752 6576 4c65 786f 7264 andom GRevLexord │ │ │ │ +000412b0: 6572 2e0a 2020 2a20 4752 6576 4c65 7853 er.. * GRevLexS │ │ │ │ +000412c0: 6d61 6c6c 6573 743a 2074 7279 2074 6f20 mallest: try to │ │ │ │ +000412d0: 6669 6e64 2073 7562 6d61 7472 6963 6573 find submatrices │ │ │ │ +000412e0: 2077 6865 7265 2065 6163 6820 726f 7720 where each row │ │ │ │ +000412f0: 616e 6420 636f 6c75 6d6e 2068 6173 2061 and column has a │ │ │ │ +00041300: 0a20 2020 2073 6d61 6c6c 2065 6e74 7279 . small entry │ │ │ │ +00041310: 2077 6974 6820 7265 7370 6563 7420 746f with respect to │ │ │ │ +00041320: 2061 2072 616e 646f 6d20 4752 6576 4c65 a random GRevLe │ │ │ │ +00041330: 786f 7264 6572 2e0a 2020 2a20 4752 6576 xorder.. * GRev │ │ │ │ +00041340: 4c65 7853 6d61 6c6c 6573 7454 6572 6d3a LexSmallestTerm: │ │ │ │ +00041350: 2066 696e 6420 7375 626d 6174 7269 6365 find submatrice │ │ │ │ +00041360: 7320 7768 6572 6520 6561 6368 2072 6f77 s where each row │ │ │ │ +00041370: 2061 6e64 2063 6f6c 756d 6e20 6861 7320 and column has │ │ │ │ +00041380: 616e 0a20 2020 2065 6e74 7279 2077 6974 an. entry wit │ │ │ │ +00041390: 6820 6120 736d 616c 6c20 7465 726d 2077 h a small term w │ │ │ │ +000413a0: 6974 6820 7265 7370 6563 7420 746f 2061 ith respect to a │ │ │ │ +000413b0: 2072 616e 646f 6d20 4752 6576 4c65 786f random GRevLexo │ │ │ │ +000413c0: 7264 6572 2e0a 2020 2a20 4c65 784c 6172 rder.. * LexLar │ │ │ │ +000413d0: 6765 7374 3a20 7472 7920 746f 2066 696e gest: try to fin │ │ │ │ +000413e0: 6420 7375 626d 6174 7269 6365 7320 7768 d submatrices wh │ │ │ │ +000413f0: 6572 6520 6561 6368 2072 6f77 2061 6e64 ere each row and │ │ │ │ +00041400: 2063 6f6c 756d 6e20 6861 7320 6120 6c61 column has a la │ │ │ │ +00041410: 7267 650a 2020 2020 656e 7472 7920 7769 rge. entry wi │ │ │ │ +00041420: 7468 2072 6573 7065 6374 2074 6f20 6120 th respect to a │ │ │ │ +00041430: 7261 6e64 6f6d 204c 6578 6f72 6465 722e random Lexorder. │ │ │ │ +00041440: 0a20 202a 204c 6578 536d 616c 6c65 7374 . * LexSmallest │ │ │ │ +00041450: 3a20 7472 7920 746f 2066 696e 6420 7375 : try to find su │ │ │ │ +00041460: 626d 6174 7269 6365 7320 7768 6572 6520 bmatrices where │ │ │ │ +00041470: 6561 6368 2072 6f77 2061 6e64 2063 6f6c each row and col │ │ │ │ +00041480: 756d 6e20 6861 7320 6120 736d 616c 6c0a umn has a small. │ │ │ │ +00041490: 2020 2020 656e 7472 7920 7769 7468 2072 entry with r │ │ │ │ +000414a0: 6573 7065 6374 2074 6f20 6120 7261 6e64 espect to a rand │ │ │ │ +000414b0: 6f6d 204c 6578 6f72 6465 722e 0a20 202a om Lexorder.. * │ │ │ │ +000414c0: 204c 6578 536d 616c 6c65 7374 5465 726d LexSmallestTerm │ │ │ │ +000414d0: 3a20 6669 6e64 2073 7562 6d61 7472 6963 : find submatric │ │ │ │ +000414e0: 6573 2077 6865 7265 2065 6163 6820 726f es where each ro │ │ │ │ +000414f0: 7720 616e 6420 636f 6c75 6d6e 2068 6173 w and column has │ │ │ │ +00041500: 2061 6e20 656e 7472 790a 2020 2020 7769 an entry. wi │ │ │ │ +00041510: 7468 2061 2073 6d61 6c6c 2074 6572 6d20 th a small term │ │ │ │ +00041520: 7769 7468 2072 6573 7065 6374 2074 6f20 with respect to │ │ │ │ +00041530: 6120 7261 6e64 6f6d 204c 6578 6f72 6465 a random Lexorde │ │ │ │ +00041540: 722e 0a20 202a 2052 616e 646f 6d3a 2066 r.. * Random: f │ │ │ │ +00041550: 696e 6420 7261 6e64 6f6d 2073 7562 6d61 ind random subma │ │ │ │ +00041560: 7472 6963 6573 200a 2020 2a20 5261 6e64 trices . * Rand │ │ │ │ +00041570: 6f6d 4e6f 6e7a 6572 6f3a 2066 696e 6420 omNonzero: find │ │ │ │ +00041580: 7261 6e64 6f6d 2073 7562 6d61 7472 6963 random submatric │ │ │ │ +00041590: 6573 2074 6861 7420 6861 7665 206e 6f6e es that have non │ │ │ │ +000415a0: 7a65 726f 2072 6f77 7320 616e 6420 636f zero rows and co │ │ │ │ +000415b0: 6c75 6d6e 730a 2020 2a20 506f 696e 7473 lumns. * Points │ │ │ │ +000415c0: 3a20 6669 6e64 2073 7562 6d61 7472 6963 : find submatric │ │ │ │ +000415d0: 6573 2074 6861 7420 6172 6520 6e6f 7420 es that are not │ │ │ │ +000415e0: 7369 6e67 756c 6172 2061 7420 7468 6520 singular at the │ │ │ │ +000415f0: 6769 7665 6e20 6964 6561 6c20 6279 0a20 given ideal by. │ │ │ │ +00041600: 2020 2066 696e 6469 6e67 2061 2070 6f69 finding a poi │ │ │ │ +00041610: 6e74 2077 6865 7265 2074 6861 7420 6964 nt where that id │ │ │ │ +00041620: 6561 6c20 7661 6e69 7368 6573 2c20 616e eal vanishes, an │ │ │ │ +00041630: 6420 6576 616c 7561 7469 6e67 2074 6865 d evaluating the │ │ │ │ +00041640: 206d 6174 7269 7820 6174 0a20 2020 2074 matrix at. t │ │ │ │ +00041650: 6861 7420 706f 696e 7420 2876 6961 2074 hat point (via t │ │ │ │ +00041660: 6865 2070 6163 6b61 6765 202a 6e6f 7465 he package *note │ │ │ │ +00041670: 2052 616e 646f 6d50 6f69 6e74 733a 2028 RandomPoints: ( │ │ │ │ +00041680: 5261 6e64 6f6d 506f 696e 7473 2954 6f70 RandomPoints)Top │ │ │ │ +00041690: 2c29 2e20 2049 660a 2020 2020 776f 726b ,). If. work │ │ │ │ +000416a0: 696e 6720 6f76 6572 2061 2063 6861 7261 ing over a chara │ │ │ │ +000416b0: 6374 6572 6973 7469 6320 7a65 726f 2066 cteristic zero f │ │ │ │ +000416c0: 6965 6c64 2c20 7468 6973 2077 696c 6c20 ield, this will │ │ │ │ +000416d0: 7365 6c65 6374 2072 616e 646f 6d0a 2020 select random. │ │ │ │ +000416e0: 2020 7375 626d 6174 7269 6365 732e 2020 submatrices. │ │ │ │ +000416f0: 546f 2061 6363 6573 7320 6f70 7469 6f6e To access option │ │ │ │ +00041700: 7320 666f 7220 7468 6174 2070 6163 6b61 s for that packa │ │ │ │ +00041710: 6765 2c20 7365 7420 7468 6520 2a6e 6f74 ge, set the *not │ │ │ │ +00041720: 650a 2020 2020 506f 696e 744f 7074 696f e. PointOptio │ │ │ │ +00041730: 6e73 3a20 506f 696e 744f 7074 696f 6e73 ns: PointOptions │ │ │ │ +00041740: 2c20 6f70 7469 6f6e 2e0a 466f 7220 6578 , option..For ex │ │ │ │ +00041750: 616d 706c 653a 0a2b 2d2d 2d2d 2d2d 2d2d ample:.+-------- │ │ │ │ +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 2b0a 7c69 ------------+.|i │ │ │ │ -00041790: 3320 3a20 7065 656b 2053 7472 6174 6567 3 : peek Strateg │ │ │ │ -000417a0: 7944 6566 6175 6c74 2020 2020 2020 2020 yDefault │ │ │ │ -000417b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00041780: 2d2d 2d2b 0a7c 6933 203a 2070 6565 6b20 ---+.|i3 : peek │ │ │ │ +00041790: 5374 7261 7465 6779 4465 6661 756c 7420 StrategyDefault │ │ │ │ +000417a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000417b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000417c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000417d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000417e0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -000417f0: 4f70 7469 6f6e 5461 626c 657b 4752 6576 OptionTable{GRev │ │ │ │ -00041800: 4c65 784c 6172 6765 7374 203d 3e20 3020 LexLargest => 0 │ │ │ │ -00041810: 2020 2020 207d 7c0a 7c20 2020 2020 2020 }|.| │ │ │ │ -00041820: 2020 2020 2020 2020 2020 4752 6576 4c65 GRevLe │ │ │ │ -00041830: 7853 6d61 6c6c 6573 7420 3d3e 2031 3620 xSmallest => 16 │ │ │ │ -00041840: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00041850: 2020 2020 2020 2020 4752 6576 4c65 7853 GRevLexS │ │ │ │ -00041860: 6d61 6c6c 6573 7454 6572 6d20 3d3e 2031 mallestTerm => 1 │ │ │ │ -00041870: 3620 7c0a 7c20 2020 2020 2020 2020 2020 6 |.| │ │ │ │ -00041880: 2020 2020 2020 4c65 784c 6172 6765 7374 LexLargest │ │ │ │ -00041890: 203d 3e20 3020 2020 2020 2020 2020 2020 => 0 │ │ │ │ -000418a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000418b0: 2020 2020 4c65 7853 6d61 6c6c 6573 7420 LexSmallest │ │ │ │ -000418c0: 3d3e 2031 3620 2020 2020 2020 2020 7c0a => 16 |. │ │ │ │ -000418d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000418e0: 2020 4c65 7853 6d61 6c6c 6573 7454 6572 LexSmallestTer │ │ │ │ -000418f0: 6d20 3d3e 2031 3620 2020 2020 7c0a 7c20 m => 16 |.| │ │ │ │ -00041900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041910: 506f 696e 7473 203d 3e20 3020 2020 2020 Points => 0 │ │ │ │ -00041920: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00041930: 2020 2020 2020 2020 2020 2020 2020 5261 Ra │ │ │ │ -00041940: 6e64 6f6d 203d 3e20 3136 2020 2020 2020 ndom => 16 │ │ │ │ -00041950: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00041960: 2020 2020 2020 2020 2020 2020 5261 6e64 Rand │ │ │ │ -00041970: 6f6d 4e6f 6e7a 6572 6f20 3d3e 2031 3620 omNonzero => 16 │ │ │ │ -00041980: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +000417d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000417e0: 0a7c 6f33 203d 204f 7074 696f 6e54 6162 .|o3 = OptionTab │ │ │ │ +000417f0: 6c65 7b47 5265 764c 6578 4c61 7267 6573 le{GRevLexLarges │ │ │ │ +00041800: 7420 3d3e 2030 2020 2020 2020 7d7c 0a7c t => 0 }|.| │ │ │ │ +00041810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041820: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ +00041830: 203d 3e20 3136 2020 2020 207c 0a7c 2020 => 16 |.| │ │ │ │ +00041840: 2020 2020 2020 2020 2020 2020 2020 2047 G │ │ │ │ +00041850: 5265 764c 6578 536d 616c 6c65 7374 5465 RevLexSmallestTe │ │ │ │ +00041860: 726d 203d 3e20 3136 207c 0a7c 2020 2020 rm => 16 |.| │ │ │ │ +00041870: 2020 2020 2020 2020 2020 2020 204c 6578 Lex │ │ │ │ +00041880: 4c61 7267 6573 7420 3d3e 2030 2020 2020 Largest => 0 │ │ │ │ +00041890: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000418a0: 2020 2020 2020 2020 2020 204c 6578 536d LexSm │ │ │ │ +000418b0: 616c 6c65 7374 203d 3e20 3136 2020 2020 allest => 16 │ │ │ │ +000418c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000418d0: 2020 2020 2020 2020 204c 6578 536d 616c LexSmal │ │ │ │ +000418e0: 6c65 7374 5465 726d 203d 3e20 3136 2020 lestTerm => 16 │ │ │ │ +000418f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041900: 2020 2020 2020 2050 6f69 6e74 7320 3d3e Points => │ │ │ │ +00041910: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +00041920: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041930: 2020 2020 2052 616e 646f 6d20 3d3e 2031 Random => 1 │ │ │ │ +00041940: 3620 2020 2020 2020 2020 2020 2020 207c 6 | │ │ │ │ +00041950: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00041960: 2020 2052 616e 646f 6d4e 6f6e 7a65 726f RandomNonzero │ │ │ │ +00041970: 203d 3e20 3136 2020 2020 2020 207c 0a2b => 16 |.+ │ │ │ │ +00041980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000419a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000419b0: 2d2d 2d2d 2b0a 4561 6368 2073 7563 6820 ----+.Each such │ │ │ │ -000419c0: 6b65 7920 7368 6f75 6c64 2070 6f69 6e74 key should point │ │ │ │ -000419d0: 2074 6f20 616e 2069 6e74 6567 6572 2e20 to an integer. │ │ │ │ -000419e0: 2054 6865 206c 6172 6765 7220 7468 6520 The larger the │ │ │ │ -000419f0: 696e 7465 6765 722c 2074 6865 206d 6f72 integer, the mor │ │ │ │ -00041a00: 650a 6c69 6b65 6c79 2074 6861 7420 7375 e.likely that su │ │ │ │ -00041a10: 6368 2061 206d 696e 6f72 2077 696c 6c20 ch a minor will │ │ │ │ -00041a20: 6265 2063 686f 7365 6e2e 0a0a 4675 6e63 be chosen...Func │ │ │ │ -00041a30: 7469 6f6e 7320 7375 6368 2061 7320 2a6e tions such as *n │ │ │ │ -00041a40: 6f74 6520 6368 6f6f 7365 476f 6f64 4d69 ote chooseGoodMi │ │ │ │ -00041a50: 6e6f 7273 3a20 6368 6f6f 7365 476f 6f64 nors: chooseGood │ │ │ │ -00041a60: 4d69 6e6f 7273 2c20 7769 6c6c 2073 656c Minors, will sel │ │ │ │ -00041a70: 6563 7420 610a 6e75 6d62 6572 206f 6620 ect a.number of │ │ │ │ -00041a80: 7261 6e64 6f6d 2073 7562 6d61 7472 6963 random submatric │ │ │ │ -00041a90: 6573 2062 6173 6564 206f 6e20 7468 6520 es based on the │ │ │ │ -00041aa0: 7661 6c75 6573 206f 6620 7468 6f73 6520 values of those │ │ │ │ -00041ab0: 6b65 7973 2e20 2046 6f72 2065 7861 6d70 keys. For examp │ │ │ │ -00041ac0: 6c65 2c0a 6966 204c 6578 536d 616c 6c65 le,.if LexSmalle │ │ │ │ -00041ad0: 7374 2061 6e64 204c 6578 4c61 7267 6573 st and LexLarges │ │ │ │ -00041ae0: 7420 6172 6520 7365 7420 746f 2035 3020 t are set to 50 │ │ │ │ -00041af0: 6170 7072 6f78 696d 6174 656c 7920 7468 approximately th │ │ │ │ -00041b00: 6520 7375 626d 6174 7269 6373 2077 696c e submatrics wil │ │ │ │ -00041b10: 6c0a 6265 2073 6d61 6c6c 6573 7420 7769 l.be smallest wi │ │ │ │ -00041b20: 7468 2072 6573 7065 6374 2074 6f20 4c65 th respect to Le │ │ │ │ -00041b30: 7820 616e 6420 7468 6520 6f74 6865 7220 x and the other │ │ │ │ -00041b40: 6861 6c66 2077 696c 6c20 6265 206c 6172 half will be lar │ │ │ │ -00041b50: 6765 7374 2077 6974 6820 7265 7370 6563 gest with respec │ │ │ │ -00041b60: 740a 746f 204c 6578 2e54 6865 2076 616c t.to Lex.The val │ │ │ │ -00041b70: 7565 7320 646f 206e 6f74 206e 6565 6420 ues do not need │ │ │ │ -00041b80: 746f 2061 6464 2075 7020 746f 2031 3030 to add up to 100 │ │ │ │ -00041b90: 2e0a 0a54 6865 2068 6575 7269 7374 6963 ...The heuristic │ │ │ │ -00041ba0: 2066 756e 6374 696f 6e73 2061 6c6c 2077 functions all w │ │ │ │ -00041bb0: 6f72 6b20 6279 2066 696e 6469 6e67 2074 ork by finding t │ │ │ │ -00041bc0: 6865 206f 7074 696d 616c 2065 6e74 7279 he optimal entry │ │ │ │ -00041bd0: 2077 6974 6820 7265 7370 6563 7420 746f with respect to │ │ │ │ -00041be0: 0a74 6865 2067 6976 656e 2073 7472 6174 .the given strat │ │ │ │ -00041bf0: 6567 792c 2072 656d 6f76 696e 6720 7468 egy, removing th │ │ │ │ -00041c00: 6174 2072 6f77 2061 6e64 2063 6f6c 756d at row and colum │ │ │ │ -00041c10: 6e2c 2061 6e64 2074 6865 6e20 6368 6f6f n, and then choo │ │ │ │ -00041c20: 7369 6e67 2074 6865 206e 6578 740a 6f70 sing the next.op │ │ │ │ -00041c30: 7469 6d61 6c20 656e 7472 792e 2020 5468 timal entry. Th │ │ │ │ -00041c40: 6973 2069 7320 646f 6e65 2075 6e74 696c is is done until │ │ │ │ -00041c50: 2061 2073 7562 6d61 7472 6978 206f 6620 a submatrix of │ │ │ │ -00041c60: 7468 6520 6465 7369 7265 6420 7369 7a65 the desired size │ │ │ │ -00041c70: 2068 6173 2062 6565 6e0a 666f 756e 642e has been.found. │ │ │ │ -00041c80: 0a0a 496e 2073 6f6d 6520 6675 6e63 7469 ..In some functi │ │ │ │ -00041c90: 6f6e 732c 2074 6865 2047 5265 764c 6578 ons, the GRevLex │ │ │ │ -00041ca0: 2076 6572 7369 6f6e 7320 6f66 2074 6869 versions of thi │ │ │ │ -00041cb0: 7320 7374 7261 7465 6779 2077 696c 6c20 s strategy will │ │ │ │ -00041cc0: 6d6f 6469 6679 2074 6865 0a77 6f72 6b69 modify the.worki │ │ │ │ -00041cd0: 6e67 206d 6174 7269 7820 696e 2061 206c ng matrix in a l │ │ │ │ -00041ce0: 6f6f 702c 2072 6570 6561 7465 646c 7920 oop, repeatedly │ │ │ │ -00041cf0: 6c6f 7765 7269 6e67 2f72 6169 7369 6e67 lowering/raising │ │ │ │ -00041d00: 2074 6865 2064 6567 7265 6520 6f66 2065 the degree of e │ │ │ │ -00041d10: 6c65 6d65 6e74 7373 6f0a 6173 2074 6f20 lementsso.as to │ │ │ │ -00041d20: 656e 7375 7265 2074 6861 7420 6469 6666 ensure that diff │ │ │ │ -00041d30: 6572 656e 7420 6368 6f69 6365 7320 6172 erent choices ar │ │ │ │ -00041d40: 6520 6d61 6465 2e0a 0a57 6520 6272 6965 e made...We brie │ │ │ │ -00041d50: 666c 7920 7375 6d6d 6172 697a 6520 7468 fly summarize th │ │ │ │ -00041d60: 6520 5374 7261 7465 6769 6573 2070 726f e Strategies pro │ │ │ │ -00041d70: 7669 6465 6420 746f 2074 6865 2075 7365 vided to the use │ │ │ │ -00041d80: 7220 6279 2064 6566 6175 6c74 2028 736f r by default (so │ │ │ │ -00041d90: 6d65 206f 660a 7768 6963 6820 7765 2068 me of.which we h │ │ │ │ -00041da0: 6176 6520 7365 656e 2069 6e20 6163 7469 ave seen in acti │ │ │ │ -00041db0: 6f6e 2061 626f 7665 290a 2020 2a20 5374 on above). * St │ │ │ │ -00041dc0: 7261 7465 6779 4465 6661 756c 743a 2031 rategyDefault: 1 │ │ │ │ -00041dd0: 3625 206f 6620 7468 6520 6d61 7472 6963 6% of the matric │ │ │ │ -00041de0: 6573 2061 7265 204c 6578 536d 616c 6c65 es are LexSmalle │ │ │ │ -00041df0: 7374 2c20 4c65 7853 6d61 6c6c 6573 7454 st, LexSmallestT │ │ │ │ -00041e00: 6572 6d2c 0a20 2020 2047 5265 764c 6578 erm,. GRevLex │ │ │ │ -00041e10: 536d 616c 6c65 7374 2c20 4752 6576 4c65 Smallest, GRevLe │ │ │ │ -00041e20: 784c 6172 6765 7374 2c20 5261 6e64 6f6d xLargest, Random │ │ │ │ -00041e30: 2c20 616e 6420 5261 6e64 6f6d 4e6f 6e5a , and RandomNonZ │ │ │ │ -00041e40: 6572 6f20 6561 6368 0a20 202a 2053 7472 ero each. * Str │ │ │ │ -00041e50: 6174 6567 7944 6566 6175 6c74 4e6f 6e52 ategyDefaultNonR │ │ │ │ -00041e60: 616e 646f 6d3a 2032 3525 206f 6620 7468 andom: 25% of th │ │ │ │ -00041e70: 6520 6d61 7472 6963 6573 2061 7265 204c e matrices are L │ │ │ │ -00041e80: 6578 536d 616c 6c65 7374 2c0a 2020 2020 exSmallest,. │ │ │ │ -00041e90: 4c65 7853 6d61 6c6c 6573 7454 6572 6d2c LexSmallestTerm, │ │ │ │ -00041ea0: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ -00041eb0: 2061 6e64 2c20 4752 6576 4c65 784c 6172 and, GRevLexLar │ │ │ │ -00041ec0: 6765 7374 2065 6163 680a 2020 2a20 5374 gest each. * St │ │ │ │ -00041ed0: 7261 7465 6779 4c65 7853 6d61 6c6c 6573 rategyLexSmalles │ │ │ │ -00041ee0: 743a 2035 3025 206f 6620 7468 6520 6d61 t: 50% of the ma │ │ │ │ -00041ef0: 7472 6963 6573 2061 7265 204c 6578 536d trices are LexSm │ │ │ │ -00041f00: 616c 6c65 7374 2061 6e64 2035 3025 2061 allest and 50% a │ │ │ │ -00041f10: 7265 0a20 2020 204c 6578 536d 616c 6c65 re. LexSmalle │ │ │ │ -00041f20: 7374 5465 726d 0a20 202a 2053 7472 6174 stTerm. * Strat │ │ │ │ -00041f30: 6567 7947 5265 764c 6578 536d 616c 6c65 egyGRevLexSmalle │ │ │ │ -00041f40: 7374 3a20 3530 2520 6f66 2074 6865 206d st: 50% of the m │ │ │ │ -00041f50: 6174 7269 6365 7320 6172 6520 4752 6576 atrices are GRev │ │ │ │ -00041f60: 4c65 7853 6d61 6c6c 6573 7420 616e 6420 LexSmallest and │ │ │ │ -00041f70: 3530 250a 2020 2020 6172 6520 4752 6576 50%. are GRev │ │ │ │ -00041f80: 4c65 784c 6172 6765 7374 0a20 202a 2053 LexLargest. * S │ │ │ │ -00041f90: 7472 6174 6567 7952 616e 646f 6d3a 2063 trategyRandom: c │ │ │ │ -00041fa0: 686f 6f73 6573 2031 3030 2520 7261 6e64 hooses 100% rand │ │ │ │ -00041fb0: 6f6d 2073 7562 6d61 7472 6963 6573 2e0a om submatrices.. │ │ │ │ -00041fc0: 2020 2a20 5374 7261 7465 6779 506f 696e * StrategyPoin │ │ │ │ -00041fd0: 7473 3a20 6368 6f6f 7365 2061 6c6c 2073 ts: choose all s │ │ │ │ -00041fe0: 7562 6d61 7472 6963 6573 2076 6961 2050 ubmatrices via P │ │ │ │ -00041ff0: 6f69 6e74 732e 0a20 202a 2053 7472 6174 oints.. * Strat │ │ │ │ -00042000: 6567 7944 6566 6175 6c74 5769 7468 506f egyDefaultWithPo │ │ │ │ -00042010: 696e 7473 3a20 6c69 6b65 2053 7472 6174 ints: like Strat │ │ │ │ -00042020: 6567 7944 6566 6175 6c74 2062 7574 2072 egyDefault but r │ │ │ │ -00042030: 6570 6c61 6365 7320 7468 6520 5261 6e64 eplaces the Rand │ │ │ │ -00042040: 6f6d 2061 6e64 0a20 2020 2052 616e 646f om and. Rando │ │ │ │ -00042050: 6d4e 6f6e 5a65 726f 2073 7562 6d61 7472 mNonZero submatr │ │ │ │ -00042060: 6963 6573 2061 7320 7769 7468 206d 6174 ices as with mat │ │ │ │ -00042070: 7269 6365 7320 6368 6f73 656e 2061 7320 rices chosen as │ │ │ │ -00042080: 696e 2050 6f69 6e74 732e 0a41 6464 6974 in Points..Addit │ │ │ │ -00042090: 696f 6e61 6c6c 792c 2061 204d 7574 6162 ionally, a Mutab │ │ │ │ -000420a0: 6c65 4861 7368 5461 626c 6520 6e61 6d65 leHashTable name │ │ │ │ -000420b0: 6420 5374 7261 7465 6779 4375 7272 656e d StrategyCurren │ │ │ │ -000420c0: 7420 6973 2061 6c73 6f20 6578 706f 7274 t is also export │ │ │ │ -000420d0: 6564 2e20 2049 740a 6265 6769 6e73 2061 ed. It.begins a │ │ │ │ -000420e0: 7320 7468 6520 6465 6661 756c 7420 7374 s the default st │ │ │ │ -000420f0: 7261 7465 6779 2c20 6275 7420 7468 6520 rategy, but the │ │ │ │ -00042100: 7573 6572 2063 616e 206d 6f64 6966 7920 user can modify │ │ │ │ -00042110: 6974 2e0a 0a55 7369 6e67 2061 2073 696e it...Using a sin │ │ │ │ -00042120: 676c 6520 6865 7572 6973 7469 6320 2041 gle heuristic A │ │ │ │ -00042130: 6c74 6572 6e61 7469 7665 6c79 2c20 6966 lternatively, if │ │ │ │ -00042140: 2074 6865 2075 7365 7220 6f6e 6c79 2077 the user only w │ │ │ │ -00042150: 616e 7473 2074 6f20 7573 6520 7361 790a ants to use say. │ │ │ │ -00042160: 4c65 7853 6d61 6c6c 6573 7454 6572 6d20 LexSmallestTerm │ │ │ │ -00042170: 7468 6579 2063 616e 2073 6574 2c20 5374 they can set, St │ │ │ │ -00042180: 7261 7465 6779 2074 6f20 706f 696e 7420 rategy to point │ │ │ │ -00042190: 746f 2074 6861 7420 7379 6d62 6f6c 2c20 to that symbol, │ │ │ │ -000421a0: 696e 7374 6561 6420 6f66 2061 0a63 7265 instead of a.cre │ │ │ │ -000421b0: 6174 696e 6720 6120 6375 7374 6f6d 2073 ating a custom s │ │ │ │ -000421c0: 7472 6174 6567 7920 4861 7368 5461 626c trategy HashTabl │ │ │ │ -000421d0: 652e 2020 466f 7220 6578 616d 706c 653a e. For example: │ │ │ │ -000421e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000419a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a45 6163 -----------+.Eac │ │ │ │ +000419b0: 6820 7375 6368 206b 6579 2073 686f 756c h such key shoul │ │ │ │ +000419c0: 6420 706f 696e 7420 746f 2061 6e20 696e d point to an in │ │ │ │ +000419d0: 7465 6765 722e 2020 5468 6520 6c61 7267 teger. The larg │ │ │ │ +000419e0: 6572 2074 6865 2069 6e74 6567 6572 2c20 er the integer, │ │ │ │ +000419f0: 7468 6520 6d6f 7265 0a6c 696b 656c 7920 the more.likely │ │ │ │ +00041a00: 7468 6174 2073 7563 6820 6120 6d69 6e6f that such a mino │ │ │ │ +00041a10: 7220 7769 6c6c 2062 6520 6368 6f73 656e r will be chosen │ │ │ │ +00041a20: 2e0a 0a46 756e 6374 696f 6e73 2073 7563 ...Functions suc │ │ │ │ +00041a30: 6820 6173 202a 6e6f 7465 2063 686f 6f73 h as *note choos │ │ │ │ +00041a40: 6547 6f6f 644d 696e 6f72 733a 2063 686f eGoodMinors: cho │ │ │ │ +00041a50: 6f73 6547 6f6f 644d 696e 6f72 732c 2077 oseGoodMinors, w │ │ │ │ +00041a60: 696c 6c20 7365 6c65 6374 2061 0a6e 756d ill select a.num │ │ │ │ +00041a70: 6265 7220 6f66 2072 616e 646f 6d20 7375 ber of random su │ │ │ │ +00041a80: 626d 6174 7269 6365 7320 6261 7365 6420 bmatrices based │ │ │ │ +00041a90: 6f6e 2074 6865 2076 616c 7565 7320 6f66 on the values of │ │ │ │ +00041aa0: 2074 686f 7365 206b 6579 732e 2020 466f those keys. Fo │ │ │ │ +00041ab0: 7220 6578 616d 706c 652c 0a69 6620 4c65 r example,.if Le │ │ │ │ +00041ac0: 7853 6d61 6c6c 6573 7420 616e 6420 4c65 xSmallest and Le │ │ │ │ +00041ad0: 784c 6172 6765 7374 2061 7265 2073 6574 xLargest are set │ │ │ │ +00041ae0: 2074 6f20 3530 2061 7070 726f 7869 6d61 to 50 approxima │ │ │ │ +00041af0: 7465 6c79 2074 6865 2073 7562 6d61 7472 tely the submatr │ │ │ │ +00041b00: 6963 7320 7769 6c6c 0a62 6520 736d 616c ics will.be smal │ │ │ │ +00041b10: 6c65 7374 2077 6974 6820 7265 7370 6563 lest with respec │ │ │ │ +00041b20: 7420 746f 204c 6578 2061 6e64 2074 6865 t to Lex and the │ │ │ │ +00041b30: 206f 7468 6572 2068 616c 6620 7769 6c6c other half will │ │ │ │ +00041b40: 2062 6520 6c61 7267 6573 7420 7769 7468 be largest with │ │ │ │ +00041b50: 2072 6573 7065 6374 0a74 6f20 4c65 782e respect.to Lex. │ │ │ │ +00041b60: 5468 6520 7661 6c75 6573 2064 6f20 6e6f The values do no │ │ │ │ +00041b70: 7420 6e65 6564 2074 6f20 6164 6420 7570 t need to add up │ │ │ │ +00041b80: 2074 6f20 3130 302e 0a0a 5468 6520 6865 to 100...The he │ │ │ │ +00041b90: 7572 6973 7469 6320 6675 6e63 7469 6f6e uristic function │ │ │ │ +00041ba0: 7320 616c 6c20 776f 726b 2062 7920 6669 s all work by fi │ │ │ │ +00041bb0: 6e64 696e 6720 7468 6520 6f70 7469 6d61 nding the optima │ │ │ │ +00041bc0: 6c20 656e 7472 7920 7769 7468 2072 6573 l entry with res │ │ │ │ +00041bd0: 7065 6374 2074 6f0a 7468 6520 6769 7665 pect to.the give │ │ │ │ +00041be0: 6e20 7374 7261 7465 6779 2c20 7265 6d6f n strategy, remo │ │ │ │ +00041bf0: 7669 6e67 2074 6861 7420 726f 7720 616e ving that row an │ │ │ │ +00041c00: 6420 636f 6c75 6d6e 2c20 616e 6420 7468 d column, and th │ │ │ │ +00041c10: 656e 2063 686f 6f73 696e 6720 7468 6520 en choosing the │ │ │ │ +00041c20: 6e65 7874 0a6f 7074 696d 616c 2065 6e74 next.optimal ent │ │ │ │ +00041c30: 7279 2e20 2054 6869 7320 6973 2064 6f6e ry. This is don │ │ │ │ +00041c40: 6520 756e 7469 6c20 6120 7375 626d 6174 e until a submat │ │ │ │ +00041c50: 7269 7820 6f66 2074 6865 2064 6573 6972 rix of the desir │ │ │ │ +00041c60: 6564 2073 697a 6520 6861 7320 6265 656e ed size has been │ │ │ │ +00041c70: 0a66 6f75 6e64 2e0a 0a49 6e20 736f 6d65 .found...In some │ │ │ │ +00041c80: 2066 756e 6374 696f 6e73 2c20 7468 6520 functions, the │ │ │ │ +00041c90: 4752 6576 4c65 7820 7665 7273 696f 6e73 GRevLex versions │ │ │ │ +00041ca0: 206f 6620 7468 6973 2073 7472 6174 6567 of this strateg │ │ │ │ +00041cb0: 7920 7769 6c6c 206d 6f64 6966 7920 7468 y will modify th │ │ │ │ +00041cc0: 650a 776f 726b 696e 6720 6d61 7472 6978 e.working matrix │ │ │ │ +00041cd0: 2069 6e20 6120 6c6f 6f70 2c20 7265 7065 in a loop, repe │ │ │ │ +00041ce0: 6174 6564 6c79 206c 6f77 6572 696e 672f atedly lowering/ │ │ │ │ +00041cf0: 7261 6973 696e 6720 7468 6520 6465 6772 raising the degr │ │ │ │ +00041d00: 6565 206f 6620 656c 656d 656e 7473 736f ee of elementsso │ │ │ │ +00041d10: 0a61 7320 746f 2065 6e73 7572 6520 7468 .as to ensure th │ │ │ │ +00041d20: 6174 2064 6966 6665 7265 6e74 2063 686f at different cho │ │ │ │ +00041d30: 6963 6573 2061 7265 206d 6164 652e 0a0a ices are made... │ │ │ │ +00041d40: 5765 2062 7269 6566 6c79 2073 756d 6d61 We briefly summa │ │ │ │ +00041d50: 7269 7a65 2074 6865 2053 7472 6174 6567 rize the Strateg │ │ │ │ +00041d60: 6965 7320 7072 6f76 6964 6564 2074 6f20 ies provided to │ │ │ │ +00041d70: 7468 6520 7573 6572 2062 7920 6465 6661 the user by defa │ │ │ │ +00041d80: 756c 7420 2873 6f6d 6520 6f66 0a77 6869 ult (some of.whi │ │ │ │ +00041d90: 6368 2077 6520 6861 7665 2073 6565 6e20 ch we have seen │ │ │ │ +00041da0: 696e 2061 6374 696f 6e20 6162 6f76 6529 in action above) │ │ │ │ +00041db0: 0a20 202a 2053 7472 6174 6567 7944 6566 . * StrategyDef │ │ │ │ +00041dc0: 6175 6c74 3a20 3136 2520 6f66 2074 6865 ault: 16% of the │ │ │ │ +00041dd0: 206d 6174 7269 6365 7320 6172 6520 4c65 matrices are Le │ │ │ │ +00041de0: 7853 6d61 6c6c 6573 742c 204c 6578 536d xSmallest, LexSm │ │ │ │ +00041df0: 616c 6c65 7374 5465 726d 2c0a 2020 2020 allestTerm,. │ │ │ │ +00041e00: 4752 6576 4c65 7853 6d61 6c6c 6573 742c GRevLexSmallest, │ │ │ │ +00041e10: 2047 5265 764c 6578 4c61 7267 6573 742c GRevLexLargest, │ │ │ │ +00041e20: 2052 616e 646f 6d2c 2061 6e64 2052 616e Random, and Ran │ │ │ │ +00041e30: 646f 6d4e 6f6e 5a65 726f 2065 6163 680a domNonZero each. │ │ │ │ +00041e40: 2020 2a20 5374 7261 7465 6779 4465 6661 * StrategyDefa │ │ │ │ +00041e50: 756c 744e 6f6e 5261 6e64 6f6d 3a20 3235 ultNonRandom: 25 │ │ │ │ +00041e60: 2520 6f66 2074 6865 206d 6174 7269 6365 % of the matrice │ │ │ │ +00041e70: 7320 6172 6520 4c65 7853 6d61 6c6c 6573 s are LexSmalles │ │ │ │ +00041e80: 742c 0a20 2020 204c 6578 536d 616c 6c65 t,. LexSmalle │ │ │ │ +00041e90: 7374 5465 726d 2c20 4752 6576 4c65 7853 stTerm, GRevLexS │ │ │ │ +00041ea0: 6d61 6c6c 6573 7420 616e 642c 2047 5265 mallest and, GRe │ │ │ │ +00041eb0: 764c 6578 4c61 7267 6573 7420 6561 6368 vLexLargest each │ │ │ │ +00041ec0: 0a20 202a 2053 7472 6174 6567 794c 6578 . * StrategyLex │ │ │ │ +00041ed0: 536d 616c 6c65 7374 3a20 3530 2520 6f66 Smallest: 50% of │ │ │ │ +00041ee0: 2074 6865 206d 6174 7269 6365 7320 6172 the matrices ar │ │ │ │ +00041ef0: 6520 4c65 7853 6d61 6c6c 6573 7420 616e e LexSmallest an │ │ │ │ +00041f00: 6420 3530 2520 6172 650a 2020 2020 4c65 d 50% are. Le │ │ │ │ +00041f10: 7853 6d61 6c6c 6573 7454 6572 6d0a 2020 xSmallestTerm. │ │ │ │ +00041f20: 2a20 5374 7261 7465 6779 4752 6576 4c65 * StrategyGRevLe │ │ │ │ +00041f30: 7853 6d61 6c6c 6573 743a 2035 3025 206f xSmallest: 50% o │ │ │ │ +00041f40: 6620 7468 6520 6d61 7472 6963 6573 2061 f the matrices a │ │ │ │ +00041f50: 7265 2047 5265 764c 6578 536d 616c 6c65 re GRevLexSmalle │ │ │ │ +00041f60: 7374 2061 6e64 2035 3025 0a20 2020 2061 st and 50%. a │ │ │ │ +00041f70: 7265 2047 5265 764c 6578 4c61 7267 6573 re GRevLexLarges │ │ │ │ +00041f80: 740a 2020 2a20 5374 7261 7465 6779 5261 t. * StrategyRa │ │ │ │ +00041f90: 6e64 6f6d 3a20 6368 6f6f 7365 7320 3130 ndom: chooses 10 │ │ │ │ +00041fa0: 3025 2072 616e 646f 6d20 7375 626d 6174 0% random submat │ │ │ │ +00041fb0: 7269 6365 732e 0a20 202a 2053 7472 6174 rices.. * Strat │ │ │ │ +00041fc0: 6567 7950 6f69 6e74 733a 2063 686f 6f73 egyPoints: choos │ │ │ │ +00041fd0: 6520 616c 6c20 7375 626d 6174 7269 6365 e all submatrice │ │ │ │ +00041fe0: 7320 7669 6120 506f 696e 7473 2e0a 2020 s via Points.. │ │ │ │ +00041ff0: 2a20 5374 7261 7465 6779 4465 6661 756c * StrategyDefaul │ │ │ │ +00042000: 7457 6974 6850 6f69 6e74 733a 206c 696b tWithPoints: lik │ │ │ │ +00042010: 6520 5374 7261 7465 6779 4465 6661 756c e StrategyDefaul │ │ │ │ +00042020: 7420 6275 7420 7265 706c 6163 6573 2074 t but replaces t │ │ │ │ +00042030: 6865 2052 616e 646f 6d20 616e 640a 2020 he Random and. │ │ │ │ +00042040: 2020 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 RandomNonZero │ │ │ │ +00042050: 7375 626d 6174 7269 6365 7320 6173 2077 submatrices as w │ │ │ │ +00042060: 6974 6820 6d61 7472 6963 6573 2063 686f ith matrices cho │ │ │ │ +00042070: 7365 6e20 6173 2069 6e20 506f 696e 7473 sen as in Points │ │ │ │ +00042080: 2e0a 4164 6469 7469 6f6e 616c 6c79 2c20 ..Additionally, │ │ │ │ +00042090: 6120 4d75 7461 626c 6548 6173 6854 6162 a MutableHashTab │ │ │ │ +000420a0: 6c65 206e 616d 6564 2053 7472 6174 6567 le named Strateg │ │ │ │ +000420b0: 7943 7572 7265 6e74 2069 7320 616c 736f yCurrent is also │ │ │ │ +000420c0: 2065 7870 6f72 7465 642e 2020 4974 0a62 exported. It.b │ │ │ │ +000420d0: 6567 696e 7320 6173 2074 6865 2064 6566 egins as the def │ │ │ │ +000420e0: 6175 6c74 2073 7472 6174 6567 792c 2062 ault strategy, b │ │ │ │ +000420f0: 7574 2074 6865 2075 7365 7220 6361 6e20 ut the user can │ │ │ │ +00042100: 6d6f 6469 6679 2069 742e 0a0a 5573 696e modify it...Usin │ │ │ │ +00042110: 6720 6120 7369 6e67 6c65 2068 6575 7269 g a single heuri │ │ │ │ +00042120: 7374 6963 2020 416c 7465 726e 6174 6976 stic Alternativ │ │ │ │ +00042130: 656c 792c 2069 6620 7468 6520 7573 6572 ely, if the user │ │ │ │ +00042140: 206f 6e6c 7920 7761 6e74 7320 746f 2075 only wants to u │ │ │ │ +00042150: 7365 2073 6179 0a4c 6578 536d 616c 6c65 se say.LexSmalle │ │ │ │ +00042160: 7374 5465 726d 2074 6865 7920 6361 6e20 stTerm they can │ │ │ │ +00042170: 7365 742c 2053 7472 6174 6567 7920 746f set, Strategy to │ │ │ │ +00042180: 2070 6f69 6e74 2074 6f20 7468 6174 2073 point to that s │ │ │ │ +00042190: 796d 626f 6c2c 2069 6e73 7465 6164 206f ymbol, instead o │ │ │ │ +000421a0: 6620 610a 6372 6561 7469 6e67 2061 2063 f a.creating a c │ │ │ │ +000421b0: 7573 746f 6d20 7374 7261 7465 6779 2048 ustom strategy H │ │ │ │ +000421c0: 6173 6854 6162 6c65 2e20 2046 6f72 2065 ashTable. For e │ │ │ │ +000421d0: 7861 6d70 6c65 3a0a 2b2d 2d2d 2d2d 2d2d xample:.+------- │ │ │ │ +000421e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000421f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00042200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042220: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -00042230: 656c 6170 7365 6454 696d 6520 7265 6775 elapsedTime regu │ │ │ │ -00042240: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ -00042250: 2831 2c20 542c 2053 7472 6174 6567 793d (1, T, Strategy= │ │ │ │ -00042260: 3e4c 6578 536d 616c 6c65 7374 5465 726d >LexSmallestTerm │ │ │ │ -00042270: 297c 0a7c 202d 2d20 312e 3032 3932 3973 )|.| -- 1.02929s │ │ │ │ -00042280: 2065 6c61 7073 6564 2020 2020 2020 2020 elapsed │ │ │ │ +00042210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00042220: 0a7c 6934 203a 2065 6c61 7073 6564 5469 .|i4 : elapsedTi │ │ │ │ +00042230: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ +00042240: 6d65 6e73 696f 6e28 312c 2054 2c20 5374 mension(1, T, St │ │ │ │ +00042250: 7261 7465 6779 3d3e 4c65 7853 6d61 6c6c rategy=>LexSmall │ │ │ │ +00042260: 6573 7454 6572 6d29 7c0a 7c20 2d2d 202e estTerm)|.| -- . │ │ │ │ +00042270: 3935 3230 3537 7320 656c 6170 7365 6420 952057s elapsed │ │ │ │ +00042280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00042290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000422a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000422b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000422b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000422c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000422d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000422e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000422f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042300: 2020 207c 0a7c 6f34 203d 2074 7275 6520 |.|o4 = true │ │ │ │ +000422f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ +00042300: 3d20 7472 7565 2020 2020 2020 2020 2020 = true │ │ │ │ 00042310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00042320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00042330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042340: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00042340: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00042350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00042360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00042370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042390: 2d2d 2d2d 2d2b 0a0a 466f 7220 7468 6520 -----+..For the │ │ │ │ -000423a0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -000423b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -000423c0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -000423d0: 5374 7261 7465 6779 4465 6661 756c 743a StrategyDefault: │ │ │ │ -000423e0: 2053 7472 6174 6567 7944 6566 6175 6c74 StrategyDefault │ │ │ │ -000423f0: 2c20 6973 2061 6e20 2a6e 6f74 6520 6f70 , is an *note op │ │ │ │ -00042400: 7469 6f6e 2074 6162 6c65 3a0a 284d 6163 tion table:.(Mac │ │ │ │ -00042410: 6175 6c61 7932 446f 6329 4f70 7469 6f6e aulay2Doc)Option │ │ │ │ -00042420: 5461 626c 652c 2e0a 0a2d 2d2d 2d2d 2d2d Table,...------- │ │ │ │ +00042380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 ------------+..F │ │ │ │ +00042390: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +000423a0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +000423b0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +000423c0: 202a 6e6f 7465 2053 7472 6174 6567 7944 *note StrategyD │ │ │ │ +000423d0: 6566 6175 6c74 3a20 5374 7261 7465 6779 efault: 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in./build/reprod │ │ │ │ +000424a0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +000424b0: 756c 6179 322d 312e 3235 2e31 312b 6473 ulay2-1.25.11+ds │ │ │ │ +000424c0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +000424d0: 636b 6167 6573 2f46 6173 744d 696e 6f72 ckages/FastMinor │ │ │ │ +000424e0: 732e 0a6d 323a 3139 3933 3a30 2e0a 1f0a s..m2:1993:0.... │ │ │ │ +000424f0: 5461 6720 5461 626c 653a 0a4e 6f64 653a Tag Table:.Node: │ │ │ │ +00042500: 2054 6f70 7f32 3338 0a4e 6f64 653a 2063 Top.238.Node: c │ │ │ │ +00042510: 686f 6f73 6547 6f6f 644d 696e 6f72 737f hooseGoodMinors. │ │ │ │ +00042520: 3134 3238 360a 4e6f 6465 3a20 6368 6f6f 14286.Node: choo │ │ │ │ +00042530: 7365 5261 6e64 6f6d 5375 626d 6174 7269 seRandomSubmatri │ │ │ │ +00042540: 787f 3138 3732 300a 4e6f 6465 3a20 6368 x.18720.Node: ch │ │ │ │ +00042550: 6f6f 7365 5375 626d 6174 7269 784c 6172 ooseSubmatrixLar │ │ │ │ +00042560: 6765 7374 4465 6772 6565 7f32 3037 3333 gestDegree.20733 │ │ │ │ +00042570: 0a4e 6f64 653a 2063 686f 6f73 6553 7562 .Node: chooseSub │ │ │ │ +00042580: 6d61 7472 6978 536d 616c 6c65 7374 4465 matrixSmallestDe │ │ │ │ +00042590: 6772 6565 7f32 3334 3130 0a4e 6f64 653a gree.23410.Node: │ │ │ │ +000425a0: 2044 6574 5374 7261 7465 6779 7f32 3630 DetStrategy.260 │ │ │ │ +000425b0: 3139 0a4e 6f64 653a 2046 6173 744d 696e 19.Node: FastMin │ │ │ │ +000425c0: 6f72 7353 7472 6174 6567 7954 7574 6f72 orsStrategyTutor │ │ │ │ +000425d0: 6961 6c7f 3237 3936 390a 4e6f 6465 3a20 ial.27969.Node: │ │ │ │ +000425e0: 6765 7453 7562 6d61 7472 6978 4f66 5261 getSubmatrixOfRa │ │ │ │ +000425f0: 6e6b 7f39 3034 3937 0a4e 6f64 653a 2069 nk.90497.Node: i │ │ │ │ +00042600: 7343 6f64 696d 4174 4c65 6173 747f 3936 sCodimAtLeast.96 │ │ │ │ +00042610: 3230 350a 4e6f 6465 3a20 6973 4469 6d41 205.Node: isDimA │ │ │ │ +00042620: 744d 6f73 747f 3130 3434 3436 0a4e 6f64 tMost.104446.Nod │ │ │ │ +00042630: 653a 2069 7352 616e 6b41 744c 6561 7374 e: isRankAtLeast │ │ │ │ +00042640: 7f31 3036 3037 310a 4e6f 6465 3a20 6973 .106071.Node: is │ │ │ │ +00042650: 5261 6e6b 4174 4c65 6173 745f 6c70 5f70 RankAtLeast_lp_p │ │ │ │ +00042660: 645f 7064 5f70 645f 636d 5468 7265 6164 d_pd_pd_cmThread │ │ │ │ +00042670: 733d 3e5f 7064 5f70 645f 7064 5f72 707f s=>_pd_pd_pd_rp. │ │ │ │ +00042680: 3131 3034 3832 0a4e 6f64 653a 204d 6178 110482.Node: Max │ │ │ │ +00042690: 4d69 6e6f 7273 7f31 3132 3032 370a 4e6f Minors.112027.No │ │ │ │ +000426a0: 6465 3a20 4d69 6e44 696d 656e 7369 6f6e de: MinDimension │ │ │ │ +000426b0: 7f31 3133 3630 390a 4e6f 6465 3a20 4d6f .113609.Node: Mo │ │ │ │ +000426c0: 6475 6c75 737f 3131 3436 3033 0a4e 6f64 dulus.114603.Nod │ │ │ │ +000426d0: 653a 2050 6f69 6e74 4f70 7469 6f6e 737f e: PointOptions. │ │ │ │ +000426e0: 3131 3536 3235 0a4e 6f64 653a 2070 726f 115625.Node: pro │ │ │ │ +000426f0: 6a44 696d 7f31 3139 3639 390a 4e6f 6465 jDim.119699.Node │ │ │ │ +00042700: 3a20 7265 6375 7273 6976 654d 696e 6f72 : recursiveMinor │ │ │ │ +00042710: 737f 3132 3531 3332 0a4e 6f64 653a 2072 s.125132.Node: r │ │ │ │ +00042720: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ +00042730: 696f 6e7f 3132 3833 3739 0a4e 6f64 653a ion.128379.Node: │ │ │ │ +00042740: 2052 6567 756c 6172 496e 436f 6469 6d65 RegularInCodime │ │ │ │ +00042750: 6e73 696f 6e54 7574 6f72 6961 6c7f 3231 nsionTutorial.21 │ │ │ │ +00042760: 3433 3834 0a4e 6f64 653a 2072 656f 7264 4384.Node: reord │ │ │ │ +00042770: 6572 506f 6c79 6e6f 6d69 616c 5269 6e67 erPolynomialRing │ │ │ │ +00042780: 7f32 3539 3134 340a 4e6f 6465 3a20 5374 .259144.Node: St │ │ │ │ +00042790: 7261 7465 6779 4465 6661 756c 747f 3236 rategyDefault.26 │ │ │ │ +000427a0: 3136 3537 0a1f 0a45 6e64 2054 6167 2054 1657...End Tag T │ │ │ │ +000427b0: 6162 6c65 0a able. │ │ ├── ./usr/share/info/FiniteFittingIdeals.info.gz │ │ │ ├── FiniteFittingIdeals.info │ │ │ │ @@ -1017,17 +1017,17 @@ │ │ │ │ 00003f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00003fa0: 0a7c 6931 3520 3a20 7469 6d65 2049 3d63 .|i15 : time I=c │ │ │ │ 00003fb0: 6f31 4669 7474 696e 6728 4b33 2920 2020 o1Fitting(K3) │ │ │ │ 00003fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00003ff0: 2075 7365 6420 302e 3030 3438 3837 3539 used 0.00488759 │ │ │ │ -00004000: 7320 2863 7075 293b 2030 2e30 3034 3838 s (cpu); 0.00488 │ │ │ │ -00004010: 3434 3473 2028 7468 7265 6164 293b 2030 444s (thread); 0 │ │ │ │ +00003ff0: 2075 7365 6420 302e 3030 3331 3439 3931 used 0.00314991 │ │ │ │ +00004000: 7320 2863 7075 293b 2030 2e30 3033 3134 s (cpu); 0.00314 │ │ │ │ +00004010: 3635 3473 2028 7468 7265 6164 293b 2030 654s (thread); 0 │ │ │ │ 00004020: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00004030: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00004040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004080: 7c0a 7c6f 3135 203d 2069 6465 616c 2028 |.|o15 = ideal ( │ │ │ │ @@ -1055,17 +1055,17 @@ │ │ │ │ 000041e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000041f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 -------+.|i16 : │ │ │ │ 00004200: 7469 6d65 204a 3d66 6974 7469 6e67 4964 time J=fittingId │ │ │ │ 00004210: 6561 6c28 322d 312c 636f 6b65 7220 4b33 eal(2-1,coker K3 │ │ │ │ 00004220: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ 00004230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004240: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00004250: 3031 3130 3831 3673 2028 6370 7529 3b20 0110816s (cpu); │ │ │ │ -00004260: 302e 3031 3130 3837 3673 2028 7468 7265 0.0110876s (thre │ │ │ │ -00004270: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +00004250: 3030 3636 3134 3037 7320 2863 7075 293b 00661407s (cpu); │ │ │ │ +00004260: 2030 2e30 3036 3631 3538 3673 2028 7468 0.00661586s (th │ │ │ │ +00004270: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00004280: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00004290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000042a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000042b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000042c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000042d0: 2020 2020 2020 2020 7c0a 7c6f 3136 203a |.|o16 : │ │ │ │ 000042e0: 2049 6465 616c 206f 6620 5220 2020 2020 Ideal of R │ │ ├── ./usr/share/info/ForeignFunctions.info.gz │ │ │ ├── ForeignFunctions.info │ │ │ │ @@ -3504,16 +3504,16 @@ │ │ │ │ 0000daf0: 2078 203d 206d 616c 6c6f 6320 3820 2020 x = malloc 8 │ │ │ │ 0000db00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000db20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0000db30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000db60: 7c0a 7c6f 3137 203d 2030 7837 6661 3437 |.|o17 = 0x7fa47 │ │ │ │ -0000db70: 3030 3661 3466 3020 2020 2020 2020 2020 006a4f0 │ │ │ │ +0000db60: 7c0a 7c6f 3137 203d 2030 7837 6632 3536 |.|o17 = 0x7f256 │ │ │ │ +0000db70: 3830 3661 3466 3020 2020 2020 2020 2020 806a4f0 │ │ │ │ 0000db80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000db90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000dbd0: 2020 2020 2020 2020 7c0a 7c6f 3137 203a |.|o17 : │ │ │ │ 0000dbe0: 2046 6f72 6569 676e 4f62 6a65 6374 206f ForeignObject o │ │ │ │ @@ -3638,15 +3638,15 @@ │ │ │ │ 0000e350: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ 0000e360: 7065 656b 206d 7066 7220 2020 2020 2020 peek mpfr │ │ │ │ 0000e370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e380: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e3a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0000e3b0: 7c6f 3220 3d20 5368 6172 6564 4c69 6272 |o2 = SharedLibr │ │ │ │ -0000e3c0: 6172 797b 3078 3766 6335 6261 3531 6435 ary{0x7fc5ba51d5 │ │ │ │ +0000e3c0: 6172 797b 3078 3766 3230 6138 3863 3435 ary{0x7f20a88c45 │ │ │ │ 0000e3d0: 3530 2c20 6d70 6672 7d7c 0a2b 2d2d 2d2d 50, mpfr}|.+---- │ │ │ │ 0000e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e400: 2d2d 2d2d 2b0a 2a20 4d65 6e75 3a0a 0a2a ----+.* Menu:..* │ │ │ │ 0000e410: 206f 7065 6e53 6861 7265 644c 6962 7261 openSharedLibra │ │ │ │ 0000e420: 7279 3a3a 2020 2020 2020 2020 2020 206f ry:: o │ │ │ │ 0000e430: 7065 6e20 6120 7368 6172 6564 206c 6962 pen a shared lib │ │ │ │ @@ -5393,29 +5393,29 @@ │ │ │ │ 00015100: 6520 706f 696e 7465 722e 0a0a 2b2d 2d2d e pointer...+--- │ │ │ │ 00015110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00015130: 6931 203a 2070 7472 203d 2061 6464 7265 i1 : ptr = addre │ │ │ │ 00015140: 7373 2069 6e74 2030 2020 2020 2020 2020 ss int 0 │ │ │ │ 00015150: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00015160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015170: 2020 207c 0a7c 6f31 203d 2030 7837 6663 |.|o1 = 0x7fc │ │ │ │ -00015180: 3561 6537 3032 3737 3020 2020 2020 2020 5ae702770 │ │ │ │ +00015170: 2020 207c 0a7c 6f31 203d 2030 7837 6632 |.|o1 = 0x7f2 │ │ │ │ +00015180: 3039 3732 6463 6530 3020 2020 2020 2020 0972dce00 │ │ │ │ 00015190: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000151a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000151b0: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ 000151c0: 2050 6f69 6e74 6572 2020 2020 2020 2020 Pointer │ │ │ │ 000151d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 000151e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000151f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00015200: 0a7c 6932 203a 2076 6f69 6473 7461 7220 .|i2 : voidstar │ │ │ │ 00015210: 7074 7220 2020 2020 2020 2020 2020 2020 ptr │ │ │ │ 00015220: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00015230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015240: 2020 2020 207c 0a7c 6f32 203d 2030 7837 |.|o2 = 0x7 │ │ │ │ -00015250: 6663 3561 6537 3032 3737 3020 2020 2020 fc5ae702770 │ │ │ │ +00015250: 6632 3039 3732 6463 6530 3020 2020 2020 f20972dce00 │ │ │ │ 00015260: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00015270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015280: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ 00015290: 203a 2046 6f72 6569 676e 4f62 6a65 6374 : ForeignObject │ │ │ │ 000152a0: 206f 6620 7479 7065 2076 6f69 642a 7c0a of type void*|. │ │ │ │ 000152b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000152c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5496,16 +5496,16 @@ │ │ │ │ 00015770: 7970 6520 696e 7433 327c 0a2b 2d2d 2d2d ype int32|.+---- │ │ │ │ 00015780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 000157a0: 3220 3a20 7074 7220 3d20 6164 6472 6573 2 : ptr = addres │ │ │ │ 000157b0: 7320 7820 2020 2020 2020 2020 2020 207c s x | │ │ │ │ 000157c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000157d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000157e0: 2020 7c0a 7c6f 3220 3d20 3078 3766 6335 |.|o2 = 0x7fc5 │ │ │ │ -000157f0: 6162 6532 3264 6530 2020 2020 2020 2020 abe22de0 │ │ │ │ +000157e0: 2020 7c0a 7c6f 3220 3d20 3078 3766 3230 |.|o2 = 0x7f20 │ │ │ │ +000157f0: 3934 3934 6134 3230 2020 2020 2020 2020 9494a420 │ │ │ │ 00015800: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00015810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015820: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ 00015830: 506f 696e 7465 7220 2020 2020 2020 2020 Pointer │ │ │ │ 00015840: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00015850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ @@ -5646,16 +5646,16 @@ │ │ │ │ 000160d0: 696e 7465 7273 2e0a 0a2b 2d2d 2d2d 2d2d inters...+------ │ │ │ │ 000160e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000160f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 00016100: 203a 2070 7472 203d 2076 6f69 6473 7461 : ptr = voidsta │ │ │ │ 00016110: 7220 6164 6472 6573 7320 696e 7420 357c r address int 5| │ │ │ │ 00016120: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00016130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016140: 2020 207c 0a7c 6f31 203d 2030 7837 6663 |.|o1 = 0x7fc │ │ │ │ -00016150: 3561 6265 3232 6234 3020 2020 2020 2020 5abe22b40 │ │ │ │ +00016140: 2020 207c 0a7c 6f31 203d 2030 7837 6632 |.|o1 = 0x7f2 │ │ │ │ +00016150: 3039 3439 3461 3138 3020 2020 2020 2020 09494a180 │ │ │ │ 00016160: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00016170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016180: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ 00016190: 203a 2046 6f72 6569 676e 4f62 6a65 6374 : ForeignObject │ │ │ │ 000161a0: 206f 6620 7479 7065 2076 6f69 642a 207c of type void* | │ │ │ │ 000161b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5744,15 +5744,15 @@ │ │ │ │ 000166f0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 00016700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016710: 2d2d 2d2d 2b0a 7c69 3120 3a20 7074 7220 ----+.|i1 : ptr │ │ │ │ 00016720: 3d20 6765 744d 656d 6f72 7920 3820 2020 = getMemory 8 │ │ │ │ 00016730: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00016740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016750: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -00016760: 3d20 3078 3766 6335 6235 6634 3532 3030 = 0x7fc5b5f45200 │ │ │ │ +00016760: 3d20 3078 3766 3230 6133 6236 3134 3930 = 0x7f20a3b61490 │ │ │ │ 00016770: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00016780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000167a0: 7c0a 7c6f 3120 3a20 466f 7265 6967 6e4f |.|o1 : ForeignO │ │ │ │ 000167b0: 626a 6563 7420 6f66 2074 7970 6520 766f bject of type vo │ │ │ │ 000167c0: 6964 2a7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d id*|.+---------- │ │ │ │ 000167d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5767,16 +5767,16 @@ │ │ │ │ 00016860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00016880: 7c69 3220 3a20 7074 7220 3d20 6765 744d |i2 : ptr = getM │ │ │ │ 00016890: 656d 6f72 7928 382c 2041 746f 6d69 6320 emory(8, Atomic │ │ │ │ 000168a0: 3d3e 2074 7275 6529 7c0a 7c20 2020 2020 => true)|.| │ │ │ │ 000168b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000168c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000168d0: 2020 7c0a 7c6f 3220 3d20 3078 3766 6335 |.|o2 = 0x7fc5 │ │ │ │ -000168e0: 6162 6464 3731 6130 2020 2020 2020 2020 abdd71a0 │ │ │ │ +000168d0: 2020 7c0a 7c6f 3220 3d20 3078 3766 3230 |.|o2 = 0x7f20 │ │ │ │ +000168e0: 3934 3934 6166 6530 2020 2020 2020 2020 9494afe0 │ │ │ │ 000168f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00016900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016920: 2020 2020 2020 7c0a 7c6f 3220 3a20 466f |.|o2 : Fo │ │ │ │ 00016930: 7265 6967 6e4f 626a 6563 7420 6f66 2074 reignObject of t │ │ │ │ 00016940: 7970 6520 766f 6964 2a20 2020 2020 2020 ype void* │ │ │ │ 00016950: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ @@ -5795,16 +5795,16 @@ │ │ │ │ 00016a20: 6361 6c6c 792e 0a0a 2b2d 2d2d 2d2d 2d2d cally...+------- │ │ │ │ 00016a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016a40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ 00016a50: 2070 7472 203d 2067 6574 4d65 6d6f 7279 ptr = getMemory │ │ │ │ 00016a60: 2069 6e74 2020 2020 2020 2020 7c0a 7c20 int |.| │ │ │ │ 00016a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016a80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016a90: 0a7c 6f33 203d 2030 7837 6663 3561 6264 .|o3 = 0x7fc5abd │ │ │ │ -00016aa0: 6437 3061 3020 2020 2020 2020 2020 2020 d70a0 │ │ │ │ +00016a90: 0a7c 6f33 203d 2030 7837 6632 3039 3439 .|o3 = 0x7f20949 │ │ │ │ +00016aa0: 3461 6564 3020 2020 2020 2020 2020 2020 4aed0 │ │ │ │ 00016ab0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00016ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ad0: 2020 2020 207c 0a7c 6f33 203a 2046 6f72 |.|o3 : For │ │ │ │ 00016ae0: 6569 676e 4f62 6a65 6374 206f 6620 7479 eignObject of ty │ │ │ │ 00016af0: 7065 2076 6f69 642a 7c0a 2b2d 2d2d 2d2d pe void*|.+----- │ │ │ │ 00016b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 -----------+..Se │ │ │ │ @@ -5965,18 +5965,18 @@ │ │ │ │ 000174c0: 7320 696e 7420 312c 2061 6464 7265 7373 s int 1, address │ │ │ │ 000174d0: 2069 6e74 2032 7d20 2020 2020 2020 2020 int 2} │ │ │ │ 000174e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000174f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017520: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017530: 0a7c 6f33 203d 207b 3078 3766 6335 6162 .|o3 = {0x7fc5ab │ │ │ │ -00017540: 6532 3230 3930 2c20 3078 3766 6335 6162 e22090, 0x7fc5ab │ │ │ │ -00017550: 6532 3230 3830 2c20 3078 3766 6335 6162 e22080, 0x7fc5ab │ │ │ │ -00017560: 6532 3230 3730 7d20 2020 2020 2020 2020 e22070} │ │ │ │ +00017530: 0a7c 6f33 203d 207b 3078 3766 3230 3934 .|o3 = {0x7f2094 │ │ │ │ +00017540: 3939 3634 3830 2c20 3078 3766 3230 3934 996480, 0x7f2094 │ │ │ │ +00017550: 3939 3634 3730 2c20 3078 3766 3230 3934 996470, 0x7f2094 │ │ │ │ +00017560: 3939 3634 3630 7d20 2020 2020 2020 2020 996460} │ │ │ │ 00017570: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00017580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000175a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000175b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000175c0: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ 000175d0: 2046 6f72 6569 676e 4f62 6a65 6374 206f ForeignObject o │ │ │ │ @@ -6431,17 +6431,17 @@ │ │ │ │ 000191e0: 7373 2069 6e74 2030 2c20 6164 6472 6573 ss int 0, addres │ │ │ │ 000191f0: 7320 696e 7420 312c 2061 6464 7265 7373 s int 1, address │ │ │ │ 00019200: 2069 6e74 2032 7d7c 0a7c 2020 2020 2020 int 2}|.| │ │ │ │ 00019210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019240: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ -00019250: 207b 3078 3766 6335 6162 6536 6365 6430 {0x7fc5abe6ced0 │ │ │ │ -00019260: 2c20 3078 3766 6335 6162 6536 6365 6330 , 0x7fc5abe6cec0 │ │ │ │ -00019270: 2c20 3078 3766 6335 6162 6536 6365 6230 , 0x7fc5abe6ceb0 │ │ │ │ +00019250: 207b 3078 3766 3230 3934 3939 3632 3730 {0x7f2094996270 │ │ │ │ +00019260: 2c20 3078 3766 3230 3934 3939 3632 3630 , 0x7f2094996260 │ │ │ │ +00019270: 2c20 3078 3766 3230 3934 3939 3632 3530 , 0x7f2094996250 │ │ │ │ 00019280: 7d20 2020 2020 2020 2020 207c 0a7c 2020 } |.| │ │ │ │ 00019290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000192a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000192b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000192c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000192d0: 6f32 203a 2046 6f72 6569 676e 4f62 6a65 o2 : ForeignObje │ │ │ │ 000192e0: 6374 206f 6620 7479 7065 2076 6f69 642a ct of type void* │ │ │ │ @@ -7909,15 +7909,15 @@ │ │ │ │ 0001ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ee50: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001ee60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ee90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0001eea0: 7c6f 3220 3d20 4861 7368 5461 626c 657b |o2 = HashTable{ │ │ │ │ -0001eeb0: 2262 6172 2220 3d3e 2036 2e39 3430 3834 "bar" => 6.94084 │ │ │ │ +0001eeb0: 2262 6172 2220 3d3e 2036 2e39 3035 3832 "bar" => 6.90582 │ │ │ │ 0001eec0: 652d 3331 307d 2020 2020 2020 2020 2020 e-310} │ │ │ │ 0001eed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001eef0: 2020 2020 2020 2020 2020 2020 2020 2266 "f │ │ │ │ 0001ef00: 6f6f 2220 3d3e 2032 3720 2020 2020 2020 oo" => 27 │ │ │ │ 0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8160,26 +8160,26 @@ │ │ │ │ 0001fdf0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7065 ------+.|i2 : pe │ │ │ │ 0001fe00: 656b 2078 2020 2020 2020 2020 2020 2020 ek x │ │ │ │ 0001fe10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0001fe20: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0001fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe40: 2020 2020 2020 7c0a 7c6f 3220 3d20 696e |.|o2 = in │ │ │ │ 0001fe50: 7433 327b 4164 6472 6573 7320 3d3e 2030 t32{Address => 0 │ │ │ │ -0001fe60: 7837 6663 3561 6264 6437 3461 307d 7c0a x7fc5abdd74a0}|. │ │ │ │ +0001fe60: 7837 6632 3039 3439 3461 6338 307d 7c0a x7f209494ac80}|. │ │ │ │ 0001fe70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0001fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fe90: 2d2d 2d2d 2d2d 2b0a 0a54 6f20 6765 7420 ------+..To get │ │ │ │ 0001fea0: 7468 6973 2c20 7573 6520 2a6e 6f74 6520 this, use *note │ │ │ │ 0001feb0: 6164 6472 6573 733a 2061 6464 7265 7373 address: address │ │ │ │ 0001fec0: 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...+----------- │ │ │ │ 0001fed0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ 0001fee0: 6164 6472 6573 7320 7820 2020 2020 7c0a address x |. │ │ │ │ 0001fef0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0001ff00: 2020 2020 7c0a 7c6f 3320 3d20 3078 3766 |.|o3 = 0x7f │ │ │ │ -0001ff10: 6335 6162 6464 3734 6130 7c0a 7c20 2020 c5abdd74a0|.| │ │ │ │ +0001ff10: 3230 3934 3934 6163 3830 7c0a 7c20 2020 209494ac80|.| │ │ │ │ 0001ff20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ff30: 7c0a 7c6f 3320 3a20 506f 696e 7465 7220 |.|o3 : Pointer │ │ │ │ 0001ff40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0001ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a55 ------------+..U │ │ │ │ 0001ff60: 7365 202a 6e6f 7465 2063 6c61 7373 3a20 se *note class: │ │ │ │ 0001ff70: 284d 6163 6175 6c61 7932 446f 6329 636c (Macaulay2Doc)cl │ │ │ │ 0001ff80: 6173 732c 2074 6f20 6465 7465 726d 696e ass, to determin │ │ │ │ @@ -8881,29 +8881,29 @@ │ │ │ │ 00022b00: 626a 6563 7473 2e0a 0a2b 2d2d 2d2d 2d2d bjects...+------ │ │ │ │ 00022b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ 00022b30: 3a20 7820 3d20 766f 6964 7374 6172 2061 : x = voidstar a │ │ │ │ 00022b40: 6464 7265 7373 2069 6e74 2035 207c 0a7c ddress int 5 |.| │ │ │ │ 00022b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b70: 7c0a 7c6f 3520 3d20 3078 3766 6335 6162 |.|o5 = 0x7fc5ab │ │ │ │ -00022b80: 6532 3262 3130 2020 2020 2020 2020 2020 e22b10 │ │ │ │ +00022b70: 7c0a 7c6f 3520 3d20 3078 3766 3230 3934 |.|o5 = 0x7f2094 │ │ │ │ +00022b80: 3934 6136 6430 2020 2020 2020 2020 2020 94a6d0 │ │ │ │ 00022b90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022bb0: 2020 2020 2020 7c0a 7c6f 3520 3a20 466f |.|o5 : Fo │ │ │ │ 00022bc0: 7265 6967 6e4f 626a 6563 7420 6f66 2074 reignObject of t │ │ │ │ 00022bd0: 7970 6520 766f 6964 2a7c 0a2b 2d2d 2d2d ype void*|.+---- │ │ │ │ 00022be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00022c00: 3620 3a20 7661 6c75 6520 7820 2020 2020 6 : value x │ │ │ │ 00022c10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00022c20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00022c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c40: 2020 7c0a 7c6f 3620 3d20 3078 3766 6335 |.|o6 = 0x7fc5 │ │ │ │ -00022c50: 6162 6532 3262 3130 2020 2020 2020 2020 abe22b10 │ │ │ │ +00022c40: 2020 7c0a 7c6f 3620 3d20 3078 3766 3230 |.|o6 = 0x7f20 │ │ │ │ +00022c50: 3934 3934 6136 6430 2020 2020 2020 2020 9494a6d0 │ │ │ │ 00022c60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00022c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c80: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ 00022c90: 506f 696e 7465 7220 2020 2020 2020 2020 Pointer │ │ │ │ 00022ca0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00022cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ @@ -9430,50 +9430,50 @@ │ │ │ │ 00024d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024d60: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2063 -------+.|i5 : c │ │ │ │ 00024d70: 6f6c 6c65 6374 4761 7262 6167 6528 2920 ollectGarbage() │ │ │ │ 00024d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024da0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00024db0: 6672 6565 696e 6720 6d65 6d6f 7279 2061 freeing memory a │ │ │ │ -00024dc0: 7420 3078 3766 6335 3963 3037 6639 6230 t 0x7fc59c07f9b0 │ │ │ │ -00024dd0: 6672 6565 696e 6720 6d65 6d6f 7279 2061 freeing memory a │ │ │ │ -00024de0: 7420 3078 3766 6335 3963 3037 6639 3930 t 0x7fc59c07f990 │ │ │ │ +00024dc0: 7420 3078 3766 3230 3763 3037 6639 3530 t 0x7f207c07f950 │ │ │ │ +00024dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024df0: 2020 207c 0a7c 6672 6565 696e 6720 6d65 |.|freeing me │ │ │ │ -00024e00: 6d6f 7279 2061 7420 3078 3766 6335 3963 mory at 0x7fc59c │ │ │ │ -00024e10: 3037 6639 3330 2020 2020 2020 2020 2020 07f930 │ │ │ │ +00024e00: 6d6f 7279 2061 7420 3078 3766 3230 3763 mory at 0x7f207c │ │ │ │ +00024e10: 3037 6639 3130 2020 2020 2020 2020 2020 07f910 │ │ │ │ 00024e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024e30: 2020 2020 2020 2020 207c 0a7c 6672 6565 |.|free │ │ │ │ 00024e40: 696e 6720 6d65 6d6f 7279 2061 7420 3078 ing memory at 0x │ │ │ │ -00024e50: 3766 6335 3963 3037 6639 3530 2020 2020 7fc59c07f950 │ │ │ │ +00024e50: 3766 3230 3763 3037 6632 3330 2020 2020 7f207c07f230 │ │ │ │ 00024e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024e70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00024e80: 0a7c 6672 6565 696e 6720 6d65 6d6f 7279 .|freeing memory │ │ │ │ -00024e90: 2061 7420 3078 3766 6335 3963 3037 6638 at 0x7fc59c07f8 │ │ │ │ -00024ea0: 6630 2020 2020 2020 2020 2020 2020 2020 f0 │ │ │ │ +00024e90: 2061 7420 3078 3766 3230 3763 3037 6639 at 0x7f207c07f9 │ │ │ │ +00024ea0: 6230 2020 2020 2020 2020 2020 2020 2020 b0 │ │ │ │ 00024eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ec0: 2020 2020 207c 0a7c 6672 6565 696e 6720 |.|freeing │ │ │ │ -00024ed0: 6d65 6d6f 7279 2061 7420 3078 3766 6335 memory at 0x7fc5 │ │ │ │ -00024ee0: 3963 3037 6632 3530 2020 2020 2020 2020 9c07f250 │ │ │ │ +00024ed0: 6d65 6d6f 7279 2061 7420 3078 3766 3230 memory at 0x7f20 │ │ │ │ +00024ee0: 3763 3037 6639 3730 2020 2020 2020 2020 7c07f970 │ │ │ │ 00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f00: 2020 2020 2020 2020 2020 207c 0a7c 6672 |.|fr │ │ │ │ 00024f10: 6565 696e 6720 6d65 6d6f 7279 2061 7420 eeing memory at │ │ │ │ -00024f20: 3078 3766 6335 3963 3037 6632 3330 2020 0x7fc59c07f230 │ │ │ │ +00024f20: 3078 3766 3230 3763 3037 6639 3930 2020 0x7f207c07f990 │ │ │ │ 00024f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f50: 207c 0a7c 6672 6565 696e 6720 6d65 6d6f |.|freeing memo │ │ │ │ -00024f60: 7279 2061 7420 3078 3766 6335 3963 3037 ry at 0x7fc59c07 │ │ │ │ -00024f70: 6639 3130 2020 2020 2020 2020 2020 2020 f910 │ │ │ │ +00024f60: 7279 2061 7420 3078 3766 3230 3763 3037 ry at 0x7f207c07 │ │ │ │ +00024f70: 6632 3530 2020 2020 2020 2020 2020 2020 f250 │ │ │ │ 00024f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f90: 2020 2020 2020 207c 0a7c 6672 6565 696e |.|freein │ │ │ │ 00024fa0: 6720 6d65 6d6f 7279 2061 7420 3078 3766 g memory at 0x7f │ │ │ │ -00024fb0: 6335 3963 3037 6639 3730 2020 2020 2020 c59c07f970 │ │ │ │ +00024fb0: 3230 3763 3037 6638 6630 2020 2020 2020 207c07f8f0 │ │ │ │ 00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00024fe0: 6672 6565 696e 6720 6d65 6d6f 7279 2061 freeing memory a │ │ │ │ -00024ff0: 7420 3078 3766 6335 3963 3037 6639 6230 t 0x7fc59c07f9b0 │ │ │ │ +00024ff0: 7420 3078 3766 3230 3763 3037 6639 3330 t 0x7f207c07f930 │ │ │ │ 00025000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025020: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00025030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025060: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ @@ -9547,49 +9547,49 @@ │ │ │ │ 000254a0: 2d2d 2d2b 0a7c 6932 203a 2070 6565 6b20 ---+.|i2 : peek │ │ │ │ 000254b0: 7820 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ 000254c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000254d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254f0: 2020 207c 0a7c 6f32 203d 2069 6e74 3332 |.|o2 = int32 │ │ │ │ 00025500: 7b41 6464 7265 7373 203d 3e20 3078 3766 {Address => 0x7f │ │ │ │ -00025510: 6335 6162 6464 3734 3630 7d7c 0a2b 2d2d c5abdd7460}|.+-- │ │ │ │ +00025510: 3230 3934 3930 3732 6430 7d7c 0a2b 2d2d 20949072d0}|.+-- │ │ │ │ 00025520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025540: 2d2d 2d2b 0a0a 5468 6573 6520 706f 696e ---+..These poin │ │ │ │ 00025550: 7465 7273 2063 616e 2062 6520 6163 6365 ters can be acce │ │ │ │ 00025560: 7373 6564 2075 7369 6e67 202a 6e6f 7465 ssed using *note │ │ │ │ 00025570: 2061 6464 7265 7373 3a20 6164 6472 6573 address: addres │ │ │ │ 00025580: 732c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d s,...+---------- │ │ │ │ 00025590: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ 000255a0: 3a20 7074 7220 3d20 6164 6472 6573 7320 : ptr = address │ │ │ │ 000255b0: 787c 0a7c 2020 2020 2020 2020 2020 2020 x|.| │ │ │ │ 000255c0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -000255d0: 3078 3766 6335 6162 6464 3734 3630 207c 0x7fc5abdd7460 | │ │ │ │ +000255d0: 3078 3766 3230 3934 3930 3732 6430 207c 0x7f20949072d0 | │ │ │ │ 000255e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000255f0: 2020 2020 2020 7c0a 7c6f 3320 3a20 506f |.|o3 : Po │ │ │ │ 00025600: 696e 7465 7220 2020 2020 2020 207c 0a2b inter |.+ │ │ │ │ 00025610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025620: 2d2d 2d2d 2b0a 0a53 696d 706c 6520 6172 ----+..Simple ar │ │ │ │ 00025630: 6974 686d 6574 6963 2063 616e 2062 6520 ithmetic can be │ │ │ │ 00025640: 7065 7266 6f72 6d65 6420 6f6e 2070 6f69 performed on poi │ │ │ │ 00025650: 6e74 6572 732e 0a0a 2b2d 2d2d 2d2d 2d2d nters...+------- │ │ │ │ 00025660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00025670: 3420 3a20 7074 7220 2b20 3520 2020 2020 4 : ptr + 5 │ │ │ │ 00025680: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00025690: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ -000256a0: 3078 3766 6335 6162 6464 3734 3635 7c0a 0x7fc5abdd7465|. │ │ │ │ +000256a0: 3078 3766 3230 3934 3930 3732 6435 7c0a 0x7f20949072d5|. │ │ │ │ 000256b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000256c0: 2020 2020 7c0a 7c6f 3420 3a20 506f 696e |.|o4 : Poin │ │ │ │ 000256d0: 7465 7220 2020 2020 2020 7c0a 2b2d 2d2d ter |.+--- │ │ │ │ 000256e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000256f0: 2b0a 7c69 3520 3a20 7074 7220 2d20 3320 +.|i5 : ptr - 3 │ │ │ │ 00025700: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00025710: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00025720: 3520 3d20 3078 3766 6335 6162 6464 3734 5 = 0x7fc5abdd74 │ │ │ │ -00025730: 3564 7c0a 7c20 2020 2020 2020 2020 2020 5d|.| │ │ │ │ +00025720: 3520 3d20 3078 3766 3230 3934 3930 3732 5 = 0x7f20949072 │ │ │ │ +00025730: 6364 7c0a 7c20 2020 2020 2020 2020 2020 cd|.| │ │ │ │ 00025740: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20 |.|o5 : │ │ │ │ 00025750: 506f 696e 7465 7220 2020 2020 2020 7c0a Pointer |. │ │ │ │ 00025760: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00025770: 2d2d 2d2d 2b0a 2a20 4d65 6e75 3a0a 0a2a ----+.* Menu:..* │ │ │ │ 00025780: 206e 756c 6c50 6f69 6e74 6572 3a3a 2020 nullPointer:: │ │ │ │ 00025790: 2020 2020 2020 2020 2020 2020 2020 2074 t │ │ │ │ 000257a0: 6865 206e 756c 6c20 706f 696e 7465 720a he null pointer. │ │ │ │ @@ -9758,15 +9758,15 @@ │ │ │ │ 000261d0: 740a 7573 6564 2062 7920 6c69 6266 6669 t.used by libffi │ │ │ │ 000261e0: 2074 6f20 6964 656e 7469 6679 2074 6865 to identify the │ │ │ │ 000261f0: 2074 7970 652e 0a0a 2b2d 2d2d 2d2d 2d2d type...+------- │ │ │ │ 00026200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00026210: 3120 3a20 6164 6472 6573 7320 696e 7420 1 : address int │ │ │ │ 00026220: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00026230: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -00026240: 3078 3536 3437 3632 6130 3362 3430 7c0a 0x564762a03b40|. │ │ │ │ +00026240: 3078 3536 3037 6138 6564 6662 3430 7c0a 0x5607a8edfb40|. │ │ │ │ 00026250: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00026260: 2020 2020 7c0a 7c6f 3120 3a20 506f 696e |.|o1 : Poin │ │ │ │ 00026270: 7465 7220 2020 2020 2020 7c0a 2b2d 2d2d ter |.+--- │ │ │ │ 00026280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026290: 2b0a 0a49 6620 7820 6973 2061 2066 6f72 +..If x is a for │ │ │ │ 000262a0: 6569 676e 206f 626a 6563 742c 2074 6865 eign object, the │ │ │ │ 000262b0: 6e20 7468 6973 2072 6574 7572 6e73 2074 n this returns t │ │ │ │ @@ -9775,16 +9775,16 @@ │ │ │ │ 000262e0: 6861 7665 7320 6c69 6b65 2074 6865 2026 haves like the & │ │ │ │ 000262f0: 2022 6164 6472 6573 732d 6f66 2220 6f70 "address-of" op │ │ │ │ 00026300: 6572 6174 6f72 2069 6e20 432e 0a0a 2b2d erator in C...+- │ │ │ │ 00026310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026320: 2d2d 2b0a 7c69 3220 3a20 6164 6472 6573 --+.|i2 : addres │ │ │ │ 00026330: 7320 696e 7420 3520 7c0a 7c20 2020 2020 s int 5 |.| │ │ │ │ 00026340: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026350: 7c6f 3220 3d20 3078 3766 6335 6162 6464 |o2 = 0x7fc5abdd │ │ │ │ -00026360: 3739 3230 7c0a 7c20 2020 2020 2020 2020 7920|.| │ │ │ │ +00026350: 7c6f 3220 3d20 3078 3766 3230 3934 3930 |o2 = 0x7f209490 │ │ │ │ +00026360: 3734 3530 7c0a 7c20 2020 2020 2020 2020 7450|.| │ │ │ │ 00026370: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ 00026380: 3a20 506f 696e 7465 7220 2020 2020 2020 : Pointer │ │ │ │ 00026390: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 000263a0: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f ------+..Ways to │ │ │ │ 000263b0: 2075 7365 2061 6464 7265 7373 3a0a 3d3d use address:.== │ │ │ │ 000263c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 000263d0: 3d3d 0a0a 2020 2a20 2261 6464 7265 7373 ==.. * "address │ │ │ │ @@ -9866,16 +9866,16 @@ │ │ │ │ 00026890: 7970 6520 696e 7433 327c 0a2b 2d2d 2d2d ype int32|.+---- │ │ │ │ 000268a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000268b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 000268c0: 3220 3a20 7074 7220 3d20 6164 6472 6573 2 : ptr = addres │ │ │ │ 000268d0: 7320 7820 2020 2020 2020 2020 2020 207c s x | │ │ │ │ 000268e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000268f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026900: 2020 7c0a 7c6f 3220 3d20 3078 3766 6335 |.|o2 = 0x7fc5 │ │ │ │ -00026910: 6162 6464 3736 3430 2020 2020 2020 2020 abdd7640 │ │ │ │ +00026900: 2020 7c0a 7c6f 3220 3d20 3078 3766 3230 |.|o2 = 0x7f20 │ │ │ │ +00026910: 3934 3934 6134 3430 2020 2020 2020 2020 9494a440 │ │ │ │ 00026920: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00026930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026940: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ 00026950: 506f 696e 7465 7220 2020 2020 2020 2020 Pointer │ │ │ │ 00026960: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00026970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ ├── ./usr/share/info/FourTiTwo.info.gz │ │ │ ├── FourTiTwo.info │ │ │ │ @@ -838,25 +838,25 @@ │ │ │ │ 00003450: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00003460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003470: 2d2d 2d2d 2b0a 7c69 3220 3a20 7320 3d20 ----+.|i2 : s = │ │ │ │ 00003480: 7465 6d70 6f72 6172 7946 696c 654e 616d temporaryFileNam │ │ │ │ 00003490: 6528 2920 2020 2020 207c 0a7c 2020 2020 e() |.| │ │ │ │ 000034a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000034c0: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3135 |o2 = /tmp/M2-15 │ │ │ │ -000034d0: 3937 302d 302f 3020 2020 2020 2020 2020 970-0/0 │ │ │ │ +000034c0: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3231 |o2 = /tmp/M2-21 │ │ │ │ +000034d0: 3039 352d 302f 3020 2020 2020 2020 2020 095-0/0 │ │ │ │ 000034e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 000034f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003500: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ 00003510: 4620 3d20 6f70 656e 4f75 7428 7329 2020 F = openOut(s) │ │ │ │ 00003520: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00003530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003550: 2020 7c0a 7c6f 3320 3d20 2f74 6d70 2f4d |.|o3 = /tmp/M │ │ │ │ -00003560: 322d 3135 3937 302d 302f 3020 2020 2020 2-15970-0/0 │ │ │ │ +00003560: 322d 3231 3039 352d 302f 3020 2020 2020 2-21095-0/0 │ │ │ │ 00003570: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00003580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003590: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 000035a0: 3320 3a20 4669 6c65 2020 2020 2020 2020 3 : File │ │ │ │ 000035b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000035c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 000035d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -866,15 +866,15 @@ │ │ │ │ 00003610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003630: 2b0a 7c69 3520 3a20 636c 6f73 6528 4629 +.|i5 : close(F) │ │ │ │ 00003640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003650: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00003660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003670: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ -00003680: 3d20 2f74 6d70 2f4d 322d 3135 3937 302d = /tmp/M2-15970- │ │ │ │ +00003680: 3d20 2f74 6d70 2f4d 322d 3231 3039 352d = /tmp/M2-21095- │ │ │ │ 00003690: 302f 3020 2020 2020 2020 2020 2020 207c 0/0 | │ │ │ │ 000036a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000036b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000036c0: 2020 2020 7c0a 7c6f 3520 3a20 4669 6c65 |.|o5 : File │ │ │ │ 000036d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000036e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 000036f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/FrobeniusThresholds.info.gz │ │ │ ├── FrobeniusThresholds.info │ │ │ │ @@ -2692,18 +2692,18 @@ │ │ │ │ 0000a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a850: 2d2d 2d2d 2d2b 0a7c 6932 3720 3a20 7469 -----+.|i27 : ti │ │ │ │ 0000a860: 6d65 206e 756d 6572 6963 2066 7074 2866 me numeric fpt(f │ │ │ │ 0000a870: 2c20 4465 7074 684f 6653 6561 7263 6820 , DepthOfSearch │ │ │ │ 0000a880: 3d3e 2033 2c20 4669 6e61 6c41 7474 656d => 3, FinalAttem │ │ │ │ 0000a890: 7074 203d 3e20 7472 7565 297c 0a7c 202d pt => true)|.| - │ │ │ │ -0000a8a0: 2d20 7573 6564 2032 2e33 3431 3435 7320 - used 2.34145s │ │ │ │ -0000a8b0: 2863 7075 293b 2031 2e35 3234 3138 7320 (cpu); 1.52418s │ │ │ │ -0000a8c0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0000a8d0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0000a8a0: 2d20 7573 6564 2032 2e39 3334 3235 7320 - used 2.93425s │ │ │ │ +0000a8b0: 2863 7075 293b 2031 2e35 3734 3673 2028 (cpu); 1.5746s ( │ │ │ │ +0000a8c0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +0000a8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a8e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a920: 2020 2020 2020 207c 0a7c 6f32 3720 3d20 |.|o27 = │ │ │ │ 0000a930: 7b2e 3134 3230 3637 2c20 2e31 3434 7d20 {.142067, .144} │ │ │ │ 0000a940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2723,16 +2723,16 @@ │ │ │ │ 0000aa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000aa40: 0a7c 6932 3820 3a20 7469 6d65 2066 7074 .|i28 : time fpt │ │ │ │ 0000aa50: 2866 2c20 4465 7074 684f 6653 6561 7263 (f, DepthOfSearc │ │ │ │ 0000aa60: 6820 3d3e 2033 2c20 4174 7465 6d70 7473 h => 3, Attempts │ │ │ │ 0000aa70: 203d 3e20 3729 2020 2020 2020 2020 2020 => 7) │ │ │ │ 0000aa80: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0000aa90: 2031 2e33 3736 3534 7320 2863 7075 293b 1.37654s (cpu); │ │ │ │ -0000aaa0: 2030 2e39 3137 3035 3273 2028 7468 7265 0.917052s (thre │ │ │ │ +0000aa90: 2031 2e37 3232 3032 7320 2863 7075 293b 1.72202s (cpu); │ │ │ │ +0000aaa0: 2030 2e39 3532 3131 3973 2028 7468 7265 0.952119s (thre │ │ │ │ 0000aab0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 0000aac0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab10: 207c 0a7c 2020 2020 2020 3120 2020 2020 |.| 1 │ │ │ │ @@ -2762,16 +2762,16 @@ │ │ │ │ 0000ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000acb0: 2d2d 2d2d 2d2b 0a7c 6932 3920 3a20 7469 -----+.|i29 : ti │ │ │ │ 0000acc0: 6d65 2066 7074 2866 2c20 4465 7074 684f me fpt(f, DepthO │ │ │ │ 0000acd0: 6653 6561 7263 6820 3d3e 2034 2920 2020 fSearch => 4) │ │ │ │ 0000ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000acf0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -0000ad00: 2d20 7573 6564 2031 2e31 3533 3534 7320 - used 1.15354s │ │ │ │ -0000ad10: 2863 7075 293b 2030 2e37 3039 3530 3673 (cpu); 0.709506s │ │ │ │ +0000ad00: 2d20 7573 6564 2031 2e33 3532 3333 7320 - used 1.35233s │ │ │ │ +0000ad10: 2863 7075 293b 2030 2e37 3831 3134 3373 (cpu); 0.781143s │ │ │ │ 0000ad20: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 0000ad30: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 0000ad40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000ad50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ad80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ @@ -3670,17 +3670,17 @@ │ │ │ │ 0000e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e580: 2b0a 7c69 3135 203a 2074 696d 6520 6672 +.|i15 : time fr │ │ │ │ 0000e590: 6f62 656e 6975 734e 7528 332c 2066 2920 obeniusNu(3, f) │ │ │ │ 0000e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e5b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0000e5c0: 7c20 2d2d 2075 7365 6420 302e 3030 3232 | -- used 0.0022 │ │ │ │ -0000e5d0: 3230 3933 7320 2863 7075 293b 2030 2e30 2093s (cpu); 0.0 │ │ │ │ -0000e5e0: 3034 3333 3438 3473 2028 7468 7265 6164 0433484s (thread │ │ │ │ +0000e5c0: 7c20 2d2d 2075 7365 6420 302e 3030 3830 | -- used 0.0080 │ │ │ │ +0000e5d0: 3536 3333 7320 2863 7075 293b 2030 2e30 5633s (cpu); 0.0 │ │ │ │ +0000e5e0: 3035 3937 3234 3873 2028 7468 7265 6164 0597248s (thread │ │ │ │ 0000e5f0: 293b 2030 7320 2867 6329 2020 7c0a 7c20 ); 0s (gc) |.| │ │ │ │ 0000e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e630: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ 0000e640: 203d 2033 3735 3620 2020 2020 2020 2020 = 3756 │ │ │ │ 0000e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3690,16 +3690,16 @@ │ │ │ │ 0000e690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e6b0: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2074 ------+.|i16 : t │ │ │ │ 0000e6c0: 696d 6520 6672 6f62 656e 6975 734e 7528 ime frobeniusNu( │ │ │ │ 0000e6d0: 332c 2066 2c20 5573 6553 7065 6369 616c 3, f, UseSpecial │ │ │ │ 0000e6e0: 416c 676f 7269 7468 6d73 203d 3e20 6661 Algorithms => fa │ │ │ │ 0000e6f0: 6c73 6529 7c0a 7c20 2d2d 2075 7365 6420 lse)|.| -- used │ │ │ │ -0000e700: 302e 3339 3831 3939 7320 2863 7075 293b 0.398199s (cpu); │ │ │ │ -0000e710: 2030 2e32 3539 3533 3473 2028 7468 7265 0.259534s (thre │ │ │ │ +0000e700: 302e 3433 3633 3032 7320 2863 7075 293b 0.436302s (cpu); │ │ │ │ +0000e710: 2030 2e32 3938 3237 3473 2028 7468 7265 0.298274s (thre │ │ │ │ 0000e720: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 0000e730: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e770: 7c0a 7c6f 3136 203d 2033 3735 3620 2020 |.|o16 = 3756 │ │ │ │ 0000e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3805,17 +3805,17 @@ │ │ │ │ 0000edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ede0: 2b0a 7c69 3139 203a 2074 696d 6520 6672 +.|i19 : time fr │ │ │ │ 0000edf0: 6f62 656e 6975 734e 7528 342c 2066 2920 obeniusNu(4, f) │ │ │ │ 0000ee00: 2d2d 2043 6f6e 7461 696e 6d65 6e74 5465 -- ContainmentTe │ │ │ │ 0000ee10: 7374 2069 7320 7365 7420 746f 2046 726f st is set to Fro │ │ │ │ 0000ee20: 6265 6e69 7573 526f 6f74 2c20 6279 2020 beniusRoot, by │ │ │ │ -0000ee30: 7c0a 7c20 2d2d 2075 7365 6420 302e 3238 |.| -- used 0.28 │ │ │ │ -0000ee40: 3831 3239 7320 2863 7075 293b 2030 2e32 8129s (cpu); 0.2 │ │ │ │ -0000ee50: 3133 3039 3773 2028 7468 7265 6164 293b 13097s (thread); │ │ │ │ +0000ee30: 7c0a 7c20 2d2d 2075 7365 6420 302e 3535 |.| -- used 0.55 │ │ │ │ +0000ee40: 3337 3032 7320 2863 7075 293b 2030 2e33 3702s (cpu); 0.3 │ │ │ │ +0000ee50: 3233 3236 3773 2028 7468 7265 6164 293b 23267s (thread); │ │ │ │ 0000ee60: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0000ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ee80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3840,18 +3840,18 @@ │ │ │ │ 0000eff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f010: 2b0a 7c69 3230 203a 2074 696d 6520 6672 +.|i20 : time fr │ │ │ │ 0000f020: 6f62 656e 6975 734e 7528 342c 2066 2c20 obeniusNu(4, f, │ │ │ │ 0000f030: 436f 6e74 6169 6e6d 656e 7454 6573 7420 ContainmentTest │ │ │ │ 0000f040: 3d3e 2053 7461 6e64 6172 6450 6f77 6572 => StandardPower │ │ │ │ 0000f050: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0000f060: 7c0a 7c20 2d2d 2075 7365 6420 312e 3530 |.| -- used 1.50 │ │ │ │ -0000f070: 3134 7320 2863 7075 293b 2031 2e31 3738 14s (cpu); 1.178 │ │ │ │ -0000f080: 3536 7320 2874 6872 6561 6429 3b20 3073 56s (thread); 0s │ │ │ │ -0000f090: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0000f060: 7c0a 7c20 2d2d 2075 7365 6420 312e 3438 |.| -- used 1.48 │ │ │ │ +0000f070: 3730 3473 2028 6370 7529 3b20 312e 3235 704s (cpu); 1.25 │ │ │ │ +0000f080: 3233 3273 2028 7468 7265 6164 293b 2030 232s (thread); 0 │ │ │ │ +0000f090: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0000f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f100: 7c0a 7c6f 3230 203d 2034 3939 2020 2020 |.|o20 = 499 │ │ │ │ @@ -4015,17 +4015,17 @@ │ │ │ │ 0000fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000fb00: 0a7c 6932 3720 3a20 7469 6d65 2066 726f .|i27 : time fro │ │ │ │ 0000fb10: 6265 6e69 7573 4e75 2835 2c20 6629 202d beniusNu(5, f) - │ │ │ │ 0000fb20: 2d20 7573 6573 2062 696e 6172 7920 7365 - uses binary se │ │ │ │ 0000fb30: 6172 6368 2028 6465 6661 756c 7429 2020 arch (default) │ │ │ │ 0000fb40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000fb50: 0a7c 202d 2d20 7573 6564 2031 2e30 3138 .| -- used 1.018 │ │ │ │ -0000fb60: 3236 7320 2863 7075 293b 2030 2e36 3730 26s (cpu); 0.670 │ │ │ │ -0000fb70: 3036 3973 2028 7468 7265 6164 293b 2030 069s (thread); 0 │ │ │ │ +0000fb50: 0a7c 202d 2d20 7573 6564 2031 2e32 3735 .| -- used 1.275 │ │ │ │ +0000fb60: 3033 7320 2863 7075 293b 2030 2e37 3733 03s (cpu); 0.773 │ │ │ │ +0000fb70: 3730 3773 2028 7468 7265 6164 293b 2030 707s (thread); 0 │ │ │ │ 0000fb80: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0000fb90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000fba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -4040,18 +4040,18 @@ │ │ │ │ 0000fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000fc90: 0a7c 6932 3820 3a20 7469 6d65 2066 726f .|i28 : time fro │ │ │ │ 0000fca0: 6265 6e69 7573 4e75 2835 2c20 662c 2053 beniusNu(5, f, S │ │ │ │ 0000fcb0: 6561 7263 6820 3d3e 204c 696e 6561 7229 earch => Linear) │ │ │ │ 0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fcd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000fce0: 0a7c 202d 2d20 7573 6564 2031 2e35 3131 .| -- used 1.511 │ │ │ │ -0000fcf0: 3173 2028 6370 7529 3b20 302e 3937 3537 1s (cpu); 0.9757 │ │ │ │ -0000fd00: 3432 7320 2874 6872 6561 6429 3b20 3073 42s (thread); 0s │ │ │ │ -0000fd10: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0000fce0: 0a7c 202d 2d20 7573 6564 2031 2e38 3735 .| -- used 1.875 │ │ │ │ +0000fcf0: 3739 7320 2863 7075 293b 2031 2e31 3339 79s (cpu); 1.139 │ │ │ │ +0000fd00: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ +0000fd10: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0000fd20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000fd30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000fd80: 0a7c 6f32 3820 3d20 3131 3234 2020 2020 .|o28 = 1124 │ │ │ │ @@ -4085,17 +4085,17 @@ │ │ │ │ 0000ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000ff60: 0a7c 6933 3020 3a20 7469 6d65 2066 726f .|i30 : time fro │ │ │ │ 0000ff70: 6265 6e69 7573 4e75 2832 2c20 4d2c 204d beniusNu(2, M, M │ │ │ │ 0000ff80: 5e32 2920 2d2d 2075 7365 7320 6269 6e61 ^2) -- uses bina │ │ │ │ 0000ff90: 7279 2073 6561 7263 6820 2864 6566 6175 ry search (defau │ │ │ │ 0000ffa0: 6c74 2920 2020 2020 2020 2020 2020 207c lt) | │ │ │ │ -0000ffb0: 0a7c 202d 2d20 7573 6564 2031 2e37 3135 .| -- used 1.715 │ │ │ │ -0000ffc0: 3631 7320 2863 7075 293b 2031 2e34 3137 61s (cpu); 1.417 │ │ │ │ -0000ffd0: 3832 7320 2874 6872 6561 6429 3b20 3073 82s (thread); 0s │ │ │ │ +0000ffb0: 0a7c 202d 2d20 7573 6564 2031 2e38 3739 .| -- used 1.879 │ │ │ │ +0000ffc0: 3039 7320 2863 7075 293b 2031 2e35 3430 09s (cpu); 1.540 │ │ │ │ +0000ffd0: 3431 7320 2874 6872 6561 6429 3b20 3073 41s (thread); 0s │ │ │ │ 0000ffe0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0000fff0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00010000: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00010010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010040: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -4110,17 +4110,17 @@ │ │ │ │ 000100d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000100e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 000100f0: 0a7c 6933 3120 3a20 7469 6d65 2066 726f .|i31 : time fro │ │ │ │ 00010100: 6265 6e69 7573 4e75 2832 2c20 4d2c 204d beniusNu(2, M, M │ │ │ │ 00010110: 5e32 2c20 5365 6172 6368 203d 3e20 4c69 ^2, Search => Li │ │ │ │ 00010120: 6e65 6172 2920 2d2d 2062 7574 206c 696e near) -- but lin │ │ │ │ 00010130: 6561 7220 7365 6172 6368 2067 6574 737c ear search gets| │ │ │ │ -00010140: 0a7c 202d 2d20 7573 6564 2030 2e36 3232 .| -- used 0.622 │ │ │ │ -00010150: 3933 3373 2028 6370 7529 3b20 302e 3531 933s (cpu); 0.51 │ │ │ │ -00010160: 3234 3138 7320 2874 6872 6561 6429 3b20 2418s (thread); │ │ │ │ +00010140: 0a7c 202d 2d20 7573 6564 2030 2e36 3039 .| -- used 0.609 │ │ │ │ +00010150: 3639 3173 2028 6370 7529 3b20 302e 3534 691s (cpu); 0.54 │ │ │ │ +00010160: 3439 3038 7320 2874 6872 6561 6429 3b20 4908s (thread); │ │ │ │ 00010170: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00010180: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00010190: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000101a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ ├── ./usr/share/info/GKMVarieties.info.gz │ │ │ ├── GKMVarieties.info │ │ │ │ @@ -17563,18 +17563,18 @@ │ │ │ │ 000449a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000449b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000449c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3720 ---------+.|i27 │ │ │ │ 000449d0: 3a20 7469 6d65 2043 203d 206f 7262 6974 : time C = orbit │ │ │ │ 000449e0: 436c 6f73 7572 6528 582c 4d61 7429 2020 Closure(X,Mat) │ │ │ │ 000449f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a00: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00044a10: 7573 6564 2030 2e36 3039 3637 3873 2028 used 0.609678s ( │ │ │ │ -00044a20: 6370 7529 3b20 302e 3338 3634 3031 7320 cpu); 0.386401s │ │ │ │ -00044a30: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00044a40: 2920 2020 2020 2020 207c 0a7c 2020 2020 ) |.| │ │ │ │ +00044a10: 7573 6564 2031 2e37 3839 3131 7320 2863 used 1.78911s (c │ │ │ │ +00044a20: 7075 293b 2030 2e35 3130 3332 3573 2028 pu); 0.510325s ( │ │ │ │ +00044a30: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00044a40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00044a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a80: 2020 2020 2020 2020 207c 0a7c 6f32 3720 |.|o27 │ │ │ │ 00044a90: 3d20 616e 2022 6571 7569 7661 7269 616e = an "equivarian │ │ │ │ 00044aa0: 7420 4b2d 636c 6173 7322 206f 6e20 6120 t K-class" on a │ │ │ │ 00044ab0: 474b 4d20 7661 7269 6574 7920 2020 2020 GKM variety │ │ │ │ @@ -17591,16 +17591,16 @@ │ │ │ │ 00044b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044b80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3820 ---------+.|i28 │ │ │ │ 00044b90: 3a20 7469 6d65 2043 203d 206f 7262 6974 : time C = orbit │ │ │ │ 00044ba0: 436c 6f73 7572 6528 582c 4d61 742c 2052 Closure(X,Mat, R │ │ │ │ 00044bb0: 5245 464d 6574 686f 6420 3d3e 2074 7275 REFMethod => tru │ │ │ │ 00044bc0: 6529 2020 2020 2020 207c 0a7c 202d 2d20 e) |.| -- │ │ │ │ -00044bd0: 7573 6564 2031 2e38 3432 3733 7320 2863 used 1.84273s (c │ │ │ │ -00044be0: 7075 293b 2031 2e30 3737 3731 7320 2874 pu); 1.07771s (t │ │ │ │ +00044bd0: 7573 6564 2033 2e35 3538 3037 7320 2863 used 3.55807s (c │ │ │ │ +00044be0: 7075 293b 2031 2e31 3234 3539 7320 2874 pu); 1.12459s (t │ │ │ │ 00044bf0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00044c00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00044c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044c40: 2020 2020 2020 2020 207c 0a7c 6f32 3820 |.|o28 │ │ │ │ 00044c50: 3d20 616e 2022 6571 7569 7661 7269 616e = an "equivarian │ │ ├── ./usr/share/info/Graphs.info.gz │ │ │ ├── Graphs.info │ │ │ │ @@ -18078,16 +18078,16 @@ │ │ │ │ 000469d0: 7973 2048 2020 2020 2020 2020 2020 2020 ys H │ │ │ │ 000469e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000469f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00046a00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00046a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a30: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00046a40: 7b6d 6170 2c20 6469 6772 6170 682c 206e {map, digraph, n │ │ │ │ -00046a50: 6577 4469 6772 6170 687d 2020 2020 2020 ewDigraph} │ │ │ │ +00046a40: 7b6e 6577 4469 6772 6170 682c 2064 6967 {newDigraph, dig │ │ │ │ +00046a50: 7261 7068 2c20 6d61 707d 2020 2020 2020 raph, map} │ │ │ │ 00046a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00046a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046aa0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ 00046ab0: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ 00046ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/GroebnerStrata.info.gz │ │ │ ├── GroebnerStrata.info │ │ │ │ @@ -8737,28 +8737,28 @@ │ │ │ │ 00022200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022210: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00022220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022260: 2020 2020 2020 7c0a 7c6f 3134 203d 207c |.|o14 = | │ │ │ │ -00022270: 2034 3220 3920 3339 2039 2033 3420 3720 42 9 39 9 34 7 │ │ │ │ -00022280: 2d31 3220 2d31 3720 2d32 3920 2d33 3520 -12 -17 -29 -35 │ │ │ │ -00022290: 3530 2032 2031 3320 3139 202d 3434 2035 50 2 13 19 -44 5 │ │ │ │ -000222a0: 3020 3220 2d32 3920 3135 2032 202d 3237 0 2 -29 15 2 -27 │ │ │ │ -000222b0: 2032 3120 2020 7c0a 7c20 2020 2020 202d 21 |.| - │ │ │ │ +00022270: 202d 3620 3438 2034 3420 2d32 3320 2d32 -6 48 44 -23 -2 │ │ │ │ +00022280: 202d 3131 202d 3335 202d 3236 2032 3720 -11 -35 -26 27 │ │ │ │ +00022290: 2d34 3320 3438 2032 3720 3135 202d 3232 -43 48 27 15 -22 │ │ │ │ +000222a0: 2032 3520 2d31 3620 3334 202d 3239 2034 25 -16 34 -29 4 │ │ │ │ +000222b0: 3620 2d32 3020 7c0a 7c20 2020 2020 202d 6 -20 |.| - │ │ │ │ 000222c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000222d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000222e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000222f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022300: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 202d ------|.| - │ │ │ │ -00022310: 3336 202d 3239 202d 3339 202d 3130 2032 36 -29 -39 -10 2 │ │ │ │ -00022320: 3420 2d31 3620 3139 202d 3239 2033 3920 4 -16 19 -29 39 │ │ │ │ -00022330: 2d33 3820 2d32 3220 2d38 202d 3330 202d -38 -22 -8 -30 - │ │ │ │ -00022340: 3234 207c 2020 2020 2020 2020 2020 2020 24 | │ │ │ │ +00022300: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2034 ------|.| 4 │ │ │ │ +00022310: 3020 3231 202d 3330 202d 3338 202d 3139 0 21 -30 -38 -19 │ │ │ │ +00022320: 202d 3820 2d33 3620 3339 2031 3920 2d32 -8 -36 39 19 -2 │ │ │ │ +00022330: 3920 2d31 3620 2d32 3920 2d31 3020 3139 9 -16 -29 -10 19 │ │ │ │ +00022340: 2032 3420 2d32 3420 7c20 2020 2020 2020 24 -24 | │ │ │ │ 00022350: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00022360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000223b0: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ @@ -8782,28 +8782,28 @@ │ │ │ │ 000224d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000224e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000224f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022530: 2020 2020 2020 7c0a 7c6f 3135 203d 207c |.|o15 = | │ │ │ │ -00022540: 2033 3020 3130 2034 3320 3230 202d 3339 30 10 43 20 -39 │ │ │ │ -00022550: 2032 3320 2d33 3020 3430 202d 3334 2032 23 -30 40 -34 2 │ │ │ │ -00022560: 3220 3436 202d 3235 2032 3120 2d31 3820 2 46 -25 21 -18 │ │ │ │ -00022570: 2d33 3520 2d31 2032 3120 2d33 3920 2d34 -35 -1 21 -39 -4 │ │ │ │ -00022580: 3520 3136 2020 7c0a 7c20 2020 2020 202d 5 16 |.| - │ │ │ │ +00022540: 202d 3438 202d 3436 2031 3620 3137 202d -48 -46 16 17 - │ │ │ │ +00022550: 3120 2d34 3320 3135 202d 3120 3132 202d 1 -43 15 -1 12 - │ │ │ │ +00022560: 3138 202d 3620 2d32 3820 3134 202d 3238 18 -6 -28 14 -28 │ │ │ │ +00022570: 202d 3920 3332 202d 3232 202d 3339 2036 -9 32 -22 -39 6 │ │ │ │ +00022580: 202d 3437 2020 7c0a 7c20 2020 2020 202d -47 |.| - │ │ │ │ 00022590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000225a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000225b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000225c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000225d0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 202d ------|.| - │ │ │ │ -000225e0: 3335 202d 3520 3139 202d 3437 202d 3230 35 -5 19 -47 -20 │ │ │ │ -000225f0: 202d 3133 2033 3420 3333 202d 3238 202d -13 34 33 -28 - │ │ │ │ -00022600: 3433 2032 3220 3220 3020 2d31 3520 2d34 43 22 2 0 -15 -4 │ │ │ │ -00022610: 3720 3338 207c 2020 2020 2020 2020 2020 7 38 | │ │ │ │ +000225d0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2032 ------|.| 2 │ │ │ │ +000225e0: 3820 2d33 3720 2d34 3720 3338 202d 3136 8 -37 -47 38 -16 │ │ │ │ +000225f0: 202d 3135 2033 3420 3237 202d 3133 202d -15 34 27 -13 - │ │ │ │ +00022600: 3433 2032 3220 3136 2030 202d 3138 2031 43 22 16 0 -18 1 │ │ │ │ +00022610: 3920 3220 7c20 2020 2020 2020 2020 2020 9 2 | │ │ │ │ 00022620: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00022630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022670: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00022680: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ @@ -8830,79 +8830,79 @@ │ │ │ │ 000227d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000227e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000227f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022800: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00022810: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ 00022820: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ 00022830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022840: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00022840: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 00022850: 2020 2020 2020 7c0a 7c6f 3136 203d 2069 |.|o16 = i │ │ │ │ -00022860: 6465 616c 2028 6120 202d 2034 3462 2a63 deal (a - 44b*c │ │ │ │ -00022870: 202d 2033 3563 2020 2b20 3261 2a64 202b - 35c + 2a*d + │ │ │ │ -00022880: 2037 622a 6420 2b20 3339 632a 6420 2b20 7b*d + 39c*d + │ │ │ │ -00022890: 3432 6420 2c20 612a 6220 2d20 3339 622a 42d , a*b - 39b* │ │ │ │ -000228a0: 6320 2b20 2020 7c0a 7c20 2020 2020 202d c + |.| - │ │ │ │ +00022860: 6465 616c 2028 6120 202b 2032 3562 2a63 deal (a + 25b*c │ │ │ │ +00022870: 202d 2034 3363 2020 2b20 3237 612a 6420 - 43c + 27a*d │ │ │ │ +00022880: 2d20 3131 622a 6420 2b20 3434 632a 6420 - 11b*d + 44c*d │ │ │ │ +00022890: 2d20 3664 202c 2061 2a62 202d 2031 3962 - 6d , a*b - 19b │ │ │ │ +000228a0: 2a63 202b 2020 7c0a 7c20 2020 2020 202d *c + |.| - │ │ │ │ 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 2d2d ---------------- │ │ │ │ 000228f0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ 00022900: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00022910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022920: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00022920: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00022930: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -00022940: 2020 2020 2020 7c0a 7c20 2020 2020 2031 |.| 1 │ │ │ │ -00022950: 3563 2020 2d20 3237 612a 6420 2b20 3133 5c - 27a*d + 13 │ │ │ │ -00022960: 622a 6420 2d20 3239 632a 6420 2b20 3964 b*d - 29c*d + 9d │ │ │ │ -00022970: 202c 2061 2a63 202d 2033 3862 2a63 202d , a*c - 38b*c - │ │ │ │ -00022980: 2031 3063 2020 2d20 3136 612a 6420 2b20 10c - 16a*d + │ │ │ │ -00022990: 3262 2a64 202b 7c0a 7c20 2020 2020 202d 2b*d +|.| - │ │ │ │ +00022940: 2020 2020 2020 7c0a 7c20 2020 2020 2034 |.| 4 │ │ │ │ +00022950: 3663 2020 2b20 3430 612a 6420 2b20 3135 6c + 40a*d + 15 │ │ │ │ +00022960: 622a 6420 2b20 3237 632a 6420 2b20 3438 b*d + 27c*d + 48 │ │ │ │ +00022970: 6420 2c20 612a 6320 2d20 3239 622a 6320 d , a*c - 29b*c │ │ │ │ +00022980: 2d20 3863 2020 2b20 3339 612a 6420 2d20 - 8c + 39a*d - │ │ │ │ +00022990: 3230 622a 6420 7c0a 7c20 2020 2020 202d 20b*d |.| - │ │ │ │ 000229a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000229b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000229c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000229d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000229e0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ -000229f0: 2020 2020 2020 2020 2020 3220 2020 3220 2 2 │ │ │ │ -00022a00: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +000229f0: 2020 2020 2020 2020 2020 2032 2020 2032 2 2 │ │ │ │ +00022a00: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ 00022a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a20: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -00022a30: 2020 3220 2020 7c0a 7c20 2020 2020 2031 2 |.| 1 │ │ │ │ -00022a40: 3963 2a64 202b 2033 3464 202c 2062 2020 9c*d + 34d , b │ │ │ │ -00022a50: 2d20 3330 622a 6320 2b20 3139 6320 202d - 30b*c + 19c - │ │ │ │ -00022a60: 2032 3261 2a64 202b 2032 3162 2a64 202b 22a*d + 21b*d + │ │ │ │ -00022a70: 2035 3063 2a64 202d 2031 3264 202c 2062 50c*d - 12d , b │ │ │ │ -00022a80: 2a63 2020 2d20 7c0a 7c20 2020 2020 202d *c - |.| - │ │ │ │ +00022a20: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00022a30: 2020 2032 2020 7c0a 7c20 2020 2020 202d 2 |.| - │ │ │ │ +00022a40: 2032 3263 2a64 202d 2032 6420 2c20 6220 22c*d - 2d , b │ │ │ │ +00022a50: 202b 2032 3462 2a63 202b 2031 3963 2020 + 24b*c + 19c │ │ │ │ +00022a60: 2d20 3130 612a 6420 2b20 3231 622a 6420 - 10a*d + 21b*d │ │ │ │ +00022a70: 2d20 3136 632a 6420 2d20 3335 6420 2c20 - 16c*d - 35d , │ │ │ │ +00022a80: 622a 6320 202d 7c0a 7c20 2020 2020 202d b*c -|.| - │ │ │ │ 00022a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022ad0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ 00022ae0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -00022af0: 2020 2020 2020 3220 2020 2020 2020 3220 2 2 │ │ │ │ -00022b00: 2020 2020 2020 2032 2020 2020 2033 2020 2 3 │ │ │ │ -00022b10: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -00022b20: 2020 3220 2020 7c0a 7c20 2020 2020 2032 2 |.| 2 │ │ │ │ -00022b30: 3962 2a63 2a64 202d 2033 3663 2064 202b 9b*c*d - 36c d + │ │ │ │ -00022b40: 2032 3461 2a64 2020 2b20 3262 2a64 2020 24a*d + 2b*d │ │ │ │ -00022b50: 2b20 3530 632a 6420 202b 2039 6420 2c20 + 50c*d + 9d , │ │ │ │ -00022b60: 6320 202d 2032 3462 2a63 2a64 202b 2033 c - 24b*c*d + 3 │ │ │ │ -00022b70: 3963 2064 202d 7c0a 7c20 2020 2020 202d 9c d -|.| - │ │ │ │ +00022af0: 2020 2020 2020 3220 2020 2020 2020 2032 2 2 │ │ │ │ +00022b00: 2020 2020 2020 2020 3220 2020 2020 2033 2 3 │ │ │ │ +00022b10: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00022b20: 2020 2020 3220 7c0a 7c20 2020 2020 2032 2 |.| 2 │ │ │ │ +00022b30: 3962 2a63 2a64 202d 2033 3063 2064 202d 9b*c*d - 30c d - │ │ │ │ +00022b40: 2033 3661 2a64 2020 2b20 3334 622a 6420 36a*d + 34b*d │ │ │ │ +00022b50: 202b 2034 3863 2a64 2020 2d20 3233 6420 + 48c*d - 23d │ │ │ │ +00022b60: 2c20 6320 202d 2032 3462 2a63 2a64 202d , c - 24b*c*d - │ │ │ │ +00022b70: 2031 3663 2064 7c0a 7c20 2020 2020 202d 16c d|.| - │ │ │ │ 00022b80: 2d2d 2d2d 2d2d 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 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ -00022bd0: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ -00022be0: 2020 2020 2032 2020 2020 2020 3320 2020 2 3 │ │ │ │ +00022bd0: 2020 2020 2020 3220 2020 2020 2020 2032 2 2 │ │ │ │ +00022be0: 2020 2020 2020 2020 3220 2020 2020 2033 2 3 │ │ │ │ 00022bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c10: 2020 2020 2020 7c0a 7c20 2020 2020 2038 |.| 8 │ │ │ │ -00022c20: 612a 6420 202d 2032 3962 2a64 2020 2d20 a*d - 29b*d - │ │ │ │ -00022c30: 3239 632a 6420 202d 2031 3764 2029 2020 29c*d - 17d ) │ │ │ │ -00022c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022c10: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +00022c20: 2031 3961 2a64 2020 2d20 3338 622a 6420 19a*d - 38b*d │ │ │ │ +00022c30: 202d 2032 3963 2a64 2020 2d20 3236 6420 - 29c*d - 26d │ │ │ │ +00022c40: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00022c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00022c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022cb0: 2020 2020 2020 7c0a 7c6f 3136 203a 2049 |.|o16 : I │ │ │ │ @@ -8973,80 +8973,80 @@ │ │ │ │ 000230c0: 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 │ │ │ │ 00023110: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00023120: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00023130: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00023130: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 00023140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023150: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00023150: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 00023160: 2020 2020 2020 7c0a 7c6f 3138 203d 2069 |.|o18 = i │ │ │ │ -00023170: 6465 616c 2028 6120 202d 2033 3562 2a63 deal (a - 35b*c │ │ │ │ -00023180: 202b 2032 3263 2020 2d20 3235 612a 6420 + 22c - 25a*d │ │ │ │ -00023190: 2b20 3233 622a 6420 2b20 3433 632a 6420 + 23b*d + 43c*d │ │ │ │ -000231a0: 2b20 3330 6420 2c20 612a 6220 2d20 3230 + 30d , a*b - 20 │ │ │ │ -000231b0: 622a 6320 2d20 7c0a 7c20 2020 2020 202d b*c - |.| - │ │ │ │ +00023170: 6465 616c 2028 6120 202d 2039 622a 6320 deal (a - 9b*c │ │ │ │ +00023180: 2d20 3138 6320 202d 2032 3861 2a64 202d - 18c - 28a*d - │ │ │ │ +00023190: 2034 3362 2a64 202b 2031 3663 2a64 202d 43b*d + 16c*d - │ │ │ │ +000231a0: 2034 3864 202c 2061 2a62 202d 2031 3662 48d , a*b - 16b │ │ │ │ +000231b0: 2a63 202b 2020 7c0a 7c20 2020 2020 202d *c + |.| - │ │ │ │ 000231c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023200: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ -00023210: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00023210: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00023220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023230: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00023230: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00023240: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -00023250: 2020 2020 2020 7c0a 7c20 2020 2020 2034 |.| 4 │ │ │ │ -00023260: 3563 2020 2d20 3335 612a 6420 2b20 3231 5c - 35a*d + 21 │ │ │ │ -00023270: 622a 6420 2d20 3334 632a 6420 2b20 3130 b*d - 34c*d + 10 │ │ │ │ -00023280: 6420 2c20 612a 6320 2b20 3262 2a63 202d d , a*c + 2b*c - │ │ │ │ -00023290: 2031 3363 2020 2b20 3333 612a 6420 2b20 13c + 33a*d + │ │ │ │ -000232a0: 3136 622a 6420 7c0a 7c20 2020 2020 202d 16b*d |.| - │ │ │ │ +00023250: 2020 2020 2020 7c0a 7c20 2020 2020 2036 |.| 6 │ │ │ │ +00023260: 6320 202b 2032 3861 2a64 202b 2031 3462 c + 28a*d + 14b │ │ │ │ +00023270: 2a64 202b 2031 3263 2a64 202d 2034 3664 *d + 12c*d - 46d │ │ │ │ +00023280: 202c 2061 2a63 202b 2031 3662 2a63 202d , a*c + 16b*c - │ │ │ │ +00023290: 2031 3563 2020 2b20 3237 612a 6420 2d20 15c + 27a*d - │ │ │ │ +000232a0: 3437 622a 6420 7c0a 7c20 2020 2020 202d 47b*d |.| - │ │ │ │ 000232b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000232c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000232d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000232e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000232f0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ -00023300: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -00023310: 3220 2020 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ +00023300: 2020 2020 2020 2020 2020 3220 2020 3220 2 2 │ │ │ │ +00023310: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ 00023320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023330: 2020 2032 2020 2020 2032 2020 2020 2020 2 2 │ │ │ │ +00023330: 2020 2020 3220 2020 2020 3220 2020 2020 2 2 │ │ │ │ 00023340: 2020 2020 2020 7c0a 7c20 2020 2020 202d |.| - │ │ │ │ -00023350: 2031 3863 2a64 202d 2033 3964 202c 2062 18c*d - 39d , b │ │ │ │ -00023360: 2020 2d20 3437 622a 6320 2d20 3238 6320 - 47b*c - 28c │ │ │ │ -00023370: 202d 2035 622a 6420 2d20 632a 6420 2d20 - 5b*d - c*d - │ │ │ │ -00023380: 3330 6420 2c20 622a 6320 202d 2034 3362 30d , b*c - 43b │ │ │ │ -00023390: 2a63 2a64 202b 7c0a 7c20 2020 2020 202d *c*d +|.| - │ │ │ │ +00023350: 2032 3863 2a64 202d 2064 202c 2062 2020 28c*d - d , b │ │ │ │ +00023360: 2b20 3139 622a 6320 2d20 3133 6320 202d + 19b*c - 13c - │ │ │ │ +00023370: 2033 3762 2a64 202b 2033 3263 2a64 202b 37b*d + 32c*d + │ │ │ │ +00023380: 2031 3564 202c 2062 2a63 2020 2d20 3433 15d , b*c - 43 │ │ │ │ +00023390: 622a 632a 6420 7c0a 7c20 2020 2020 202d b*c*d |.| - │ │ │ │ 000233a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000233b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000233c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000233d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000233e0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ -000233f0: 2020 3220 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ -00023400: 2020 2020 2032 2020 2020 2020 2020 3220 2 2 │ │ │ │ -00023410: 2020 2020 2033 2020 2033 2020 2020 2020 3 3 │ │ │ │ +000233f0: 2020 2020 3220 2020 2020 2020 2020 3220 2 2 │ │ │ │ +00023400: 2020 2020 2020 2032 2020 2020 2020 2032 2 2 │ │ │ │ +00023410: 2020 2020 2020 3320 2020 3320 2020 2020 3 3 │ │ │ │ 00023420: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -00023430: 2020 2020 3220 7c0a 7c20 2020 2020 2031 2 |.| 1 │ │ │ │ -00023440: 3963 2064 202b 2033 3461 2a64 2020 2b20 9c d + 34a*d + │ │ │ │ -00023450: 3231 622a 6420 202b 2034 3663 2a64 2020 21b*d + 46c*d │ │ │ │ -00023460: 2b20 3230 6420 2c20 6320 202b 2033 3862 + 20d , c + 38b │ │ │ │ +00023430: 2020 2020 3220 7c0a 7c20 2020 2020 202d 2 |.| - │ │ │ │ +00023440: 2034 3763 2064 202b 2033 3461 2a64 2020 47c d + 34a*d │ │ │ │ +00023450: 2d20 3232 622a 6420 202d 2036 632a 6420 - 22b*d - 6c*d │ │ │ │ +00023460: 202b 2031 3764 202c 2063 2020 2b20 3262 + 17d , c + 2b │ │ │ │ 00023470: 2a63 2a64 202b 2032 3263 2064 202d 2031 *c*d + 22c d - 1 │ │ │ │ -00023480: 3561 2a64 2020 7c0a 7c20 2020 2020 202d 5a*d |.| - │ │ │ │ +00023480: 3861 2a64 2020 7c0a 7c20 2020 2020 202d 8a*d |.| - │ │ │ │ 00023490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000234a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000234b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000234c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000234d0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ 000234e0: 2020 2020 2020 3220 2020 2020 2020 2032 2 2 │ │ │ │ -000234f0: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ +000234f0: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ 00023500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023520: 2020 2020 2020 7c0a 7c20 2020 2020 202d |.| - │ │ │ │ -00023530: 2034 3762 2a64 2020 2d20 3339 632a 6420 47b*d - 39c*d │ │ │ │ -00023540: 202b 2034 3064 2029 2020 2020 2020 2020 + 40d ) │ │ │ │ +00023520: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +00023530: 2033 3862 2a64 2020 2d20 3339 632a 6420 38b*d - 39c*d │ │ │ │ +00023540: 202d 2064 2029 2020 2020 2020 2020 2020 - d ) │ │ │ │ 00023550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023570: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00023580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000235a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000235b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9127,3605 +9127,3615 @@ │ │ │ │ 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: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00023aa0: 2b2d 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 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -00023ae0: 2020 2020 2020 207c 0a7c 6f32 3020 3d20 |.|o20 = │ │ │ │ -00023af0: 7c69 6465 616c 2028 6320 2d20 3133 642c |ideal (c - 13d, │ │ │ │ -00023b00: 2062 202b 2033 3264 2c20 6120 2b20 3336 b + 32d, a + 36 │ │ │ │ +00023ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ae0: 2d2d 2d2d 2d2d 2d7c 0a7c 6f32 3020 3d20 -------|.|o20 = │ │ │ │ +00023af0: 7c69 6465 616c 2028 6320 2b20 3339 642c |ideal (c + 39d, │ │ │ │ +00023b00: 2062 202b 2032 3764 2c20 6120 2d20 3138 b + 27d, a - 18 │ │ │ │ 00023b10: 6429 2020 2020 2020 2020 2020 2020 2020 d) │ │ │ │ -00023b20: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ +00023b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00023b40: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00023b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023b70: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -00023b80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00023b90: 7c69 6465 616c 2028 6320 2d20 3136 642c |ideal (c - 16d, │ │ │ │ -00023ba0: 2062 202b 2064 2c20 6120 2b20 3136 6429 b + d, a + 16d) │ │ │ │ -00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bc0: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ -00023bd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00023be0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00023bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023c10: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -00023c20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00023c30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00023c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c50: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00023c60: 2020 3220 2020 207c 2020 2020 2020 2020 2 | │ │ │ │ -00023c70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00023c80: 7c69 6465 616c 2028 6220 2d20 3663 202b |ideal (b - 6c + │ │ │ │ -00023c90: 2033 3364 2c20 6120 2d20 3336 6320 2b20 33d, a - 36c + │ │ │ │ -00023ca0: 3264 2c20 6320 202b 2034 3363 2a64 202d 2d, c + 43c*d - │ │ │ │ -00023cb0: 2064 2029 2020 207c 2020 2020 2020 2020 d ) | │ │ │ │ -00023cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00023cd0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00023b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023b80: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ +00023b90: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00023ba0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00023bb0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023bd0: 2020 3220 2020 337c 0a7c 2020 2020 2020 2 3|.| │ │ │ │ +00023be0: 7c69 6465 616c 2028 6120 2d20 3239 6220 |ideal (a - 29b │ │ │ │ +00023bf0: 2d20 3863 202d 2031 3364 2c20 6220 202b - 8c - 13d, b + │ │ │ │ +00023c00: 2032 3462 2a63 202b 2031 3963 2020 2b20 24b*c + 19c + │ │ │ │ +00023c10: 3334 622a 6420 2b20 3563 2a64 202b 2033 34b*d + 5c*d + 3 │ │ │ │ +00023c20: 3764 202c 2063 207c 0a7c 2020 2020 2020 7d , c |.| │ │ │ │ +00023c30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00023c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023c70: 2d2d 2d2d 2d2d 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ +00023c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023cc0: 2d2d 2d2d 2d2d 2d7c 0a7c 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023d00: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -00023d10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00023d20: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00023d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023d10: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ +00023d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d40: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00023d50: 2020 2020 2032 207c 2020 2020 2020 2020 2 | │ │ │ │ -00023d60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00023d70: 7c69 6465 616c 2028 6220 2b20 3239 6320 |ideal (b + 29c │ │ │ │ -00023d80: 2b20 3764 2c20 6120 2d20 3139 6320 2b20 + 7d, a - 19c + │ │ │ │ -00023d90: 3234 642c 2063 2020 2d20 3230 632a 6420 24d, c - 20c*d │ │ │ │ -00023da0: 2d20 3330 6420 297c 2020 2020 2020 2020 - 30d )| │ │ │ │ -00023db0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00023dc0: 2b2d 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 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -00023e00: 2020 2020 2020 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023e50: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 -------+.|i21 : │ │ │ │ -00023e60: 6e65 744c 6973 7420 6465 636f 6d70 6f73 netList decompos │ │ │ │ -00023e70: 6520 4632 2020 2020 2020 2020 2020 2020 e F2 │ │ │ │ -00023e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ea0: 2020 2020 2020 207c 0a7c 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 2020 2020 │ │ │ │ -00023ef0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00023f00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00023f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023f40: 2d2d 2d2d 2d2d 2d7c 0a7c 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 207c 0a7c 2d2d 2d2d 2d2d |.|------ │ │ │ │ +00023d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023db0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ +00023dc0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00023dd0: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ +00023de0: 2020 2033 2020 2020 2032 2020 2020 2020 3 2 │ │ │ │ +00023df0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00023e00: 2020 2020 3220 207c 0a7c 202d 2032 3462 2 |.| - 24b │ │ │ │ +00023e10: 2a63 2a64 202d 2031 3663 2064 202b 2038 *c*d - 16c d + 8 │ │ │ │ +00023e20: 622a 6420 202b 2032 3263 2a64 2020 2b20 b*d + 22c*d + │ │ │ │ +00023e30: 3139 6420 2c20 622a 6320 202d 2032 3962 19d , b*c - 29b │ │ │ │ +00023e40: 2a63 2a64 202d 2033 3063 2064 202d 2033 *c*d - 30c d - 3 │ │ │ │ +00023e50: 3863 2a64 2020 2b7c 0a7c 2d2d 2d2d 2d2d 8c*d +|.|------ │ │ │ │ +00023e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ea0: 2d2d 2d2d 2d2d 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ +00023eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ef0: 2d2d 2d2d 2d2d 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ +00023f00: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +00023f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023f40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00023f50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00023f60: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00023f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f80: 2020 2020 2020 2032 2020 2032 2020 2020 2 2 │ │ │ │ -00023f90: 2020 2020 2020 207c 0a7c 6f32 3120 3d20 |.|o21 = │ │ │ │ -00023fa0: 7c69 6465 616c 2028 612a 6320 2b20 3262 |ideal (a*c + 2b │ │ │ │ -00023fb0: 2a63 202d 2031 3363 2020 2b20 3333 612a *c - 13c + 33a* │ │ │ │ -00023fc0: 6420 2b20 3136 622a 6420 2d20 3138 632a d + 16b*d - 18c* │ │ │ │ -00023fd0: 6420 2d20 3339 6420 2c20 6220 202d 2034 d - 39d , b - 4 │ │ │ │ -00023fe0: 3762 2a63 202d 207c 0a7c 2020 2020 2020 7b*c - |.| │ │ │ │ -00023ff0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00024000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024030: 2d2d 2d2d 2d2d 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ -00024040: 2d2d 2d2d 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 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ -00024090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000240a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000240b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000240c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000240d0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2032 2020 -------|.| 2 │ │ │ │ -000240e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000240f0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00024100: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00024110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024120: 2020 2020 3220 207c 0a7c 3238 6320 202d 2 |.|28c - │ │ │ │ -00024130: 2035 622a 6420 2d20 632a 6420 2d20 3330 5b*d - c*d - 30 │ │ │ │ -00024140: 6420 2c20 612a 6220 2d20 3230 622a 6320 d , a*b - 20b*c │ │ │ │ -00024150: 2d20 3435 6320 202d 2033 3561 2a64 202b - 45c - 35a*d + │ │ │ │ -00024160: 2032 3162 2a64 202d 2033 3463 2a64 202b 21b*d - 34c*d + │ │ │ │ -00024170: 2031 3064 202c 207c 0a7c 2d2d 2d2d 2d2d 10d , |.|------ │ │ │ │ -00024180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000241a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000241b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000241c0: 2d2d 2d2d 2d2d 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ -000241d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023f90: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d |.|------ │ │ │ │ +00023fa0: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +00023fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023fe0: 2020 2020 2020 207c 0a7c 2020 2020 3320 |.| 3 │ │ │ │ +00023ff0: 7c20 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 2020 2020 │ │ │ │ +00024030: 2020 2020 2020 207c 0a7c 2031 3464 2029 |.| 14d ) │ │ │ │ +00024040: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00024050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024080: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d |.|------ │ │ │ │ +00024090: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +000240a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000240b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000240c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000240d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000240e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000240f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024120: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 -------+.|i21 : │ │ │ │ +00024130: 6e65 744c 6973 7420 6465 636f 6d70 6f73 netList decompos │ │ │ │ +00024140: 6520 4632 2020 2020 2020 2020 2020 2020 e F2 │ │ │ │ +00024150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024170: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000241a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000241b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000241c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000241d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000241e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000241f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024210: 2d2d 2d2d 2d2d 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ -00024220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024260: 2d2d 2d2d 2d2d 2d7c 0a7c 2032 2020 2020 -------|.| 2 │ │ │ │ -00024270: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -00024280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024290: 2020 2020 2020 2020 2032 2020 2033 2020 2 3 │ │ │ │ -000242a0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -000242b0: 2020 2020 2020 207c 0a7c 6120 202d 2033 |.|a - 3 │ │ │ │ -000242c0: 3562 2a63 202b 2032 3263 2020 2d20 3235 5b*c + 22c - 25 │ │ │ │ -000242d0: 612a 6420 2b20 3233 622a 6420 2b20 3433 a*d + 23b*d + 43 │ │ │ │ -000242e0: 632a 6420 2b20 3330 6420 2c20 6320 202b c*d + 30d , c + │ │ │ │ -000242f0: 2033 3862 2a63 2a64 202b 2032 3263 2064 38b*c*d + 22c d │ │ │ │ -00024300: 202d 2031 3561 2a7c 0a7c 2d2d 2d2d 2d2d - 15a*|.|------ │ │ │ │ -00024310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024200: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020 --------+ │ │ │ │ +00024210: 2020 2020 2020 207c 0a7c 6f32 3120 3d20 |.|o21 = │ │ │ │ +00024220: 7c69 6465 616c 2028 6320 2d20 3332 642c |ideal (c - 32d, │ │ │ │ +00024230: 2062 202d 2035 642c 2061 202d 2032 3964 b - 5d, a - 29d │ │ │ │ +00024240: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00024250: 2020 2020 2020 2020 7c20 2020 2020 2020 | │ │ │ │ +00024260: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024270: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00024280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000242a0: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020 --------+ │ │ │ │ +000242b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000242c0: 7c69 6465 616c 2028 6320 2b20 3433 642c |ideal (c + 43d, │ │ │ │ +000242d0: 2062 202d 2034 3764 2c20 6120 2d20 3237 b - 47d, a - 27 │ │ │ │ +000242e0: 6429 2020 2020 2020 2020 2020 2020 2020 d) │ │ │ │ +000242f0: 2020 2020 2020 2020 7c20 2020 2020 2020 | │ │ │ │ +00024300: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024310: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00024320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024350: 2d2d 2d2d 2d2d 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ -00024360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000243a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ -000243b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024340: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020 --------+ │ │ │ │ +00024350: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024360: 7c69 6465 616c 2028 6320 2b20 3234 642c |ideal (c + 24d, │ │ │ │ +00024370: 2062 202d 2034 3964 2c20 6129 2020 2020 b - 49d, a) │ │ │ │ +00024380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024390: 2020 2020 2020 2020 7c20 2020 2020 2020 | │ │ │ │ +000243a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000243b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000243c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000243d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000243e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000243f0: 2d2d 2d2d 2d2d 2d7c 0a7c 2032 2020 2020 -------|.| 2 │ │ │ │ -00024400: 2020 2020 3220 2020 2020 2020 2032 2020 2 2 │ │ │ │ -00024410: 2020 2020 3320 2020 2020 3220 2020 2020 3 2 │ │ │ │ -00024420: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00024430: 2020 2020 2032 2020 2020 2020 2020 3220 2 2 │ │ │ │ -00024440: 2020 2020 2020 207c 0a7c 6420 202d 2034 |.|d - 4 │ │ │ │ -00024450: 3762 2a64 2020 2d20 3339 632a 6420 202b 7b*d - 39c*d + │ │ │ │ -00024460: 2034 3064 202c 2062 2a63 2020 2d20 3433 40d , b*c - 43 │ │ │ │ -00024470: 622a 632a 6420 2b20 3139 6320 6420 2b20 b*c*d + 19c d + │ │ │ │ -00024480: 3334 612a 6420 202b 2032 3162 2a64 2020 34a*d + 21b*d │ │ │ │ -00024490: 2b20 3436 632a 647c 0a7c 2d2d 2d2d 2d2d + 46c*d|.|------ │ │ │ │ -000244a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000244b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000244c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000244d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000244e0: 2d2d 2d2d 2d2d 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ -000244f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024530: 2d2d 2d2d 2d2d 2d7c 0a7c 2d2d 2d2d 2d2d -------|.|------ │ │ │ │ -00024540: 2d2d 2d2b 2020 2020 2020 2020 2020 2020 ---+ │ │ │ │ -00024550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024580: 2020 2020 2020 207c 0a7c 3220 2020 2020 |.|2 │ │ │ │ -00024590: 2033 207c 2020 2020 2020 2020 2020 2020 3 | │ │ │ │ -000245a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000245b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000245c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000245d0: 2020 2020 2020 207c 0a7c 2020 2b20 3230 |.| + 20 │ │ │ │ -000245e0: 6420 297c 2020 2020 2020 2020 2020 2020 d )| │ │ │ │ -000245f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024620: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d |.|------ │ │ │ │ -00024630: 2d2d 2d2b 2020 2020 2020 2020 2020 2020 ---+ │ │ │ │ -00024640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024670: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00024680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000246a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000246b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000246c0: 2d2d 2d2d 2d2d 2d2b 0a0a 5765 2063 616e -------+..We can │ │ │ │ -000246d0: 2064 6574 6572 6d69 6e65 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-00024800: 6e64 6f6d 2070 6f69 6e74 206f 6e20 610a ndom point on a. │ │ │ │ -00024810: 2020 2020 7661 7269 6574 7920 7468 6174 variety that │ │ │ │ -00024820: 2063 616e 2062 6520 6465 7465 6374 6564 can be detected │ │ │ │ -00024830: 2074 6f20 6265 2072 6174 696f 6e61 6c0a to be rational. │ │ │ │ -00024840: 0a57 6179 7320 746f 2075 7365 206e 6f6e .Ways to use non │ │ │ │ -00024850: 6d69 6e69 6d61 6c4d 6170 733a 0a3d 3d3d minimalMaps:.=== │ │ │ │ -00024860: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00024870: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226e ========.. * "n │ │ │ │ -00024880: 6f6e 6d69 6e69 6d61 6c4d 6170 7328 4964 onminimalMaps(Id │ │ │ │ -00024890: 6561 6c29 220a 0a46 6f72 2074 6865 2070 eal)"..For the p │ │ │ │ -000248a0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -000248b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -000248c0: 6520 6f62 6a65 6374 202a 6e6f 7465 206e e object *note n │ │ │ │ -000248d0: 6f6e 6d69 6e69 6d61 6c4d 6170 733a 206e onminimalMaps: n │ │ │ │ -000248e0: 6f6e 6d69 6e69 6d61 6c4d 6170 732c 2069 onminimalMaps, i │ │ │ │ -000248f0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ -00024900: 2066 756e 6374 696f 6e3a 0a28 4d61 6361 function:.(Maca │ │ │ │ -00024910: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -00024920: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ -00024930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024970: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -00024980: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00024990: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -000249a0: 7569 6c64 2f72 6570 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2c20 5072 deal_cmZZ_rp, Pr │ │ │ │ -00024a80: 6576 3a20 6e6f 6e6d 696e 696d 616c 4d61 ev: nonminimalMa │ │ │ │ -00024a90: 7073 2c20 5570 3a20 546f 700a 0a72 616e ps, Up: Top..ran │ │ │ │ -00024aa0: 646f 6d50 6f69 6e74 4f6e 5261 7469 6f6e domPointOnRation │ │ │ │ -00024ab0: 616c 5661 7269 6574 7928 4964 6561 6c29 alVariety(Ideal) │ │ │ │ -00024ac0: 202d 2d20 6669 6e64 2061 2072 616e 646f -- find a rando │ │ │ │ -00024ad0: 6d20 706f 696e 7420 6f6e 2061 2076 6172 m point on a var │ │ │ │ -00024ae0: 6965 7479 2074 6861 7420 6361 6e20 6265 iety that can be │ │ │ │ -00024af0: 2064 6574 6563 7465 6420 746f 2062 6520 detected to be │ │ │ │ -00024b00: 7261 7469 6f6e 616c 0a2a 2a2a 2a2a 2a2a rational.******* │ │ │ │ -00024b10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024b20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024b60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024b70: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469 ****.. * Functi │ │ │ │ -00024b80: 6f6e 3a20 2a6e 6f74 6520 7261 6e64 6f6d on: *note random │ │ │ │ -00024b90: 506f 696e 744f 6e52 6174 696f 6e61 6c56 PointOnRationalV │ │ │ │ -00024ba0: 6172 6965 7479 3a0a 2020 2020 7261 6e64 ariety:. rand │ │ │ │ -00024bb0: 6f6d 506f 696e 744f 6e52 6174 696f 6e61 omPointOnRationa │ │ │ │ -00024bc0: 6c56 6172 6965 7479 5f6c 7049 6465 616c lVariety_lpIdeal │ │ │ │ -00024bd0: 5f72 702c 0a20 202a 2055 7361 6765 3a20 _rp,. * Usage: │ │ │ │ -00024be0: 0a20 2020 2020 2020 2072 616e 646f 6d50 . randomP │ │ │ │ -00024bf0: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ -00024c00: 7269 6574 7920 490a 2020 2020 2020 2020 riety I. │ │ │ │ -00024c10: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ -00024c20: 696f 6e61 6c56 6172 6965 7479 0a20 202a ionalVariety. * │ │ │ │ -00024c30: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00024c40: 2049 2c20 616e 202a 6e6f 7465 2069 6465 I, an *note ide │ │ │ │ -00024c50: 616c 3a20 284d 6163 6175 6c61 7932 446f al: (Macaulay2Do │ │ │ │ -00024c60: 6329 4964 6561 6c2c 2c20 416e 2069 6465 c)Ideal,, An ide │ │ │ │ -00024c70: 616c 2069 6e20 6120 706f 6c79 6e6f 6d69 al in a polynomi │ │ │ │ -00024c80: 616c 2072 696e 670a 2020 2020 2020 2020 al ring. │ │ │ │ -00024c90: 2453 2420 6f76 6572 2061 2066 6965 6c64 $S$ over a field │ │ │ │ -00024ca0: 2c20 7768 6963 6820 6465 6669 6e65 7320 , which defines │ │ │ │ -00024cb0: 6120 7072 696d 6520 6964 6561 6c0a 2020 a prime ideal. │ │ │ │ -00024cc0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ -00024cd0: 202a 2061 202a 6e6f 7465 206d 6174 7269 * a *note matri │ │ │ │ -00024ce0: 783a 2028 4d61 6361 756c 6179 3244 6f63 x: (Macaulay2Doc │ │ │ │ -00024cf0: 294d 6174 7269 782c 2c20 4120 6f6e 6520 )Matrix,, A one │ │ │ │ -00024d00: 726f 7720 6d61 7472 6978 206f 7665 7220 row matrix over │ │ │ │ -00024d10: 7468 6520 6261 7365 0a20 2020 2020 2020 the base. │ │ │ │ -00024d20: 2066 6965 6c64 206f 6620 2453 242c 2072 field of $S$, r │ │ │ │ -00024d30: 6570 7265 7365 6e74 696e 6720 6120 7261 epresenting a ra │ │ │ │ -00024d40: 6e64 6f6d 6c79 2063 686f 7365 6e20 706f ndomly chosen po │ │ │ │ -00024d50: 696e 7420 6f6e 2074 6865 207a 6572 6f20 int on the zero │ │ │ │ -00024d60: 6c6f 6375 7320 6f66 0a20 2020 2020 2020 locus of. │ │ │ │ -00024d70: 2024 4924 2e20 206e 756c 6c20 6973 2072 $I$. null is r │ │ │ │ -00024d80: 6574 7572 6e65 6420 696e 2074 6865 2063 eturned in the c │ │ │ │ -00024d90: 6173 6520 7768 656e 2074 6865 2072 6f75 ase when the rou │ │ │ │ -00024da0: 7469 6e65 2063 616e 6e6f 7420 6465 7465 tine cannot dete │ │ │ │ -00024db0: 726d 696e 6520 6966 0a20 2020 2020 2020 rmine if. │ │ │ │ -00024dc0: 2074 6865 2076 6172 6965 7479 2069 7320 the variety is │ │ │ │ -00024dd0: 7261 7469 6f6e 616c 2061 6e64 2069 7272 rational and irr │ │ │ │ -00024de0: 6564 7563 6962 6c65 2e0a 0a44 6573 6372 educible...Descr │ │ │ │ -00024df0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00024e00: 3d3d 0a0a 4173 2061 2066 6972 7374 2065 ==..As a first e │ │ │ │ -00024e10: 7861 6d70 6c65 2c20 7765 2066 696e 6420 xample, we find │ │ │ │ -00024e20: 6120 7261 6e64 6f6d 2070 6f69 6e74 206f a random point o │ │ │ │ -00024e30: 6e20 7468 6520 5665 726f 6e65 7365 2073 n the Veronese s │ │ │ │ -00024e40: 7572 6661 6365 2069 6e20 245c 5050 5e35 urface in $\PP^5 │ │ │ │ -00024e50: 242e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d $...+----------- │ │ │ │ -00024e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024ea0: 2d2d 2b0a 7c69 3120 3a20 6b6b 203d 205a --+.|i1 : kk = Z │ │ │ │ -00024eb0: 5a2f 3130 313b 2020 2020 2020 2020 2020 Z/101; │ │ │ │ -00024ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ef0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00024f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024f40: 2d2d 2b0a 7c69 3220 3a20 5320 3d20 6b6b --+.|i2 : S = kk │ │ │ │ -00024f50: 5b61 2e2e 665d 3b20 2020 2020 2020 2020 [a..f]; │ │ │ │ +000243e0: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020 --------+ │ │ │ │ +000243f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024400: 7c69 6465 616c 2028 6320 2b20 3134 642c |ideal (c + 14d, │ │ │ │ +00024410: 2062 202b 2033 3164 2c20 6120 2d20 3136 b + 31d, a - 16 │ │ │ │ +00024420: 6429 2020 2020 2020 2020 2020 2020 2020 d) │ │ │ │ +00024430: 2020 2020 2020 2020 7c20 2020 2020 2020 | │ │ │ │ +00024440: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024450: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00024460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024480: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020 --------+ │ │ │ │ +00024490: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000244a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000244b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000244c0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +000244d0: 2020 2020 2020 3220 7c20 2020 2020 2020 2 | │ │ │ │ +000244e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000244f0: 7c69 6465 616c 2028 6220 2b20 3131 6320 |ideal (b + 11c │ │ │ │ +00024500: 2b20 3232 642c 2061 202b 2031 3163 202b + 22d, a + 11c + │ │ │ │ +00024510: 2034 3264 2c20 6320 202d 2034 3363 2a64 42d, c - 43c*d │ │ │ │ +00024520: 202b 2033 3164 2029 7c20 2020 2020 2020 + 31d )| │ │ │ │ +00024530: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00024540: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00024550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024570: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020 --------+ │ │ │ │ +00024580: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00024590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000245a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000245b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000245c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000245d0: 2d2d 2d2d 2d2d 2d2b 0a0a 5765 2063 616e -------+..We can │ │ │ │ +000245e0: 2064 6574 6572 6d69 6e65 2077 6861 7420 determine what │ │ │ │ +000245f0: 7468 6573 6520 7265 7072 6573 656e 742e these represent. │ │ │ │ +00024600: 2020 4f6e 6520 7368 6f75 6c64 2062 6520 One should be │ │ │ │ +00024610: 6120 7365 7420 6f66 2036 2070 6f69 6e74 a set of 6 point │ │ │ │ +00024620: 732c 2077 6865 7265 0a35 206c 6965 206f s, where.5 lie o │ │ │ │ +00024630: 6e20 6120 706c 616e 652e 2020 5468 6520 n a plane. The │ │ │ │ +00024640: 6f74 6865 7220 7368 6f75 6c64 2062 6520 other should be │ │ │ │ +00024650: 3620 706f 696e 7473 2077 6974 6820 3320 6 points with 3 │ │ │ │ +00024660: 706f 696e 7473 206f 6e20 6f6e 6520 6c69 points on one li │ │ │ │ +00024670: 6e65 2c20 616e 640a 7468 6520 6f74 6865 ne, and.the othe │ │ │ │ +00024680: 7220 3320 706f 696e 7473 206f 6e20 6120 r 3 points on a │ │ │ │ +00024690: 736b 6577 206c 696e 652e 0a0a 5365 6520 skew line...See │ │ │ │ +000246a0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ +000246b0: 202a 202a 6e6f 7465 2072 616e 646f 6d50 * *note randomP │ │ │ │ +000246c0: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ +000246d0: 7269 6574 793a 0a20 2020 2072 616e 646f riety:. rando │ │ │ │ +000246e0: 6d50 6f69 6e74 4f6e 5261 7469 6f6e 616c mPointOnRational │ │ │ │ +000246f0: 5661 7269 6574 795f 6c70 4964 6561 6c5f Variety_lpIdeal_ │ │ │ │ +00024700: 7270 2c20 2d2d 2066 696e 6420 6120 7261 rp, -- find a ra │ │ │ │ +00024710: 6e64 6f6d 2070 6f69 6e74 206f 6e20 610a ndom point on a. │ │ │ │ +00024720: 2020 2020 7661 7269 6574 7920 7468 6174 variety that │ │ │ │ +00024730: 2063 616e 2062 6520 6465 7465 6374 6564 can be detected │ │ │ │ +00024740: 2074 6f20 6265 2072 6174 696f 6e61 6c0a to be rational. │ │ │ │ +00024750: 0a57 6179 7320 746f 2075 7365 206e 6f6e .Ways to use non │ │ │ │ +00024760: 6d69 6e69 6d61 6c4d 6170 733a 0a3d 3d3d minimalMaps:.=== │ │ │ │ +00024770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00024780: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226e ========.. * "n │ │ │ │ +00024790: 6f6e 6d69 6e69 6d61 6c4d 6170 7328 4964 onminimalMaps(Id │ │ │ │ +000247a0: 6561 6c29 220a 0a46 6f72 2074 6865 2070 eal)"..For the p │ │ │ │ +000247b0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +000247c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +000247d0: 6520 6f62 6a65 6374 202a 6e6f 7465 206e e object *note n │ │ │ │ +000247e0: 6f6e 6d69 6e69 6d61 6c4d 6170 733a 206e onminimalMaps: n │ │ │ │ +000247f0: 6f6e 6d69 6e69 6d61 6c4d 6170 732c 2069 onminimalMaps, i │ │ │ │ +00024800: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +00024810: 2066 756e 6374 696f 6e3a 0a28 4d61 6361 function:.(Maca │ │ │ │ +00024820: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ +00024830: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +00024840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024880: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +00024890: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +000248a0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +000248b0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +000248c0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +000248d0: 2d31 2e32 352e 3131 2b64 732f 4d32 2f4d -1.25.11+ds/M2/M │ │ │ │ +000248e0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +000248f0: 732f 0a47 726f 6562 6e65 7253 7472 6174 s/.GroebnerStrat │ │ │ │ +00024900: 612e 6d32 3a31 3031 323a 302e 0a1f 0a46 a.m2:1012:0....F │ │ │ │ +00024910: 696c 653a 2047 726f 6562 6e65 7253 7472 ile: GroebnerStr │ │ │ │ +00024920: 6174 612e 696e 666f 2c20 4e6f 6465 3a20 ata.info, Node: │ │ │ │ +00024930: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ +00024940: 696f 6e61 6c56 6172 6965 7479 5f6c 7049 ionalVariety_lpI │ │ │ │ +00024950: 6465 616c 5f72 702c 204e 6578 743a 2072 deal_rp, Next: r │ │ │ │ +00024960: 616e 646f 6d50 6f69 6e74 734f 6e52 6174 andomPointsOnRat │ │ │ │ +00024970: 696f 6e61 6c56 6172 6965 7479 5f6c 7049 ionalVariety_lpI │ │ │ │ +00024980: 6465 616c 5f63 6d5a 5a5f 7270 2c20 5072 deal_cmZZ_rp, Pr │ │ │ │ +00024990: 6576 3a20 6e6f 6e6d 696e 696d 616c 4d61 ev: nonminimalMa │ │ │ │ +000249a0: 7073 2c20 5570 3a20 546f 700a 0a72 616e ps, Up: Top..ran │ │ │ │ +000249b0: 646f 6d50 6f69 6e74 4f6e 5261 7469 6f6e domPointOnRation │ │ │ │ +000249c0: 616c 5661 7269 6574 7928 4964 6561 6c29 alVariety(Ideal) │ │ │ │ +000249d0: 202d 2d20 6669 6e64 2061 2072 616e 646f -- find a rando │ │ │ │ +000249e0: 6d20 706f 696e 7420 6f6e 2061 2076 6172 m point on a var │ │ │ │ +000249f0: 6965 7479 2074 6861 7420 6361 6e20 6265 iety that can be │ │ │ │ +00024a00: 2064 6574 6563 7465 6420 746f 2062 6520 detected to be │ │ │ │ +00024a10: 7261 7469 6f6e 616c 0a2a 2a2a 2a2a 2a2a rational.******* │ │ │ │ +00024a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024a30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024a40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024a50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024a60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024a70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024a80: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469 ****.. * Functi │ │ │ │ +00024a90: 6f6e 3a20 2a6e 6f74 6520 7261 6e64 6f6d on: *note random │ │ │ │ +00024aa0: 506f 696e 744f 6e52 6174 696f 6e61 6c56 PointOnRationalV │ │ │ │ +00024ab0: 6172 6965 7479 3a0a 2020 2020 7261 6e64 ariety:. rand │ │ │ │ +00024ac0: 6f6d 506f 696e 744f 6e52 6174 696f 6e61 omPointOnRationa │ │ │ │ +00024ad0: 6c56 6172 6965 7479 5f6c 7049 6465 616c lVariety_lpIdeal │ │ │ │ +00024ae0: 5f72 702c 0a20 202a 2055 7361 6765 3a20 _rp,. * Usage: │ │ │ │ +00024af0: 0a20 2020 2020 2020 2072 616e 646f 6d50 . randomP │ │ │ │ +00024b00: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ +00024b10: 7269 6574 7920 490a 2020 2020 2020 2020 riety I. │ │ │ │ +00024b20: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ +00024b30: 696f 6e61 6c56 6172 6965 7479 0a20 202a ionalVariety. * │ │ │ │ +00024b40: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +00024b50: 2049 2c20 616e 202a 6e6f 7465 2069 6465 I, an *note ide │ │ │ │ +00024b60: 616c 3a20 284d 6163 6175 6c61 7932 446f al: (Macaulay2Do │ │ │ │ +00024b70: 6329 4964 6561 6c2c 2c20 416e 2069 6465 c)Ideal,, An ide │ │ │ │ +00024b80: 616c 2069 6e20 6120 706f 6c79 6e6f 6d69 al in a polynomi │ │ │ │ +00024b90: 616c 2072 696e 670a 2020 2020 2020 2020 al ring. │ │ │ │ +00024ba0: 2453 2420 6f76 6572 2061 2066 6965 6c64 $S$ over a field │ │ │ │ +00024bb0: 2c20 7768 6963 6820 6465 6669 6e65 7320 , which defines │ │ │ │ +00024bc0: 6120 7072 696d 6520 6964 6561 6c0a 2020 a prime ideal. │ │ │ │ +00024bd0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00024be0: 202a 2061 202a 6e6f 7465 206d 6174 7269 * a *note matri │ │ │ │ +00024bf0: 783a 2028 4d61 6361 756c 6179 3244 6f63 x: (Macaulay2Doc │ │ │ │ +00024c00: 294d 6174 7269 782c 2c20 4120 6f6e 6520 )Matrix,, A one │ │ │ │ +00024c10: 726f 7720 6d61 7472 6978 206f 7665 7220 row matrix over │ │ │ │ +00024c20: 7468 6520 6261 7365 0a20 2020 2020 2020 the base. │ │ │ │ +00024c30: 2066 6965 6c64 206f 6620 2453 242c 2072 field of $S$, r │ │ │ │ +00024c40: 6570 7265 7365 6e74 696e 6720 6120 7261 epresenting a ra │ │ │ │ +00024c50: 6e64 6f6d 6c79 2063 686f 7365 6e20 706f ndomly chosen po │ │ │ │ +00024c60: 696e 7420 6f6e 2074 6865 207a 6572 6f20 int on the zero │ │ │ │ +00024c70: 6c6f 6375 7320 6f66 0a20 2020 2020 2020 locus of. │ │ │ │ +00024c80: 2024 4924 2e20 206e 756c 6c20 6973 2072 $I$. null is r │ │ │ │ +00024c90: 6574 7572 6e65 6420 696e 2074 6865 2063 eturned in the c │ │ │ │ +00024ca0: 6173 6520 7768 656e 2074 6865 2072 6f75 ase when the rou │ │ │ │ +00024cb0: 7469 6e65 2063 616e 6e6f 7420 6465 7465 tine cannot dete │ │ │ │ +00024cc0: 726d 696e 6520 6966 0a20 2020 2020 2020 rmine if. │ │ │ │ +00024cd0: 2074 6865 2076 6172 6965 7479 2069 7320 the variety is │ │ │ │ +00024ce0: 7261 7469 6f6e 616c 2061 6e64 2069 7272 rational and irr │ │ │ │ +00024cf0: 6564 7563 6962 6c65 2e0a 0a44 6573 6372 educible...Descr │ │ │ │ +00024d00: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +00024d10: 3d3d 0a0a 4173 2061 2066 6972 7374 2065 ==..As a first e │ │ │ │ +00024d20: 7861 6d70 6c65 2c20 7765 2066 696e 6420 xample, we find │ │ │ │ +00024d30: 6120 7261 6e64 6f6d 2070 6f69 6e74 206f a random point o │ │ │ │ +00024d40: 6e20 7468 6520 5665 726f 6e65 7365 2073 n the Veronese s │ │ │ │ +00024d50: 7572 6661 6365 2069 6e20 245c 5050 5e35 urface in $\PP^5 │ │ │ │ +00024d60: 242e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d $...+----------- │ │ │ │ +00024d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024db0: 2d2d 2b0a 7c69 3120 3a20 6b6b 203d 205a --+.|i1 : kk = Z │ │ │ │ +00024dc0: 5a2f 3130 313b 2020 2020 2020 2020 2020 Z/101; │ │ │ │ +00024dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024e00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00024e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024e50: 2d2d 2b0a 7c69 3220 3a20 5320 3d20 6b6b --+.|i2 : S = kk │ │ │ │ +00024e60: 5b61 2e2e 665d 3b20 2020 2020 2020 2020 [a..f]; │ │ │ │ +00024e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024ea0: 2020 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024ef0: 2d2d 2b0a 7c69 3320 3a20 4920 3d20 6d69 --+.|i3 : I = mi │ │ │ │ +00024f00: 6e6f 7273 2832 2c20 6765 6e65 7269 6353 nors(2, genericS │ │ │ │ +00024f10: 796d 6d65 7472 6963 4d61 7472 6978 2853 ymmetricMatrix(S │ │ │ │ +00024f20: 2c20 3329 2920 2020 2020 2020 2020 2020 , 3)) │ │ │ │ +00024f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024f40: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00024fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024fe0: 2d2d 2b0a 7c69 3320 3a20 4920 3d20 6d69 --+.|i3 : I = mi │ │ │ │ -00024ff0: 6e6f 7273 2832 2c20 6765 6e65 7269 6353 nors(2, genericS │ │ │ │ -00025000: 796d 6d65 7472 6963 4d61 7472 6978 2853 ymmetricMatrix(S │ │ │ │ -00025010: 2c20 3329 2920 2020 2020 2020 2020 2020 , 3)) │ │ │ │ -00025020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025030: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025080: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025090: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00024f90: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024fa0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00024fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024fd0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00024fe0: 2020 7c0a 7c6f 3320 3d20 6964 6561 6c20 |.|o3 = ideal │ │ │ │ +00024ff0: 282d 2062 2020 2b20 612a 642c 202d 2062 (- b + a*d, - b │ │ │ │ +00025000: 2a63 202b 2061 2a65 2c20 2d20 632a 6420 *c + a*e, - c*d │ │ │ │ +00025010: 2b20 622a 652c 202d 2062 2a63 202b 2061 + b*e, - b*c + a │ │ │ │ +00025020: 2a65 2c20 2d20 6320 202b 2061 2a66 2c20 *e, - c + a*f, │ │ │ │ +00025030: 2d20 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d - |.| ------ │ │ │ │ +00025040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025080: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ +00025090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000250a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250c0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -000250d0: 2020 7c0a 7c6f 3320 3d20 6964 6561 6c20 |.|o3 = ideal │ │ │ │ -000250e0: 282d 2062 2020 2b20 612a 642c 202d 2062 (- b + a*d, - b │ │ │ │ -000250f0: 2a63 202b 2061 2a65 2c20 2d20 632a 6420 *c + a*e, - c*d │ │ │ │ -00025100: 2b20 622a 652c 202d 2062 2a63 202b 2061 + b*e, - b*c + a │ │ │ │ -00025110: 2a65 2c20 2d20 6320 202b 2061 2a66 2c20 *e, - c + a*f, │ │ │ │ -00025120: 2d20 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d - |.| ------ │ │ │ │ -00025130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025170: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00025180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000250b0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000250c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000250d0: 2020 7c0a 7c20 2020 2020 632a 6520 2b20 |.| c*e + │ │ │ │ +000250e0: 622a 662c 202d 2063 2a64 202b 2062 2a65 b*f, - c*d + b*e │ │ │ │ +000250f0: 2c20 2d20 632a 6520 2b20 622a 662c 202d , - c*e + b*f, - │ │ │ │ +00025100: 2065 2020 2b20 642a 6629 2020 2020 2020 e + d*f) │ │ │ │ +00025110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025120: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025170: 2020 7c0a 7c6f 3320 3a20 4964 6561 6c20 |.|o3 : Ideal │ │ │ │ +00025180: 6f66 2053 2020 2020 2020 2020 2020 2020 of S │ │ │ │ 00025190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000251a0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000251a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000251b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000251c0: 2020 7c0a 7c20 2020 2020 632a 6520 2b20 |.| c*e + │ │ │ │ -000251d0: 622a 662c 202d 2063 2a64 202b 2062 2a65 b*f, - c*d + b*e │ │ │ │ -000251e0: 2c20 2d20 632a 6520 2b20 622a 662c 202d , - c*e + b*f, - │ │ │ │ -000251f0: 2065 2020 2b20 642a 6629 2020 2020 2020 e + d*f) │ │ │ │ -00025200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025210: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000251c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000251d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000251e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000251f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025210: 2d2d 2b0a 7c69 3420 3a20 7074 203d 2072 --+.|i4 : pt = r │ │ │ │ +00025220: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ +00025230: 6f6e 616c 5661 7269 6574 7920 4920 2020 onalVariety I │ │ │ │ 00025240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025260: 2020 7c0a 7c6f 3320 3a20 4964 6561 6c20 |.|o3 : Ideal │ │ │ │ -00025270: 6f66 2053 2020 2020 2020 2020 2020 2020 of S │ │ │ │ +00025260: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000252b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -000252c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000252d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000252e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000252f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025300: 2d2d 2b0a 7c69 3420 3a20 7074 203d 2072 --+.|i4 : pt = r │ │ │ │ -00025310: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ -00025320: 6f6e 616c 5661 7269 6574 7920 4920 2020 onalVariety I │ │ │ │ +000252b0: 2020 7c0a 7c6f 3420 3d20 7c20 3120 3439 |.|o4 = | 1 49 │ │ │ │ +000252c0: 2032 3420 2d32 3320 2d33 3620 2d33 3020 24 -23 -36 -30 │ │ │ │ +000252d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000252e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000252f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025300: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025350: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025360: 2020 2031 2020 2020 2020 2036 2020 2020 1 6 │ │ │ │ 00025370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000253a0: 2020 7c0a 7c6f 3420 3d20 7c20 3120 3439 |.|o4 = | 1 49 │ │ │ │ -000253b0: 2032 3420 2d32 3320 2d33 3620 2d33 3020 24 -23 -36 -30 │ │ │ │ -000253c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000253a0: 2020 7c0a 7c6f 3420 3a20 4d61 7472 6978 |.|o4 : Matrix │ │ │ │ +000253b0: 206b 6b20 203c 2d2d 206b 6b20 2020 2020 kk <-- kk │ │ │ │ +000253c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000253d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000253e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000253f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025440: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025450: 2020 2031 2020 2020 2020 2036 2020 2020 1 6 │ │ │ │ +000253f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00025400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025440: 2d2d 2b0a 7c69 3520 3a20 7375 6228 492c --+.|i5 : sub(I, │ │ │ │ +00025450: 2070 7429 203d 3d20 3020 2020 2020 2020 pt) == 0 │ │ │ │ 00025460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025490: 2020 7c0a 7c6f 3420 3a20 4d61 7472 6978 |.|o4 : Matrix │ │ │ │ -000254a0: 206b 6b20 203c 2d2d 206b 6b20 2020 2020 kk <-- kk │ │ │ │ +00025490: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000254a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000254e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -000254f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025530: 2d2d 2b0a 7c69 3520 3a20 7375 6228 492c --+.|i5 : sub(I, │ │ │ │ -00025540: 2070 7429 203d 3d20 3020 2020 2020 2020 pt) == 0 │ │ │ │ -00025550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025580: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000255a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000255b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000255c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000255d0: 2020 7c0a 7c6f 3520 3d20 7472 7565 2020 |.|o5 = true │ │ │ │ -000255e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000254e0: 2020 7c0a 7c6f 3520 3d20 7472 7565 2020 |.|o5 = true │ │ │ │ +000254f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025530: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00025540: 2d2d 2d2d 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 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d --+.+----------- │ │ │ │ +00025590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000255a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000255b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000255c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000255d0: 2d2d 2b0a 7c69 3620 3a20 5320 3d20 6b6b --+.|i6 : S = kk │ │ │ │ +000255e0: 5b61 2e2e 645d 3b20 2020 2020 2020 2020 [a..d]; │ │ │ │ 000255f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025620: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00025630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025670: 2d2d 2b0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d --+.+----------- │ │ │ │ -00025680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000256a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000256b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000256c0: 2d2d 2b0a 7c69 3620 3a20 5320 3d20 6b6b --+.|i6 : S = kk │ │ │ │ -000256d0: 5b61 2e2e 645d 3b20 2020 2020 2020 2020 [a..d]; │ │ │ │ +00025670: 2d2d 2b0a 7c69 3720 3a20 4620 3d20 6772 --+.|i7 : F = gr │ │ │ │ +00025680: 6f65 626e 6572 4661 6d69 6c79 2069 6465 oebnerFamily ide │ │ │ │ +00025690: 616c 2261 322c 6162 2c61 632c 6232 2220 al"a2,ab,ac,b2" │ │ │ │ +000256a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000256b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000256c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000256d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000256e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000256f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025710: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00025720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025760: 2d2d 2b0a 7c69 3720 3a20 4620 3d20 6772 --+.|i7 : F = gr │ │ │ │ -00025770: 6f65 626e 6572 4661 6d69 6c79 2069 6465 oebnerFamily ide │ │ │ │ -00025780: 616c 2261 322c 6162 2c61 632c 6232 2220 al"a2,ab,ac,b2" │ │ │ │ -00025790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000257a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025710: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025720: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00025730: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00025740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025750: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00025760: 2020 7c0a 7c6f 3720 3d20 6964 6561 6c20 |.|o7 = ideal │ │ │ │ +00025770: 2861 2020 2b20 7420 622a 6320 2b20 7420 (a + t b*c + t │ │ │ │ +00025780: 612a 6420 2b20 7420 6320 202b 2074 2062 a*d + t c + t b │ │ │ │ +00025790: 2a64 202b 2074 2063 2a64 202b 2074 2064 *d + t c*d + t d │ │ │ │ +000257a0: 202c 2061 2a62 202b 2074 2062 2a63 202b , a*b + t b*c + │ │ │ │ 000257b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -000257c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000257d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000257e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000257f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025800: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025810: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00025820: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00025830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025840: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00025850: 2020 7c0a 7c6f 3720 3d20 6964 6561 6c20 |.|o7 = ideal │ │ │ │ -00025860: 2861 2020 2b20 7420 622a 6320 2b20 7420 (a + t b*c + t │ │ │ │ -00025870: 612a 6420 2b20 7420 6320 202b 2074 2062 a*d + t c + t b │ │ │ │ -00025880: 2a64 202b 2074 2063 2a64 202b 2074 2064 *d + t c*d + t d │ │ │ │ -00025890: 202c 2061 2a62 202b 2074 2062 2a63 202b , a*b + t b*c + │ │ │ │ -000258a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -000258b0: 2020 2020 2020 2031 2020 2020 2020 2033 1 3 │ │ │ │ -000258c0: 2020 2020 2020 2032 2020 2020 2020 3420 2 4 │ │ │ │ -000258d0: 2020 2020 2020 3520 2020 2020 2020 3620 5 6 │ │ │ │ -000258e0: 2020 2020 2020 2020 2020 3720 2020 2020 7 │ │ │ │ -000258f0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00025900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025940: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00025950: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00025960: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00025970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025980: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00025990: 2020 7c0a 7c20 2020 2020 7420 612a 6420 |.| t a*d │ │ │ │ -000259a0: 2b20 7420 6320 202b 2074 2020 622a 6420 + t c + t b*d │ │ │ │ -000259b0: 2b20 7420 2063 2a64 202b 2074 2020 6420 + t c*d + t d │ │ │ │ -000259c0: 2c20 612a 6320 2b20 7420 2062 2a63 202b , a*c + t b*c + │ │ │ │ -000259d0: 2074 2020 612a 6420 2b20 7420 2063 2020 t a*d + t c │ │ │ │ -000259e0: 2b20 7c0a 7c20 2020 2020 2039 2020 2020 + |.| 9 │ │ │ │ -000259f0: 2020 2038 2020 2020 2020 3130 2020 2020 8 10 │ │ │ │ -00025a00: 2020 2031 3120 2020 2020 2020 3132 2020 11 12 │ │ │ │ -00025a10: 2020 2020 2020 2020 2031 3320 2020 2020 13 │ │ │ │ -00025a20: 2020 3135 2020 2020 2020 2031 3420 2020 15 14 │ │ │ │ -00025a30: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00025a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025a80: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00025a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025aa0: 3220 2020 3220 2020 2020 2020 2020 2020 2 2 │ │ │ │ -00025ab0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00025ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025ad0: 2020 7c0a 7c20 2020 2020 7420 2062 2a64 |.| t b*d │ │ │ │ -00025ae0: 202b 2074 2020 632a 6420 2b20 7420 2064 + t c*d + t d │ │ │ │ -00025af0: 202c 2062 2020 2b20 7420 2062 2a63 202b , b + t b*c + │ │ │ │ -00025b00: 2074 2020 612a 6420 2b20 7420 2063 2020 t a*d + t c │ │ │ │ -00025b10: 2b20 7420 2062 2a64 202b 2074 2020 632a + t b*d + t c* │ │ │ │ -00025b20: 6420 7c0a 7c20 2020 2020 2031 3620 2020 d |.| 16 │ │ │ │ -00025b30: 2020 2020 3137 2020 2020 2020 2031 3820 17 18 │ │ │ │ -00025b40: 2020 2020 2020 2020 2031 3920 2020 2020 19 │ │ │ │ -00025b50: 2020 3231 2020 2020 2020 2032 3020 2020 21 20 │ │ │ │ -00025b60: 2020 2032 3220 2020 2020 2020 3233 2020 22 23 │ │ │ │ -00025b70: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00025b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025bc0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00025bd0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000257c0: 2020 2020 2020 2031 2020 2020 2020 2033 1 3 │ │ │ │ +000257d0: 2020 2020 2020 2032 2020 2020 2020 3420 2 4 │ │ │ │ +000257e0: 2020 2020 2020 3520 2020 2020 2020 3620 5 6 │ │ │ │ +000257f0: 2020 2020 2020 2020 2020 3720 2020 2020 7 │ │ │ │ +00025800: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00025810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025850: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ +00025860: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00025870: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +00025880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025890: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +000258a0: 2020 7c0a 7c20 2020 2020 7420 612a 6420 |.| t a*d │ │ │ │ +000258b0: 2b20 7420 6320 202b 2074 2020 622a 6420 + t c + t b*d │ │ │ │ +000258c0: 2b20 7420 2063 2a64 202b 2074 2020 6420 + t c*d + t d │ │ │ │ +000258d0: 2c20 612a 6320 2b20 7420 2062 2a63 202b , a*c + t b*c + │ │ │ │ +000258e0: 2074 2020 612a 6420 2b20 7420 2063 2020 t a*d + t c │ │ │ │ +000258f0: 2b20 7c0a 7c20 2020 2020 2039 2020 2020 + |.| 9 │ │ │ │ +00025900: 2020 2038 2020 2020 2020 3130 2020 2020 8 10 │ │ │ │ +00025910: 2020 2031 3120 2020 2020 2020 3132 2020 11 12 │ │ │ │ +00025920: 2020 2020 2020 2020 2031 3320 2020 2020 13 │ │ │ │ +00025930: 2020 3135 2020 2020 2020 2031 3420 2020 15 14 │ │ │ │ +00025940: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00025950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025990: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ +000259a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000259b0: 3220 2020 3220 2020 2020 2020 2020 2020 2 2 │ │ │ │ +000259c0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +000259d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000259e0: 2020 7c0a 7c20 2020 2020 7420 2062 2a64 |.| t b*d │ │ │ │ +000259f0: 202b 2074 2020 632a 6420 2b20 7420 2064 + t c*d + t d │ │ │ │ +00025a00: 202c 2062 2020 2b20 7420 2062 2a63 202b , b + t b*c + │ │ │ │ +00025a10: 2074 2020 612a 6420 2b20 7420 2063 2020 t a*d + t c │ │ │ │ +00025a20: 2b20 7420 2062 2a64 202b 2074 2020 632a + t b*d + t c* │ │ │ │ +00025a30: 6420 7c0a 7c20 2020 2020 2031 3620 2020 d |.| 16 │ │ │ │ +00025a40: 2020 2020 3137 2020 2020 2020 2031 3820 17 18 │ │ │ │ +00025a50: 2020 2020 2020 2020 2031 3920 2020 2020 19 │ │ │ │ +00025a60: 2020 3231 2020 2020 2020 2032 3020 2020 21 20 │ │ │ │ +00025a70: 2020 2032 3220 2020 2020 2020 3233 2020 22 23 │ │ │ │ +00025a80: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00025a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025ad0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ +00025ae0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00025af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b20: 2020 7c0a 7c20 2020 2020 2b20 7420 2064 |.| + t d │ │ │ │ +00025b30: 2029 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00025b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b70: 2020 7c0a 7c20 2020 2020 2020 2032 3420 |.| 24 │ │ │ │ +00025b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025bc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c10: 2020 7c0a 7c20 2020 2020 2b20 7420 2064 |.| + t d │ │ │ │ -00025c20: 2029 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -00025c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c60: 2020 7c0a 7c20 2020 2020 2020 2032 3420 |.| 24 │ │ │ │ -00025c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025cb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d00: 2020 7c0a 7c6f 3720 3a20 4964 6561 6c20 |.|o7 : Ideal │ │ │ │ -00025d10: 6f66 206b 6b5b 7420 2c20 7420 2c20 7420 of kk[t , t , t │ │ │ │ -00025d20: 202c 2074 202c 2074 202c 2074 2020 2c20 , t , t , t , │ │ │ │ -00025d30: 7420 202c 2074 2020 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ -00025d40: 2c20 7420 2c20 7420 202c 2074 2020 2c20 , t , t , t , │ │ │ │ -00025d50: 7420 7c0a 7c20 2020 2020 2020 2020 2020 t |.| │ │ │ │ -00025d60: 2020 2020 2020 2036 2020 2035 2020 2031 6 5 1 │ │ │ │ -00025d70: 3220 2020 3220 2020 3420 2020 3131 2020 2 2 4 11 │ │ │ │ -00025d80: 2031 3820 2020 3234 2020 2031 2020 2033 18 24 1 3 │ │ │ │ -00025d90: 2020 2038 2020 2031 3020 2020 3137 2020 8 10 17 │ │ │ │ -00025da0: 2032 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2|.|----------- │ │ │ │ +00025c10: 2020 7c0a 7c6f 3720 3a20 4964 6561 6c20 |.|o7 : Ideal │ │ │ │ +00025c20: 6f66 206b 6b5b 7420 2c20 7420 2c20 7420 of kk[t , t , t │ │ │ │ +00025c30: 202c 2074 202c 2074 202c 2074 2020 2c20 , t , t , t , │ │ │ │ +00025c40: 7420 202c 2074 2020 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ +00025c50: 2c20 7420 2c20 7420 202c 2074 2020 2c20 , t , t , t , │ │ │ │ +00025c60: 7420 7c0a 7c20 2020 2020 2020 2020 2020 t |.| │ │ │ │ +00025c70: 2020 2020 2020 2036 2020 2035 2020 2031 6 5 1 │ │ │ │ +00025c80: 3220 2020 3220 2020 3420 2020 3131 2020 2 2 4 11 │ │ │ │ +00025c90: 2031 3820 2020 3234 2020 2031 2020 2033 18 24 1 3 │ │ │ │ +00025ca0: 2020 2038 2020 2031 3020 2020 3137 2020 8 10 17 │ │ │ │ +00025cb0: 2032 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2|.|----------- │ │ │ │ +00025cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025d00: 2d2d 7c0a 7c20 2c20 7420 2c20 7420 2c20 --|.| , t , t , │ │ │ │ +00025d10: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00025d20: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ +00025d30: 202c 2074 2020 5d5b 612e 2e64 5d20 2020 , t ][a..d] │ │ │ │ +00025d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025d50: 2020 7c0a 7c33 2020 2037 2020 2039 2020 |.|3 7 9 │ │ │ │ +00025d60: 2031 3420 2020 3136 2020 2032 3020 2020 14 16 20 │ │ │ │ +00025d70: 3232 2020 2031 3320 2020 3135 2020 2031 22 13 15 1 │ │ │ │ +00025d80: 3920 2020 3231 2020 2020 2020 2020 2020 9 21 │ │ │ │ +00025d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025da0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00025db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025df0: 2d2d 7c0a 7c20 2c20 7420 2c20 7420 2c20 --|.| , t , t , │ │ │ │ -00025e00: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00025e10: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ -00025e20: 202c 2074 2020 5d5b 612e 2e64 5d20 2020 , t ][a..d] │ │ │ │ +00025df0: 2d2d 2b0a 7c69 3820 3a20 4a20 3d20 6772 --+.|i8 : J = gr │ │ │ │ +00025e00: 6f65 626e 6572 5374 7261 7475 6d20 4620 oebnerStratum F │ │ │ │ +00025e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025e40: 2020 7c0a 7c33 2020 2037 2020 2039 2020 |.|3 7 9 │ │ │ │ -00025e50: 2031 3420 2020 3136 2020 2032 3020 2020 14 16 20 │ │ │ │ -00025e60: 3232 2020 2031 3320 2020 3135 2020 2031 22 13 15 1 │ │ │ │ -00025e70: 3920 2020 3231 2020 2020 2020 2020 2020 9 21 │ │ │ │ +00025e40: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025e90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00025ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025ee0: 2d2d 2b0a 7c69 3820 3a20 4a20 3d20 6772 --+.|i8 : J = gr │ │ │ │ -00025ef0: 6f65 626e 6572 5374 7261 7475 6d20 4620 oebnerStratum F │ │ │ │ -00025f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025e90: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ee0: 2020 7c0a 7c6f 3820 3d20 6964 6561 6c20 |.|o8 = ideal │ │ │ │ +00025ef0: 282d 2074 2020 2b20 7420 2020 2d20 7420 (- t + t - t │ │ │ │ +00025f00: 2074 2020 2c20 2d20 7420 202d 2074 2020 t , - t - t │ │ │ │ +00025f10: 7420 202c 202d 2074 2020 202b 2074 2020 t , - t + t │ │ │ │ +00025f20: 202b 2074 2074 2020 202d 2074 2020 7420 + t t - t t │ │ │ │ 00025f30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025f80: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025fd0: 2020 7c0a 7c6f 3820 3d20 6964 6561 6c20 |.|o8 = ideal │ │ │ │ -00025fe0: 282d 2074 2020 2b20 7420 2020 2d20 7420 (- t + t - t │ │ │ │ -00025ff0: 2074 2020 2c20 2d20 7420 202d 2074 2020 t , - t - t │ │ │ │ -00026000: 7420 202c 202d 2074 2020 202b 2074 2020 t , - t + t │ │ │ │ -00026010: 202b 2074 2074 2020 202d 2074 2020 7420 + t t - t t │ │ │ │ -00026020: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00026030: 2020 2020 3720 2020 2031 3420 2020 2031 7 14 1 │ │ │ │ -00026040: 3320 3139 2020 2020 2038 2020 2020 3230 3 19 8 20 │ │ │ │ -00026050: 2031 3320 2020 2020 3130 2020 2020 3137 13 10 17 │ │ │ │ -00026060: 2020 2020 3920 3133 2020 2020 3232 2031 9 13 22 1 │ │ │ │ -00026070: 3320 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 3 |.| ------ │ │ │ │ -00026080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000260a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000260b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000260c0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -000260d0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -000260e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000260f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026110: 2020 7c0a 7c20 2020 2020 2d20 7420 7420 |.| - t t │ │ │ │ -00026120: 2020 2d20 7420 2074 2020 202b 2074 2020 - t t + t │ │ │ │ -00026130: 7420 202c 202d 2074 2020 7420 2020 2b20 t , - t t + │ │ │ │ -00026140: 7420 2074 2020 7420 202c 202d 2074 2020 t t t , - t │ │ │ │ -00026150: 202b 2074 2074 2020 202d 2074 2020 7420 + t t - t t │ │ │ │ -00026160: 2020 7c0a 7c20 2020 2020 2020 2037 2031 |.| 7 1 │ │ │ │ -00026170: 3520 2020 2031 3620 3139 2020 2020 3133 5 16 19 13 │ │ │ │ -00026180: 2032 3120 2020 2020 3136 2032 3120 2020 21 16 21 │ │ │ │ -00026190: 2031 3320 3135 2032 3120 2020 2020 3131 13 15 21 11 │ │ │ │ -000261a0: 2020 2020 3920 3134 2020 2020 3136 2032 9 14 16 2 │ │ │ │ -000261b0: 3020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 0 |.| ------ │ │ │ │ -000261c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000261d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000261e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000261f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026200: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00026210: 2020 2020 2020 2020 2020 2020 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 7c0a 7c20 2020 2020 2d20 7420 2074 |.| - t t │ │ │ │ -00026260: 2020 202d 2074 2074 2020 202b 2074 2020 - t t + t │ │ │ │ -00026270: 7420 2074 2020 2c20 7420 2020 2b20 7420 t t , t + t │ │ │ │ -00026280: 7420 2020 2d20 7420 2074 2020 202d 2074 t - t t - t │ │ │ │ -00026290: 2020 7420 2020 2b20 7420 2074 2020 7420 t + t t t │ │ │ │ -000262a0: 202c 7c0a 7c20 2020 2020 2020 2032 3320 ,|.| 23 │ │ │ │ -000262b0: 3133 2020 2020 3820 3135 2020 2020 3134 13 8 15 14 │ │ │ │ -000262c0: 2031 3320 3231 2020 2031 3820 2020 2039 13 21 18 9 │ │ │ │ -000262d0: 2031 3620 2020 2031 3620 3232 2020 2020 16 16 22 │ │ │ │ -000262e0: 3130 2031 3520 2020 2031 3620 3133 2032 10 15 16 13 2 │ │ │ │ -000262f0: 3120 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 1 |.| ------ │ │ │ │ -00026300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026340: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00026350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026390: 2020 7c0a 7c20 2020 2020 2d20 7420 2020 |.| - t │ │ │ │ -000263a0: 2b20 7420 2074 2020 2d20 7420 2074 2020 + t t - t t │ │ │ │ -000263b0: 202d 2074 2020 7420 2020 2d20 7420 2074 - t t - t t │ │ │ │ -000263c0: 2020 202b 2074 2020 7420 2074 2020 2c20 + t t t , │ │ │ │ -000263d0: 7420 2074 2020 2d20 7420 2074 2020 202d t t - t t - │ │ │ │ -000263e0: 2020 7c0a 7c20 2020 2020 2020 2031 3220 |.| 12 │ │ │ │ -000263f0: 2020 2031 3720 3920 2020 2032 3320 3136 17 9 23 16 │ │ │ │ -00026400: 2020 2020 3234 2031 3320 2020 2031 3120 24 13 11 │ │ │ │ -00026410: 3135 2020 2020 3137 2031 3320 3231 2020 15 17 13 21 │ │ │ │ -00026420: 2031 3820 3920 2020 2032 3420 3136 2020 18 9 24 16 │ │ │ │ -00026430: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00026440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026480: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00026490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000264a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000264b0: 3220 2020 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ -000264c0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -000264d0: 2020 7c0a 7c20 2020 2020 7420 2074 2020 |.| t t │ │ │ │ -000264e0: 202b 2074 2020 7420 2074 2020 2c20 2d20 + t t t , - │ │ │ │ -000264f0: 7420 202d 2032 7420 2074 2020 202b 2074 t - 2t t + t │ │ │ │ -00026500: 2020 7420 202c 202d 2074 2020 2d20 7420 t , - t - t │ │ │ │ -00026510: 2020 2b20 7420 2074 2020 2c20 2d20 7420 + t t , - t │ │ │ │ -00026520: 202d 7c0a 7c20 2020 2020 2031 3220 3135 -|.| 12 15 │ │ │ │ -00026530: 2020 2020 3138 2031 3320 3231 2020 2020 18 13 21 │ │ │ │ -00026540: 2031 2020 2020 2031 3420 3133 2020 2020 1 14 13 │ │ │ │ -00026550: 3133 2031 3920 2020 2020 3220 2020 2031 13 19 2 1 │ │ │ │ -00026560: 3420 2020 2032 3020 3133 2020 2020 2034 4 20 13 4 │ │ │ │ -00026570: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00026580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265c0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -000265d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000265e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000265f0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00026600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026610: 2020 7c0a 7c20 2020 2020 7420 7420 2020 |.| t t │ │ │ │ -00026620: 2d20 7420 2074 2020 202b 2074 2074 2020 - t t + t t │ │ │ │ -00026630: 202d 2032 7420 2074 2020 202b 2074 2020 - 2t t + t │ │ │ │ -00026640: 7420 2020 2d20 7420 7420 2020 2b20 7420 t - t t + t │ │ │ │ -00026650: 7420 2074 2020 202b 2074 2020 7420 2074 t t + t t t │ │ │ │ -00026660: 2020 7c0a 7c20 2020 2020 2037 2031 3620 |.| 7 16 │ │ │ │ -00026670: 2020 2031 3420 3136 2020 2020 3320 3133 14 16 3 13 │ │ │ │ -00026680: 2020 2020 2031 3720 3133 2020 2020 3232 17 13 22 │ │ │ │ -00026690: 2031 3320 2020 2031 2031 3520 2020 2037 13 1 15 7 │ │ │ │ -000266a0: 2031 3320 3135 2020 2020 3134 2031 3320 13 15 14 13 │ │ │ │ -000266b0: 3135 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 15|.| ------ │ │ │ │ -000266c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000266d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000266e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000266f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026700: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00026710: 2020 2020 2020 2020 2033 2020 2020 2020 3 │ │ │ │ -00026720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026740: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00026750: 2020 7c0a 7c20 2020 2020 2b20 7420 2074 |.| + t t │ │ │ │ -00026760: 2020 7420 2020 2d20 7420 2074 2020 2c20 t - t t , │ │ │ │ -00026770: 7420 2020 2d20 7420 7420 2020 2d20 7420 t - t t - t │ │ │ │ -00026780: 2074 2020 202b 2074 2074 2020 7420 2020 t + t t t │ │ │ │ -00026790: 2b20 7420 2074 2020 202b 2020 2020 2020 + t t + │ │ │ │ -000267a0: 2020 7c0a 7c20 2020 2020 2020 2031 3620 |.| 16 │ │ │ │ -000267b0: 3133 2031 3920 2020 2031 3320 3231 2020 13 19 13 21 │ │ │ │ -000267c0: 2031 3820 2020 2039 2031 3620 2020 2031 18 9 16 1 │ │ │ │ -000267d0: 3720 3135 2020 2020 3920 3133 2031 3520 7 15 9 13 15 │ │ │ │ -000267e0: 2020 2031 3420 3135 2020 2020 2020 2020 14 15 │ │ │ │ -000267f0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00026800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026840: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00026850: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00026860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026890: 2020 7c0a 7c20 2020 2020 7420 2074 2020 |.| t t │ │ │ │ -000268a0: 7420 2020 2d20 7420 2074 2020 7420 202c t - t t t , │ │ │ │ -000268b0: 202d 2074 2020 2b20 7420 7420 2020 2d20 - t + t t - │ │ │ │ -000268c0: 3274 2020 7420 2020 2d20 7420 7420 2020 2t t - t t │ │ │ │ -000268d0: 2b20 7420 2074 2020 7420 2020 2b20 2020 + t t t + │ │ │ │ -000268e0: 2020 7c0a 7c20 2020 2020 2031 3620 3133 |.| 16 13 │ │ │ │ -000268f0: 2032 3120 2020 2031 3320 3135 2032 3120 21 13 15 21 │ │ │ │ -00026900: 2020 2020 3520 2020 2033 2031 3420 2020 5 3 14 │ │ │ │ -00026910: 2020 3137 2031 3420 2020 2038 2031 3620 17 14 8 16 │ │ │ │ -00026920: 2020 2031 3620 3230 2031 3320 2020 2020 16 20 13 │ │ │ │ -00026930: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00026940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026960: 2d2d 2d2d 2d2d 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3134 2031 3320 3231 2020 2033 2031 3620 14 13 21 3 16 │ │ │ │ -00026a60: 2020 2031 3020 3136 2020 2020 3137 2031 10 16 17 1 │ │ │ │ -00026a70: 3620 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 6 |.| ------ │ │ │ │ -00026a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026ac0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00026ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b00: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00026b10: 2020 7c0a 7c20 2020 2020 2d20 7420 2074 |.| - t t │ │ │ │ -00026b20: 2020 202b 2074 2020 7420 2074 2020 202d + t t t - │ │ │ │ -00026b30: 2074 2074 2020 202b 2074 2020 7420 2074 t t + t t t │ │ │ │ -00026b40: 2020 202b 2074 2020 7420 2074 2020 202d + t t t - │ │ │ │ -00026b50: 2074 2020 7420 2074 2020 2c20 2d20 7420 t t t , - t │ │ │ │ -00026b60: 202b 7c0a 7c20 2020 2020 2020 2031 3820 +|.| 18 │ │ │ │ -00026b70: 3133 2020 2020 3136 2032 3220 3133 2020 13 16 22 13 │ │ │ │ -00026b80: 2020 3420 3135 2020 2020 3134 2031 3620 4 15 14 16 │ │ │ │ -00026b90: 3135 2020 2020 3130 2031 3320 3135 2020 15 10 13 15 │ │ │ │ -00026ba0: 2020 3136 2031 3320 3231 2020 2020 2036 16 13 21 6 │ │ │ │ -00026bb0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00026bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026c00: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00026c10: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00026c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c30: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -00026c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c50: 2020 7c0a 7c20 2020 2020 7420 7420 2020 |.| t t │ │ │ │ -00026c60: 2d20 7420 2020 2d20 7420 2074 2020 202d - t - t t - │ │ │ │ -00026c70: 2074 2020 7420 2020 2b20 7420 2074 2020 t t + t t │ │ │ │ -00026c80: 7420 2020 2b20 7420 2074 2020 202d 2074 t + t t - t │ │ │ │ -00026c90: 2074 2020 202b 2074 2020 7420 2074 2020 t + t t t │ │ │ │ -00026ca0: 202b 7c0a 7c20 2020 2020 2033 2031 3720 +|.| 3 17 │ │ │ │ -00026cb0: 2020 2031 3720 2020 2031 3820 3134 2020 17 18 14 │ │ │ │ -00026cc0: 2020 3131 2031 3620 2020 2032 3320 3136 11 16 23 16 │ │ │ │ -00026cd0: 2031 3320 2020 2032 3420 3133 2020 2020 13 24 13 │ │ │ │ -00026ce0: 3520 3135 2020 2020 3137 2031 3420 3135 5 15 17 14 15 │ │ │ │ -00026cf0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00026d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026d40: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00026d50: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -00026d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026d90: 2020 7c0a 7c20 2020 2020 7420 2074 2020 |.| t t │ │ │ │ -00026da0: 7420 2020 2d20 7420 2074 2020 7420 202c t - t t t , │ │ │ │ -00026db0: 2074 2020 7420 202d 2074 2020 7420 2020 t t - t t │ │ │ │ -00026dc0: 2d20 7420 2074 2020 202b 2074 2020 7420 - t t + t t │ │ │ │ -00026dd0: 2074 2020 202d 2074 2074 2020 202b 2020 t - t t + │ │ │ │ -00026de0: 2020 7c0a 7c20 2020 2020 2031 3120 3133 |.| 11 13 │ │ │ │ -00026df0: 2031 3520 2020 2031 3720 3133 2032 3120 15 17 13 21 │ │ │ │ -00026e00: 2020 3138 2033 2020 2020 3138 2031 3720 18 3 18 17 │ │ │ │ -00026e10: 2020 2031 3220 3136 2020 2020 3234 2031 12 16 24 1 │ │ │ │ -00026e20: 3620 3133 2020 2020 3620 3135 2020 2020 6 13 6 15 │ │ │ │ -00026e30: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00026e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026e80: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00026e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ea0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00026eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ed0: 2020 7c0a 7c20 2020 2020 7420 2074 2020 |.| t t │ │ │ │ -00026ee0: 7420 2020 2b20 7420 2074 2020 7420 2020 t + t t t │ │ │ │ -00026ef0: 2d20 7420 2074 2020 7420 202c 202d 2074 - t t t , - t │ │ │ │ -00026f00: 2020 2d20 7420 2074 2020 202b 2074 2074 - t t + t t │ │ │ │ -00026f10: 2020 202d 2074 2020 7420 2020 2b20 2020 - t t + │ │ │ │ -00026f20: 2020 7c0a 7c20 2020 2020 2031 3820 3134 |.| 18 14 │ │ │ │ -00026f30: 2031 3520 2020 2031 3220 3133 2031 3520 15 12 13 15 │ │ │ │ -00026f40: 2020 2031 3820 3133 2032 3120 2020 2020 18 13 21 │ │ │ │ -00026f50: 3820 2020 2032 3020 3133 2020 2020 3720 8 20 13 7 │ │ │ │ -00026f60: 3139 2020 2020 3134 2031 3920 2020 2020 19 14 19 │ │ │ │ -00026f70: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00026f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026fc0: 2d2d 7c0a 7c20 2020 2020 2020 2020 3220 --|.| 2 │ │ │ │ -00026fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027010: 2020 7c0a 7c20 2020 2020 7420 2074 2020 |.| t t │ │ │ │ -00027020: 2c20 7420 7420 2020 2d20 7420 2074 2020 , t t - t t │ │ │ │ -00027030: 202b 2074 2020 7420 2074 2020 2c20 2d20 + t t t , - │ │ │ │ -00027040: 7420 2020 2b20 7420 7420 202d 2074 2020 t + t t - t │ │ │ │ -00027050: 7420 2020 2d20 7420 2074 2020 202b 2020 t - t t + │ │ │ │ -00027060: 2020 7c0a 7c20 2020 2020 2031 3320 3139 |.| 13 19 │ │ │ │ -00027070: 2020 2037 2032 3020 2020 2031 3420 3230 7 20 14 20 │ │ │ │ -00027080: 2020 2020 3230 2031 3320 3139 2020 2020 20 13 19 │ │ │ │ -00027090: 2031 3120 2020 2037 2039 2020 2020 3136 11 7 9 16 │ │ │ │ -000270a0: 2032 3020 2020 2032 3320 3133 2020 2020 20 23 13 │ │ │ │ -000270b0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -000270c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000270d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000270e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000270f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027100: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00027110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027140: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00027150: 2020 7c0a 7c20 2020 2020 7420 2074 2020 |.| t t │ │ │ │ -00027160: 7420 2020 2b20 7420 2074 2020 202d 2074 t + t t - t │ │ │ │ -00027170: 2020 7420 2020 2b20 7420 2074 2020 7420 t + t t t │ │ │ │ -00027180: 2020 2b20 7420 7420 2074 2020 202b 2074 + t t t + t │ │ │ │ -00027190: 2020 7420 2020 2d20 7420 7420 2020 2d20 t - t t - │ │ │ │ -000271a0: 2020 7c0a 7c20 2020 2020 2032 3020 3133 |.| 20 13 │ │ │ │ -000271b0: 2031 3520 2020 2031 3020 3139 2020 2020 15 10 19 │ │ │ │ -000271c0: 3137 2031 3920 2020 2032 3220 3133 2031 17 19 22 13 1 │ │ │ │ -000271d0: 3920 2020 2037 2031 3520 3139 2020 2020 9 7 15 19 │ │ │ │ -000271e0: 3136 2031 3920 2020 2031 2032 3120 2020 16 19 1 21 │ │ │ │ -000271f0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00027200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027240: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00027250: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00027260: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00027270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027280: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00027290: 2020 7c0a 7c20 2020 2020 7420 7420 2074 |.| t t t │ │ │ │ -000272a0: 2020 202d 2074 2020 7420 2074 2020 2c20 - t t t , │ │ │ │ -000272b0: 7420 2020 2b20 7420 202d 2074 2074 2020 t + t - t t │ │ │ │ -000272c0: 202d 2074 2020 7420 2020 2b20 7420 2074 - t t + t t │ │ │ │ -000272d0: 2020 202b 2074 2074 2020 7420 2020 2d20 + t t t - │ │ │ │ -000272e0: 2020 7c0a 7c20 2020 2020 2037 2031 3320 |.| 7 13 │ │ │ │ -000272f0: 3231 2020 2020 3133 2031 3920 3231 2020 21 13 19 21 │ │ │ │ -00027300: 2032 3420 2020 2039 2020 2020 3920 3232 24 9 9 22 │ │ │ │ -00027310: 2020 2020 3233 2031 3520 2020 2032 3020 23 15 20 │ │ │ │ -00027320: 3135 2020 2020 3920 3135 2031 3920 2020 15 9 15 19 │ │ │ │ -00027330: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00027340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027380: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00027390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000273a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000273b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000273c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000273d0: 2020 7c0a 7c20 2020 2020 7420 7420 2020 |.| t t │ │ │ │ -000273e0: 2b20 7420 2074 2020 202d 2074 2074 2020 + t t - t t │ │ │ │ -000273f0: 7420 2020 2b20 7420 2074 2020 7420 2020 t + t t t │ │ │ │ -00027400: 2d20 7420 2074 2020 7420 2074 2020 2c20 - t t t t , │ │ │ │ -00027410: 7420 2074 2020 2b20 7420 7420 202d 2020 t t + t t - │ │ │ │ -00027420: 2020 7c0a 7c20 2020 2020 2033 2032 3120 |.| 3 21 │ │ │ │ -00027430: 2020 2031 3020 3231 2020 2020 3720 3135 10 21 7 15 │ │ │ │ -00027440: 2032 3120 2020 2031 3620 3139 2032 3120 21 16 19 21 │ │ │ │ -00027450: 2020 2031 3320 3135 2031 3920 3231 2020 13 15 19 21 │ │ │ │ -00027460: 2032 3320 3720 2020 2038 2039 2020 2020 23 7 8 9 │ │ │ │ -00027470: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00027480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000274a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000274b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000274c0: 2d2d 7c0a 7c20 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 2020 2020 2020 │ │ │ │ -00027500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027510: 2020 7c0a 7c20 2020 2020 7420 2074 2020 |.| t t │ │ │ │ -00027520: 202b 2074 2020 7420 2020 2d20 7420 2074 + t t - t t │ │ │ │ -00027530: 2020 202d 2074 2074 2020 202b 2074 2020 - t t + t │ │ │ │ -00027540: 7420 2074 2020 202b 2074 2020 7420 2074 t t + t t t │ │ │ │ -00027550: 2020 202b 2074 2020 7420 2074 2020 202b + t t t + │ │ │ │ -00027560: 2020 7c0a 7c20 2020 2020 2032 3320 3134 |.| 23 14 │ │ │ │ -00027570: 2020 2020 3130 2032 3020 2020 2031 3720 10 20 17 │ │ │ │ -00027580: 3230 2020 2020 3820 3232 2020 2020 3134 20 8 22 14 │ │ │ │ -00027590: 2032 3020 3135 2020 2020 3136 2032 3020 20 15 16 20 │ │ │ │ -000275a0: 3139 2020 2020 3233 2031 3320 3139 2020 19 23 13 19 │ │ │ │ -000275b0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -000275c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000275d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000275e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000275f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027600: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00027610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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---------------- │ │ │ │ -00027710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027740: 2d2d 7c0a 7c20 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 2020 2020 2020 │ │ │ │ -00027780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027790: 2020 7c0a 7c20 2020 2020 7420 2074 2020 |.| t t │ │ │ │ -000277a0: 7420 2020 2d20 7420 2074 2020 202b 2074 t - t t + t │ │ │ │ -000277b0: 2020 7420 2074 2020 202b 2074 2020 7420 t t + t t │ │ │ │ -000277c0: 2074 2020 202d 2074 2074 2020 202d 2074 t - t t - t │ │ │ │ -000277d0: 2074 2020 7420 2020 2d20 2020 2020 2020 t t - │ │ │ │ -000277e0: 2020 7c0a 7c20 2020 2020 2031 3620 3230 |.| 16 20 │ │ │ │ -000277f0: 2031 3520 2020 2031 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7420 2074 2020 2c20 7420 2074 2020 202d t t , t t - │ │ │ │ -000278f0: 2074 2020 7420 2020 2b20 7420 2074 2020 t t + t t │ │ │ │ -00027900: 2b20 7420 2074 2020 2d20 7420 2074 2020 + t t - t t │ │ │ │ -00027910: 202d 2074 2020 7420 2020 2b20 2020 2020 - t t + │ │ │ │ -00027920: 2020 7c0a 7c20 2020 2020 2031 3620 3133 |.| 16 13 │ │ │ │ -00027930: 2031 3920 3231 2020 2031 3020 3233 2020 19 21 10 23 │ │ │ │ -00027940: 2020 3137 2032 3320 2020 2032 3420 3720 17 23 24 7 │ │ │ │ -00027950: 2020 2031 3120 3920 2020 2031 3820 3230 11 9 18 20 │ │ │ │ -00027960: 2020 2020 3131 2032 3220 2020 2020 2020 11 22 │ │ │ │ -00027970: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000279b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000279c0: 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│ │ │ -00027ba0: 202b 7c0a 7c20 2020 2020 2031 3720 3133 +|.| 17 13 │ │ │ │ -00027bb0: 2031 3920 3231 2020 2032 3420 3130 2020 19 21 24 10 │ │ │ │ -00027bc0: 2020 3138 2032 3320 2020 2031 3220 3920 18 23 12 9 │ │ │ │ -00027bd0: 2020 2031 3220 3232 2020 2020 3138 2032 12 22 18 2 │ │ │ │ -00027be0: 3020 3135 2020 2020 3234 2031 3620 3139 0 15 24 16 19 │ │ │ │ -00027bf0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ -00027c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ -00027c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027c80: 2020 2020 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│ │ +00026200: 3120 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 1 |.| ------ │ │ │ │ +00026210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026250: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ +00026260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 7c0a 7c20 2020 2020 2d20 7420 2020 |.| - t │ │ │ │ +000262b0: 2b20 7420 2074 2020 2d20 7420 2074 2020 + t t - t t │ │ │ │ +000262c0: 202d 2074 2020 7420 2020 2d20 7420 2074 - t t - t t │ │ │ │ +000262d0: 2020 202b 2074 2020 7420 2074 2020 2c20 + t t t , │ │ │ │ +000262e0: 7420 2074 2020 2d20 7420 2074 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│ │ │ +00028000: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00028010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028050: 2d2d 7c0a 7c20 2020 2020 2020 2032 2020 --|.| 2 │ │ │ │ +00028060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000280a0: 2020 7c0a 7c20 2020 2020 7420 7420 2020 |.| t t │ │ │ │ +000280b0: 2b20 7420 7420 2020 2b20 7420 7420 2074 + t t + t t t │ │ │ │ +000280c0: 2020 202d 2074 2074 2020 7420 2020 2d20 - t t t - │ │ │ │ +000280d0: 7420 7420 2074 2020 7420 202c 2074 2074 t t t t , t t │ │ │ │ +000280e0: 2020 2d20 7420 7420 2020 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2020 2020 3420 3232 2020 2020 3720 3136 4 22 7 16 │ │ │ │ -00028570: 2032 3220 2020 2031 3020 3720 3135 2020 22 10 7 15 │ │ │ │ -00028580: 2020 3820 3136 2031 3520 2020 2031 2031 8 16 15 1 1 │ │ │ │ -00028590: 3620 3231 2020 2020 2020 2020 2020 2020 6 21 │ │ │ │ +00028500: 2d2d 7c0a 7c20 2020 2020 7420 7420 2074 --|.| t t t │ │ │ │ +00028510: 2020 7420 202c 2074 2020 7420 202b 2074 t , t t + t │ │ │ │ +00028520: 2020 7420 202d 2074 2020 7420 202d 2074 t - t t - t │ │ │ │ +00028530: 2020 7420 2020 2d20 7420 2074 2020 202b t - t t + │ │ │ │ +00028540: 2074 2074 2020 202d 2074 2074 2020 2b20 t t - t t + │ │ │ │ +00028550: 2020 7c0a 7c20 2020 2020 2037 2031 3620 |.| 7 16 │ │ │ │ +00028560: 3133 2032 3120 2020 3234 2031 2020 2020 13 21 24 1 │ │ │ │ +00028570: 3131 2033 2020 2020 3138 2038 2020 2020 11 3 18 8 │ │ │ │ +00028580: 3131 2031 3020 2020 2031 3120 3137 2020 11 10 11 17 │ │ │ │ +00028590: 2020 3420 3233 2020 2020 3520 3920 2020 4 23 5 9 │ │ │ │ 000285a0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ 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 2d2d ---------------- │ │ │ │ -000285f0: 2d2d 7c0a 7c20 2020 2020 7420 7420 2074 --|.| t t t │ │ │ │ -00028600: 2020 7420 202c 2074 2020 7420 202b 2074 t , t t + t │ │ │ │ -00028610: 2020 7420 202d 2074 2020 7420 202d 2074 t - t t - t │ │ │ │ -00028620: 2020 7420 2020 2d20 7420 2074 2020 202b t - t t + │ │ │ │ -00028630: 2074 2074 2020 202d 2074 2074 2020 2b20 t t - t t + │ │ │ │ -00028640: 2020 7c0a 7c20 2020 2020 2037 2031 3620 |.| 7 16 │ │ │ │ -00028650: 3133 2032 3120 2020 3234 2031 2020 2020 13 21 24 1 │ │ │ │ -00028660: 3131 2033 2020 2020 3138 2038 2020 2020 11 3 18 8 │ │ │ │ -00028670: 3131 2031 3020 2020 2031 3120 3137 2020 11 10 11 17 │ │ │ │ -00028680: 2020 3420 3233 2020 2020 3520 3920 2020 4 23 5 9 │ │ │ │ +000285f0: 2d2d 7c0a 7c20 2020 2020 7420 2074 2074 --|.| t t t │ │ │ │ +00028600: 2020 202b 2074 2020 7420 7420 2020 2b20 + t t t + │ │ │ │ +00028610: 7420 7420 2074 2020 202b 2074 2020 7420 t t t + t t │ │ │ │ +00028620: 7420 2020 2d20 7420 7420 2074 2020 202d t - t t t - │ │ │ │ +00028630: 2074 2020 7420 7420 2074 2020 2c20 2d20 t t t t , - │ │ │ │ +00028640: 2020 7c0a 7c20 2020 2020 2032 3320 3720 |.| 23 7 │ │ │ │ +00028650: 3136 2020 2020 3234 2037 2031 3320 2020 16 24 7 13 │ │ │ │ +00028660: 2038 2031 3720 3135 2020 2020 3131 2037 8 17 15 11 7 │ │ │ │ +00028670: 2031 3520 2020 2031 2031 3720 3231 2020 15 1 17 21 │ │ │ │ +00028680: 2020 3137 2037 2031 3320 3231 2020 2020 17 7 13 21 │ │ │ │ 00028690: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ 000286a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000286b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000286c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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2020 --|.| t t │ │ │ │ -000287e0: 202b 2074 2074 2020 202b 2074 2020 7420 + t t + t t │ │ │ │ -000287f0: 202d 2074 2020 7420 2020 2d20 7420 7420 - t t - t t │ │ │ │ -00028800: 202b 2074 2020 7420 7420 2020 2b20 7420 + t t t + t │ │ │ │ -00028810: 2074 2074 2020 202b 2074 2020 7420 7420 t t + t t t │ │ │ │ -00028820: 2020 7c0a 7c20 2020 2020 2031 3120 3138 |.| 11 18 │ │ │ │ -00028830: 2020 2020 3420 3234 2020 2020 3132 2033 4 24 12 3 │ │ │ │ -00028840: 2020 2020 3132 2031 3020 2020 2036 2039 12 10 6 9 │ │ │ │ -00028850: 2020 2020 3234 2037 2031 3620 2020 2031 24 7 16 1 │ │ │ │ -00028860: 3820 3820 3135 2020 2020 3132 2037 2031 8 8 15 12 7 1 │ │ │ │ -00028870: 3520 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 5 |.| ------ │ │ │ │ -00028880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000288a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000288b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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2020 │ │ │ │ +00028810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028820: 2020 7c0a 7c20 2020 2020 2020 2031 3820 |.| 18 │ │ │ │ +00028830: 3120 3231 2020 2020 3138 2037 2031 3320 1 21 18 7 13 │ │ │ │ +00028840: 3231 2020 2020 2020 2020 2020 2020 2020 21 │ │ │ │ +00028850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028870: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288c0: 2020 7c0a 7c6f 3820 3a20 4964 6561 6c20 |.|o8 : Ideal │ │ │ │ +000288d0: 6f66 206b 6b5b 7420 2c20 7420 2c20 7420 of kk[t , t , t │ │ │ │ +000288e0: 202c 2074 202c 2074 202c 2074 2020 2c20 , t , t , t , │ │ │ │ +000288f0: 7420 202c 2074 2020 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ +00028900: 2c20 7420 2c20 7420 202c 2074 2020 2c20 , t , t , t , │ │ │ │ +00028910: 7420 7c0a 7c20 2020 2020 2020 2020 2020 t |.| │ │ │ │ +00028920: 2020 2020 2020 2036 2020 2035 2020 2031 6 5 1 │ │ │ │ +00028930: 3220 2020 3220 2020 3420 2020 3131 2020 2 2 4 11 │ │ │ │ +00028940: 2031 3820 2020 3234 2020 2031 2020 2033 18 24 1 3 │ │ │ │ +00028950: 2020 2038 2020 2031 3020 2020 3137 2020 8 10 17 │ │ │ │ +00028960: 2032 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2|.|----------- │ │ │ │ +00028970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000289a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000289b0: 2d2d 7c0a 7c20 2c20 7420 2c20 7420 2c20 --|.| , t , t , │ │ │ │ +000289c0: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +000289d0: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ +000289e0: 202c 2074 2020 5d20 2020 2020 2020 2020 , t ] │ │ │ │ +000289f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028a00: 2020 7c0a 7c33 2020 2037 2020 2039 2020 |.|3 7 9 │ │ │ │ +00028a10: 2031 3420 2020 3136 2020 2032 3020 2020 14 16 20 │ │ │ │ +00028a20: 3232 2020 2031 3320 2020 3135 2020 2031 22 13 15 1 │ │ │ │ +00028a30: 3920 2020 3231 2020 2020 2020 2020 2020 9 21 │ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028a50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00028a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028aa0: 2d2d 7c0a 7c20 2c20 7420 2c20 7420 2c20 --|.| , t , t , │ │ │ │ -00028ab0: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00028ac0: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ -00028ad0: 202c 2074 2020 5d20 2020 2020 2020 2020 , t ] │ │ │ │ +00028aa0: 2d2d 2b0a 7c69 3920 3a20 636f 6d70 734a --+.|i9 : compsJ │ │ │ │ +00028ab0: 203d 2064 6563 6f6d 706f 7365 204a 3b20 = decompose J; │ │ │ │ +00028ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028af0: 2020 7c0a 7c33 2020 2037 2020 2039 2020 |.|3 7 9 │ │ │ │ -00028b00: 2031 3420 2020 3136 2020 2032 3020 2020 14 16 20 │ │ │ │ -00028b10: 3232 2020 2031 3320 2020 3135 2020 2031 22 13 15 1 │ │ │ │ -00028b20: 3920 2020 3231 2020 2020 2020 2020 2020 9 21 │ │ │ │ -00028b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00028b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028b90: 2d2d 2b0a 7c69 3920 3a20 636f 6d70 734a --+.|i9 : compsJ │ │ │ │ -00028ba0: 203d 2064 6563 6f6d 706f 7365 204a 3b20 = decompose J; │ │ │ │ -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 2020 │ │ │ │ -00028be0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00028bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028c30: 2d2d 2b0a 7c69 3130 203a 2063 6f6d 7073 --+.|i10 : comps │ │ │ │ -00028c40: 4a20 3d20 636f 6d70 734a 2f74 7269 6d3b J = compsJ/trim; │ │ │ │ +00028af0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00028b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028b40: 2d2d 2b0a 7c69 3130 203a 2063 6f6d 7073 --+.|i10 : comps │ │ │ │ +00028b50: 4a20 3d20 636f 6d70 734a 2f74 7269 6d3b J = compsJ/trim; │ │ │ │ +00028b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028b90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00028ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028be0: 2d2d 2b0a 7c69 3131 203a 2023 636f 6d70 --+.|i11 : #comp │ │ │ │ +00028bf0: 734a 203d 3d20 3220 2020 2020 2020 2020 sJ == 2 │ │ │ │ +00028c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 2020 2020 │ │ │ │ -00028c80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00028c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028cd0: 2d2d 2b0a 7c69 3131 203a 2023 636f 6d70 --+.|i11 : #comp │ │ │ │ -00028ce0: 734a 203d 3d20 3220 2020 2020 2020 2020 sJ == 2 │ │ │ │ -00028cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d20: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00028d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c80: 2020 7c0a 7c6f 3131 203d 2074 7275 6520 |.|o11 = true │ │ │ │ +00028c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028cd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00028ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028d20: 2d2d 2b0a 7c69 3132 203a 2063 6f6d 7073 --+.|i12 : comps │ │ │ │ +00028d30: 4a2f 6469 6d20 2020 2020 2020 2020 2020 J/dim │ │ │ │ 00028d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d70: 2020 7c0a 7c6f 3131 203d 2074 7275 6520 |.|o11 = true │ │ │ │ +00028d70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00028d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028dc0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00028dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028e10: 2d2d 2b0a 7c69 3132 203a 2063 6f6d 7073 --+.|i12 : comps │ │ │ │ -00028e20: 4a2f 6469 6d20 2020 2020 2020 2020 2020 J/dim │ │ │ │ +00028dc0: 2020 7c0a 7c6f 3132 203d 207b 3131 2c20 |.|o12 = {11, │ │ │ │ +00028dd0: 387d 2020 2020 2020 2020 2020 2020 2020 8} │ │ │ │ +00028de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028e10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028e60: 2020 7c0a 7c6f 3132 203a 204c 6973 7420 |.|o12 : List │ │ │ │ 00028e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028eb0: 2020 7c0a 7c6f 3132 203d 207b 3131 2c20 |.|o12 = {11, │ │ │ │ -00028ec0: 387d 2020 2020 2020 2020 2020 2020 2020 8} │ │ │ │ -00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00028f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f50: 2020 7c0a 7c6f 3132 203a 204c 6973 7420 |.|o12 : List │ │ │ │ -00028f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028fa0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00028fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ff0: 2d2d 2b0a 0a54 6865 7265 2061 7265 2032 --+..There are 2 │ │ │ │ -00029000: 2063 6f6d 706f 6e65 6e74 732e 2020 5765 components. We │ │ │ │ -00029010: 2061 7474 656d 7074 2074 6f20 6669 6e64 attempt to find │ │ │ │ -00029020: 2061 2070 6f69 6e74 206f 6e20 7468 6520 a point on the │ │ │ │ -00029030: 6669 7273 7420 636f 6d70 6f6e 656e 740a first component. │ │ │ │ -00029040: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00029050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00029090: 0a7c 6931 3320 3a20 7074 3120 3d20 7261 .|i13 : pt1 = ra │ │ │ │ -000290a0: 6e64 6f6d 506f 696e 744f 6e52 6174 696f ndomPointOnRatio │ │ │ │ -000290b0: 6e61 6c56 6172 6965 7479 2063 6f6d 7073 nalVariety comps │ │ │ │ -000290c0: 4a5f 3020 2020 2020 2020 2020 2020 2020 J_0 │ │ │ │ -000290d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000290e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000290f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028eb0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00028ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f00: 2d2d 2b0a 0a54 6865 7265 2061 7265 2032 --+..There are 2 │ │ │ │ +00028f10: 2063 6f6d 706f 6e65 6e74 732e 2020 5765 components. We │ │ │ │ +00028f20: 2061 7474 656d 7074 2074 6f20 6669 6e64 attempt to find │ │ │ │ +00028f30: 2061 2070 6f69 6e74 206f 6e20 7468 6520 a point on the │ │ │ │ +00028f40: 6669 7273 7420 636f 6d70 6f6e 656e 740a first component. │ │ │ │ +00028f50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00028f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00028fa0: 0a7c 6931 3320 3a20 7074 3120 3d20 7261 .|i13 : pt1 = ra │ │ │ │ +00028fb0: 6e64 6f6d 506f 696e 744f 6e52 6174 696f ndomPointOnRatio │ │ │ │ +00028fc0: 6e61 6c56 6172 6965 7479 2063 6f6d 7073 nalVariety comps │ │ │ │ +00028fd0: 4a5f 3020 2020 2020 2020 2020 2020 2020 J_0 │ │ │ │ +00028fe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028ff0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00029000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029030: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00029040: 0a7c 6f31 3320 3d20 7c20 3133 2034 3820 .|o13 = | 13 48 │ │ │ │ +00029050: 3433 2032 3320 3431 2033 3620 2d34 202d 43 23 41 36 -4 - │ │ │ │ +00029060: 3132 202d 3330 202d 3136 202d 3333 202d 12 -30 -16 -33 - │ │ │ │ +00029070: 3336 2031 3920 3139 2033 3020 2d31 3020 36 19 19 30 -10 │ │ │ │ +00029080: 2d33 3820 3332 202d 3239 202d 3820 207c -38 32 -29 -8 | │ │ │ │ +00029090: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ +000290a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000290b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000290c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000290d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000290e0: 0a7c 2020 2020 2020 2d32 3920 2d32 3220 .| -29 -22 │ │ │ │ +000290f0: 2d32 3920 2d32 3420 7c20 2020 2020 2020 -29 -24 | │ │ │ │ 00029100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029120: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029130: 0a7c 6f31 3320 3d20 7c20 3530 2031 3520 .|o13 = | 50 15 │ │ │ │ -00029140: 3436 202d 3333 2032 202d 3433 202d 3436 46 -33 2 -43 -46 │ │ │ │ -00029150: 2038 2033 3320 3139 202d 3220 2d31 3820 8 33 19 -2 -18 │ │ │ │ -00029160: 2d38 202d 3232 2034 3320 2d32 3920 3139 -8 -22 43 -29 19 │ │ │ │ -00029170: 2033 202d 3136 202d 3239 202d 3338 207c 3 -16 -29 -38 | │ │ │ │ -00029180: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ -00029190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000291a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000291b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000291c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -000291d0: 0a7c 2020 2020 2020 2d32 3420 2d31 3020 .| -24 -10 │ │ │ │ -000291e0: 2d32 3920 7c20 2020 2020 2020 2020 2020 -29 | │ │ │ │ +00029130: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00029140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029170: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00029180: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00029190: 2031 2020 2020 2020 2032 3420 2020 2020 1 24 │ │ │ │ +000291a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000291b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000291c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000291d0: 0a7c 6f31 3320 3a20 4d61 7472 6978 206b .|o13 : Matrix k │ │ │ │ +000291e0: 6b20 203c 2d2d 206b 6b20 2020 2020 2020 k <-- kk │ │ │ │ 000291f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029210: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029260: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029270: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029280: 2031 2020 2020 2020 2032 3420 2020 2020 1 24 │ │ │ │ -00029290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029220: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00029230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00029270: 0a7c 6931 3420 3a20 4631 203d 2073 7562 .|i14 : F1 = sub │ │ │ │ +00029280: 2846 2c20 2876 6172 7320 5329 7c70 7431 (F, (vars S)|pt1 │ │ │ │ +00029290: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 000292a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000292b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000292c0: 0a7c 6f31 3320 3a20 4d61 7472 6978 206b .|o13 : Matrix k │ │ │ │ -000292d0: 6b20 203c 2d2d 206b 6b20 2020 2020 2020 k <-- kk │ │ │ │ +000292c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000292d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000292e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000292f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029310: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00029320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00029360: 0a7c 6931 3420 3a20 4631 203d 2073 7562 .|i14 : F1 = sub │ │ │ │ -00029370: 2846 2c20 2876 6172 7320 5329 7c70 7431 (F, (vars S)|pt1 │ │ │ │ -00029380: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -00029390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000293a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000293b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000293c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000293d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000293e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000293f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029400: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029410: 3220 2020 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ -00029420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029430: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00029310: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00029320: 3220 2020 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ +00029330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029340: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00029350: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00029360: 0a7c 6f31 3420 3d20 6964 6561 6c20 2861 .|o14 = ideal (a │ │ │ │ +00029370: 2020 2d20 3330 622a 6320 2b20 3233 6320 - 30b*c + 23c │ │ │ │ +00029380: 202d 2031 3661 2a64 202b 2034 3162 2a64 - 16a*d + 41b*d │ │ │ │ +00029390: 202b 2034 3863 2a64 202b 2031 3364 202c + 48c*d + 13d , │ │ │ │ +000293a0: 2061 2a62 202b 2033 3062 2a63 202d 207c a*b + 30b*c - | │ │ │ │ +000293b0: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ +000293c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000293d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000293e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000293f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00029400: 0a7c 2020 2020 2020 2020 2032 2020 2020 .| 2 │ │ │ │ +00029410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029420: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00029430: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ 00029440: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029450: 0a7c 6f31 3420 3d20 6964 6561 6c20 2861 .|o14 = ideal (a │ │ │ │ -00029460: 2020 2b20 3333 622a 6320 2d20 3333 6320 + 33b*c - 33c │ │ │ │ -00029470: 202b 2031 3961 2a64 202b 2032 622a 6420 + 19a*d + 2b*d │ │ │ │ -00029480: 2b20 3135 632a 6420 2b20 3530 6420 2c20 + 15c*d + 50d , │ │ │ │ -00029490: 612a 6220 2b20 3433 622a 6320 2d20 207c a*b + 43b*c - | │ │ │ │ +00029450: 0a7c 2020 2020 2020 3333 6320 202d 2031 .| 33c - 1 │ │ │ │ +00029460: 3061 2a64 202d 2033 3662 2a64 202b 2033 0a*d - 36b*d + 3 │ │ │ │ +00029470: 3663 2a64 202b 2034 3364 202c 2061 2a63 6c*d + 43d , a*c │ │ │ │ +00029480: 202d 2032 3962 2a63 202d 2033 3863 2020 - 29b*c - 38c │ │ │ │ +00029490: 2d20 3232 612a 6420 2b20 3332 622a 647c - 22a*d + 32b*d| │ │ │ │ 000294a0: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ 000294b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000294c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000294d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000294e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -000294f0: 0a7c 2020 2020 2020 2020 3220 2020 2020 .| 2 │ │ │ │ -00029500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029510: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00029520: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -00029530: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029540: 0a7c 2020 2020 2020 3263 2020 2d20 3239 .| 2c - 29 │ │ │ │ -00029550: 612a 6420 2d20 3138 622a 6420 2d20 3433 a*d - 18b*d - 43 │ │ │ │ -00029560: 632a 6420 2b20 3436 6420 2c20 612a 6320 c*d + 46d , a*c │ │ │ │ -00029570: 2d20 3338 622a 6320 2b20 3139 6320 202d - 38b*c + 19c - │ │ │ │ -00029580: 2032 3461 2a64 202b 2033 622a 6420 2d7c 24a*d + 3b*d -| │ │ │ │ -00029590: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ -000295a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000295b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000295c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000295d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -000295e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000295f0: 2020 3220 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ -00029600: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +000294f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00029500: 2020 2020 3220 2020 3220 2020 2020 2020 2 2 │ │ │ │ +00029510: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00029520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029530: 2020 2020 2032 2020 2020 2020 2020 207c 2 | │ │ │ │ +00029540: 0a7c 2020 2020 2020 2b20 3139 632a 6420 .| + 19c*d │ │ │ │ +00029550: 2d20 3464 202c 2062 2020 2d20 3239 622a - 4d , b - 29b* │ │ │ │ +00029560: 6320 2d20 3239 6320 202d 2032 3461 2a64 c - 29c - 24a*d │ │ │ │ +00029570: 202d 2038 622a 6420 2b20 3139 632a 6420 - 8b*d + 19c*d │ │ │ │ +00029580: 2d20 3132 6420 2920 2020 2020 2020 207c - 12d ) | │ │ │ │ +00029590: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000295a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000295b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000295c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000295d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000295e0: 0a7c 6f31 3420 3a20 4964 6561 6c20 6f66 .|o14 : Ideal of │ │ │ │ +000295f0: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00029600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029620: 2020 2032 2020 2020 2020 2020 2020 207c 2 | │ │ │ │ -00029630: 0a7c 2020 2020 2020 3863 2a64 202d 2034 .| 8c*d - 4 │ │ │ │ -00029640: 3664 202c 2062 2020 2d20 3130 622a 6320 6d , b - 10b*c │ │ │ │ -00029650: 2d20 3136 6320 202d 2032 3961 2a64 202d - 16c - 29a*d - │ │ │ │ -00029660: 2032 3962 2a64 202d 2032 3263 2a64 202b 29b*d - 22c*d + │ │ │ │ -00029670: 2038 6420 2920 2020 2020 2020 2020 207c 8d ) | │ │ │ │ -00029680: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029620: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00029630: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00029640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00029680: 0a7c 6931 3520 3a20 6465 636f 6d70 6f73 .|i15 : decompos │ │ │ │ +00029690: 6520 4631 2020 2020 2020 2020 2020 2020 e F1 │ │ │ │ 000296a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000296b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000296c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000296d0: 0a7c 6f31 3420 3a20 4964 6561 6c20 6f66 .|o14 : Ideal of │ │ │ │ -000296e0: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +000296d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000296e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000296f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029710: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029720: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00029730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00029770: 0a7c 6931 3520 3a20 6465 636f 6d70 6f73 .|i15 : decompos │ │ │ │ -00029780: 6520 4631 2020 2020 2020 2020 2020 2020 e F1 │ │ │ │ -00029790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000297a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000297b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000297c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000297d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000297e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000297f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029800: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029810: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029830: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00029840: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00029850: 2020 2020 2020 2020 2020 2020 3220 207c 2 | │ │ │ │ -00029860: 0a7c 6f31 3520 3d20 7b69 6465 616c 2028 .|o15 = {ideal ( │ │ │ │ -00029870: 6120 2d20 3338 6220 2b20 3139 6320 2b20 a - 38b + 19c + │ │ │ │ -00029880: 3434 642c 2062 2020 2d20 3130 622a 6320 44d, b - 10b*c │ │ │ │ -00029890: 2d20 3136 6320 202d 2032 3062 2a64 202b - 16c - 20b*d + │ │ │ │ -000298a0: 2032 3463 2a64 202d 2032 3964 2029 2c7c 24c*d - 29d ),| │ │ │ │ -000298b0: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ -000298c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000298d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000298e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000298f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -00029900: 0a7c 2020 2020 2020 6964 6561 6c20 2863 .| ideal (c │ │ │ │ -00029910: 202d 2032 3464 2c20 6220 2d20 3338 642c - 24d, b - 38d, │ │ │ │ -00029920: 2061 202b 2031 3564 297d 2020 2020 2020 a + 15d)} │ │ │ │ -00029930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029940: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00029950: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00029960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029990: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000299a0: 0a7c 6f31 3520 3a20 4c69 7374 2020 2020 .|o15 : List │ │ │ │ -000299b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000299c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000299d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000299e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000299f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00029a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00029a40: 0a0a 5765 2061 7474 656d 7074 2074 6f20 ..We attempt to │ │ │ │ -00029a50: 6669 6e64 2061 2070 6f69 6e74 206f 6e20 find a point on │ │ │ │ -00029a60: 7468 6520 7365 636f 6e64 2063 6f6d 706f the second compo │ │ │ │ -00029a70: 6e65 6e74 2069 6e20 7061 7261 6d65 7465 nent in paramete │ │ │ │ -00029a80: 7220 7370 6163 652c 2061 6e64 2069 7473 r space, and its │ │ │ │ -00029a90: 0a63 6f72 7265 7370 6f6e 6469 6e67 2069 .corresponding i │ │ │ │ -00029aa0: 6465 616c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d deal...+-------- │ │ │ │ -00029ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029af0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 7074 -----+.|i16 : pt │ │ │ │ -00029b00: 3220 3d20 7261 6e64 6f6d 506f 696e 744f 2 = randomPointO │ │ │ │ -00029b10: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ -00029b20: 2063 6f6d 7073 4a5f 3120 2020 2020 2020 compsJ_1 │ │ │ │ -00029b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029b40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00029b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029720: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00029730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029740: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00029750: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00029760: 2020 2020 2020 2020 2020 3220 2020 207c 2 | │ │ │ │ +00029770: 0a7c 6f31 3520 3d20 7b69 6465 616c 2028 .|o15 = {ideal ( │ │ │ │ +00029780: 6120 2d20 3239 6220 2d20 3338 6320 2d20 a - 29b - 38c - │ │ │ │ +00029790: 3964 2c20 6220 202d 2032 3962 2a63 202d 9d, b - 29b*c - │ │ │ │ +000297a0: 2032 3963 2020 2b20 3362 2a64 202b 2031 29c + 3b*d + 1 │ │ │ │ +000297b0: 3663 2a64 202d 2032 3664 2029 2c20 207c 6c*d - 26d ), | │ │ │ │ +000297c0: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ +000297d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000297e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000297f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00029810: 0a7c 2020 2020 2020 6964 6561 6c20 2863 .| ideal (c │ │ │ │ +00029820: 202d 2032 3264 2c20 6220 2d20 3231 642c - 22d, b - 21d, │ │ │ │ +00029830: 2061 202b 2038 6429 7d20 2020 2020 2020 a + 8d)} │ │ │ │ +00029840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029850: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00029860: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00029870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000298a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000298b0: 0a7c 6f31 3520 3a20 4c69 7374 2020 2020 .|o15 : List │ │ │ │ +000298c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000298d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000298e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000298f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00029900: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00029910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00029950: 0a0a 5765 2061 7474 656d 7074 2074 6f20 ..We attempt to │ │ │ │ +00029960: 6669 6e64 2061 2070 6f69 6e74 206f 6e20 find a point on │ │ │ │ +00029970: 7468 6520 7365 636f 6e64 2063 6f6d 706f the second compo │ │ │ │ +00029980: 6e65 6e74 2069 6e20 7061 7261 6d65 7465 nent in paramete │ │ │ │ +00029990: 7220 7370 6163 652c 2061 6e64 2069 7473 r space, and its │ │ │ │ +000299a0: 0a63 6f72 7265 7370 6f6e 6469 6e67 2069 .corresponding i │ │ │ │ +000299b0: 6465 616c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d deal...+-------- │ │ │ │ +000299c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000299d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000299e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000299f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029a00: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 7074 -----+.|i16 : pt │ │ │ │ +00029a10: 3220 3d20 7261 6e64 6f6d 506f 696e 744f 2 = randomPointO │ │ │ │ +00029a20: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ +00029a30: 2063 6f6d 7073 4a5f 3120 2020 2020 2020 compsJ_1 │ │ │ │ +00029a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029a50: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029aa0: 2020 2020 207c 0a7c 6f31 3620 3d20 7c20 |.|o16 = | │ │ │ │ +00029ab0: 3436 202d 3220 3136 202d 3230 202d 3120 46 -2 16 -20 -1 │ │ │ │ +00029ac0: 2d33 3020 2d34 3320 2d34 3120 3137 202d -30 -43 -41 17 - │ │ │ │ +00029ad0: 3420 2d31 3620 2d32 3920 2d33 3920 3430 4 -16 -29 -39 40 │ │ │ │ +00029ae0: 2034 3920 2d33 3920 2d31 3820 2d31 3320 49 -39 -18 -13 │ │ │ │ +00029af0: 2d34 3720 207c 0a7c 2020 2020 2020 2d2d -47 |.| -- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029b40: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3334 -----|.| 34 │ │ │ │ +00029b50: 2031 3920 3231 2033 3920 3020 7c20 2020 19 21 39 0 | │ │ │ │ 00029b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029b90: 2020 2020 207c 0a7c 6f31 3620 3d20 7c20 |.|o16 = | │ │ │ │ -00029ba0: 2d31 3420 3430 202d 3520 3236 202d 3438 -14 40 -5 26 -48 │ │ │ │ -00029bb0: 202d 3236 202d 3335 2034 3120 2d38 202d -26 -35 41 -8 - │ │ │ │ -00029bc0: 3135 202d 3338 2033 3120 2d31 3320 3239 15 -38 31 -13 29 │ │ │ │ -00029bd0: 2032 3120 3136 2033 3920 3231 202d 3138 21 16 39 21 -18 │ │ │ │ -00029be0: 2031 3920 207c 0a7c 2020 2020 2020 2d2d 19 |.| -- │ │ │ │ -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 2d2d 2d2d ---------------- │ │ │ │ -00029c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029c30: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2d34 -----|.| -4 │ │ │ │ -00029c40: 3720 2d33 3920 3334 2030 207c 2020 2020 7 -39 34 0 | │ │ │ │ +00029b90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029be0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029bf0: 2020 2020 2020 2031 2020 2020 2020 2032 1 2 │ │ │ │ +00029c00: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +00029c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c30: 2020 2020 207c 0a7c 6f31 3620 3a20 4d61 |.|o16 : Ma │ │ │ │ +00029c40: 7472 6978 206b 6b20 203c 2d2d 206b 6b20 trix kk <-- kk │ │ │ │ 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 2020 2020 │ │ │ │ -00029c80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00029c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00029ce0: 2020 2020 2020 2031 2020 2020 2020 2032 1 2 │ │ │ │ -00029cf0: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +00029c80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00029c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029cd0: 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20 4632 -----+.|i17 : F2 │ │ │ │ +00029ce0: 203d 2073 7562 2846 2c20 2876 6172 7320 = sub(F, (vars │ │ │ │ +00029cf0: 5329 7c70 7432 2920 2020 2020 2020 2020 S)|pt2) │ │ │ │ 00029d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d20: 2020 2020 207c 0a7c 6f31 3620 3a20 4d61 |.|o16 : Ma │ │ │ │ -00029d30: 7472 6978 206b 6b20 203c 2d2d 206b 6b20 trix kk <-- kk │ │ │ │ +00029d20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00029d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029dc0: 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20 4632 -----+.|i17 : F2 │ │ │ │ -00029dd0: 203d 2073 7562 2846 2c20 2876 6172 7320 = sub(F, (vars │ │ │ │ -00029de0: 5329 7c70 7432 2920 2020 2020 2020 2020 S)|pt2) │ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e10: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00029e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00029e70: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00029e80: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -00029e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ea0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00029eb0: 2020 2020 207c 0a7c 6f31 3720 3d20 6964 |.|o17 = id │ │ │ │ -00029ec0: 6561 6c20 2861 2020 2d20 3862 2a63 202b eal (a - 8b*c + │ │ │ │ -00029ed0: 2032 3663 2020 2d20 3135 612a 6420 2d20 26c - 15a*d - │ │ │ │ -00029ee0: 3438 622a 6420 2b20 3430 632a 6420 2d20 48b*d + 40c*d - │ │ │ │ -00029ef0: 3134 6420 2c20 612a 6220 2b20 3231 622a 14d , a*b + 21b* │ │ │ │ -00029f00: 6320 2d20 207c 0a7c 2020 2020 2020 2d2d c - |.| -- │ │ │ │ +00029d70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029d80: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00029d90: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00029da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029db0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00029dc0: 2020 2020 327c 0a7c 6f31 3720 3d20 6964 2|.|o17 = id │ │ │ │ +00029dd0: 6561 6c20 2861 2020 2b20 3137 622a 6320 eal (a + 17b*c │ │ │ │ +00029de0: 2d20 3230 6320 202d 2034 612a 6420 2d20 - 20c - 4a*d - │ │ │ │ +00029df0: 622a 6420 2d20 3263 2a64 202b 2034 3664 b*d - 2c*d + 46d │ │ │ │ +00029e00: 202c 2061 2a62 202b 2034 3962 2a63 202d , a*b + 49b*c - │ │ │ │ +00029e10: 2031 3663 207c 0a7c 2020 2020 2020 2d2d 16c |.| -- │ │ │ │ +00029e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029e60: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +00029e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e80: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +00029e90: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +00029ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029eb0: 2020 2020 207c 0a7c 2020 2020 2020 2d20 |.| - │ │ │ │ +00029ec0: 3339 612a 6420 2d20 3239 622a 6420 2d20 39a*d - 29b*d - │ │ │ │ +00029ed0: 3330 632a 6420 2b20 3136 6420 2c20 612a 30c*d + 16d , a* │ │ │ │ +00029ee0: 6320 2b20 3139 622a 6320 2d20 3138 6320 c + 19b*c - 18c │ │ │ │ +00029ef0: 202b 2032 3161 2a64 202d 2031 3362 2a64 + 21a*d - 13b*d │ │ │ │ +00029f00: 202d 2020 207c 0a7c 2020 2020 2020 2d2d - |.| -- │ │ │ │ 00029f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029f50: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -00029f60: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00029f70: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +00029f60: 2020 2020 2020 2020 2032 2020 2032 2020 2 2 │ │ │ │ +00029f70: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ 00029f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029f90: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00029fa0: 2020 2020 207c 0a7c 2020 2020 2020 3338 |.| 38 │ │ │ │ -00029fb0: 6320 202b 2031 3661 2a64 202b 2033 3162 c + 16a*d + 31b │ │ │ │ -00029fc0: 2a64 202d 2032 3663 2a64 202d 2035 6420 *d - 26c*d - 5d │ │ │ │ -00029fd0: 2c20 612a 6320 2d20 3437 622a 6320 2b20 , a*c - 47b*c + │ │ │ │ -00029fe0: 3339 6320 202d 2033 3961 2a64 202b 2032 39c - 39a*d + 2 │ │ │ │ -00029ff0: 3162 2a64 207c 0a7c 2020 2020 2020 2d2d 1b*d |.| -- │ │ │ │ -0002a000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 2d2d 2d2d ---------------- │ │ │ │ -0002a040: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0002a050: 2020 2020 2020 2020 2020 2032 2020 2032 2 2 │ │ │ │ -0002a060: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00029fa0: 2020 2020 207c 0a7c 2020 2020 2020 3339 |.| 39 │ │ │ │ +00029fb0: 632a 6420 2d20 3433 6420 2c20 6220 202b c*d - 43d , b + │ │ │ │ +00029fc0: 2033 3962 2a63 202d 2034 3763 2020 2b20 39b*c - 47c + │ │ │ │ +00029fd0: 3334 622a 6420 2b20 3430 632a 6420 2d20 34b*d + 40c*d - │ │ │ │ +00029fe0: 3431 6420 2920 2020 2020 2020 2020 2020 41d ) │ │ │ │ +00029ff0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a040: 2020 2020 207c 0a7c 6f31 3720 3a20 4964 |.|o17 : Id │ │ │ │ +0002a050: 6561 6c20 6f66 2053 2020 2020 2020 2020 eal of S │ │ │ │ +0002a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a080: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0002a090: 2020 2020 207c 0a7c 2020 2020 2020 2d20 |.| - │ │ │ │ -0002a0a0: 3133 632a 6420 2d20 3335 6420 2c20 6220 13c*d - 35d , b │ │ │ │ -0002a0b0: 202b 2033 3462 2a63 202d 2031 3863 2020 + 34b*c - 18c │ │ │ │ -0002a0c0: 2b20 3139 622a 6420 2b20 3239 632a 6420 + 19b*d + 29c*d │ │ │ │ -0002a0d0: 2b20 3431 6420 2920 2020 2020 2020 2020 + 41d ) │ │ │ │ -0002a0e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a090: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a0e0: 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20 6465 -----+.|i18 : de │ │ │ │ +0002a0f0: 636f 6d70 6f73 6520 4632 2020 2020 2020 compose F2 │ │ │ │ 0002a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a130: 2020 2020 207c 0a7c 6f31 3720 3a20 4964 |.|o17 : Id │ │ │ │ -0002a140: 6561 6c20 6f66 2053 2020 2020 2020 2020 eal of S │ │ │ │ +0002a130: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a180: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -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 2d2d 2d2d ---------------- │ │ │ │ -0002a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a1d0: 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20 6465 -----+.|i18 : de │ │ │ │ -0002a1e0: 636f 6d70 6f73 6520 4632 2020 2020 2020 compose F2 │ │ │ │ -0002a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a220: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0002a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a270: 2020 2020 207c 0a7c 6f31 3820 3d20 7b69 |.|o18 = {i │ │ │ │ -0002a280: 6465 616c 2028 6220 2b20 3139 6320 2d20 deal (b + 19c - │ │ │ │ -0002a290: 3138 642c 2061 202b 2032 3363 202b 2034 18d, a + 23c + 4 │ │ │ │ -0002a2a0: 3364 292c 2069 6465 616c 2028 6220 2b20 3d), ideal (b + │ │ │ │ -0002a2b0: 3135 6320 2b20 3337 642c 2061 202b 2033 15c + 37d, a + 3 │ │ │ │ -0002a2c0: 3763 202b 207c 0a7c 2020 2020 2020 2d2d 7c + |.| -- │ │ │ │ -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 2d7c 0a7c 2020 2020 2020 3236 -----|.| 26 │ │ │ │ -0002a320: 6429 7d20 2020 2020 2020 2020 2020 2020 d)} │ │ │ │ -0002a330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a360: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0002a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a180: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a1a0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002a1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a1c0: 2020 2020 2020 3220 2020 3220 2020 2020 2 2 │ │ │ │ +0002a1d0: 2020 2020 207c 0a7c 6f31 3820 3d20 7b69 |.|o18 = {i │ │ │ │ +0002a1e0: 6465 616c 2028 612a 6320 2b20 3139 622a deal (a*c + 19b* │ │ │ │ +0002a1f0: 6320 2d20 3138 6320 202b 2032 3161 2a64 c - 18c + 21a*d │ │ │ │ +0002a200: 202d 2031 3362 2a64 202d 2033 3963 2a64 - 13b*d - 39c*d │ │ │ │ +0002a210: 202d 2034 3364 202c 2062 2020 2b20 3339 - 43d , b + 39 │ │ │ │ +0002a220: 622a 6320 2d7c 0a7c 2020 2020 2020 2d2d b*c -|.| -- │ │ │ │ +0002a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a240: 2d2d 2d2d 2d2d 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 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +0002a280: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a290: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0002a2a0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +0002a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a2c0: 2020 2020 207c 0a7c 2020 2020 2020 3437 |.| 47 │ │ │ │ +0002a2d0: 6320 202b 2033 3462 2a64 202b 2034 3063 c + 34b*d + 40c │ │ │ │ +0002a2e0: 2a64 202d 2034 3164 202c 2061 2a62 202b *d - 41d , a*b + │ │ │ │ +0002a2f0: 2034 3962 2a63 202d 2031 3663 2020 2d20 49b*c - 16c - │ │ │ │ +0002a300: 3339 612a 6420 2d20 3239 622a 6420 2d20 39a*d - 29b*d - │ │ │ │ +0002a310: 3330 632a 647c 0a7c 2020 2020 2020 2d2d 30c*d|.| -- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a360: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +0002a370: 2020 2032 2020 2032 2020 2020 2020 2020 2 2 │ │ │ │ +0002a380: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ 0002a390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a3b0: 2020 2020 207c 0a7c 6f31 3820 3a20 4c69 |.|o18 : Li │ │ │ │ -0002a3c0: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ -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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0002a410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a450: 2d2d 2d2d 2d2b 0a0a 4974 2074 7572 6e73 -----+..It turns │ │ │ │ -0002a460: 206f 7574 2074 6861 7420 7468 6973 2069 out that this i │ │ │ │ -0002a470: 7320 7468 6520 6964 6561 6c20 6f66 2032 s the ideal of 2 │ │ │ │ -0002a480: 2073 6b65 7720 6c69 6e65 732c 206a 7573 skew lines, jus │ │ │ │ -0002a490: 7420 6e6f 7420 6465 6669 6e65 6420 6f76 t not defined ov │ │ │ │ -0002a4a0: 6572 2074 6869 730a 6669 656c 642e 0a0a er this.field... │ │ │ │ -0002a4b0: 4361 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54 Caveat.======..T │ │ │ │ -0002a4c0: 6869 7320 726f 7574 696e 6520 6578 7065 his routine expe │ │ │ │ -0002a4d0: 6374 7320 7468 6520 696e 7075 7420 746f cts the input to │ │ │ │ -0002a4e0: 2072 6570 7265 7365 6e74 2061 6e20 6972 represent an ir │ │ │ │ -0002a4f0: 7265 6475 6369 626c 6520 7661 7269 6574 reducible variet │ │ │ │ -0002a500: 790a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d y..See also.==== │ │ │ │ -0002a510: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -0002a520: 7261 6e64 6f6d 506f 696e 7473 4f6e 5261 randomPointsOnRa │ │ │ │ -0002a530: 7469 6f6e 616c 5661 7269 6574 793a 0a20 tionalVariety:. │ │ │ │ -0002a540: 2020 2072 616e 646f 6d50 6f69 6e74 734f randomPointsO │ │ │ │ -0002a550: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ -0002a560: 5f6c 7049 6465 616c 5f63 6d5a 5a5f 7270 _lpIdeal_cmZZ_rp │ │ │ │ -0002a570: 2c20 2d2d 2066 696e 6420 7261 6e64 6f6d , -- find random │ │ │ │ -0002a580: 2070 6f69 6e74 7320 6f6e 2061 0a20 2020 points on a. │ │ │ │ -0002a590: 2076 6172 6965 7479 2074 6861 7420 6361 variety that ca │ │ │ │ -0002a5a0: 6e20 6265 2064 6574 6563 7465 6420 746f n be detected to │ │ │ │ -0002a5b0: 2062 6520 7261 7469 6f6e 616c 0a0a 5761 be rational..Wa │ │ │ │ -0002a5c0: 7973 2074 6f20 7573 6520 7468 6973 206d ys to use this m │ │ │ │ -0002a5d0: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d ethod:.========= │ │ │ │ -0002a5e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0002a5f0: 0a20 202a 202a 6e6f 7465 2072 616e 646f . * *note rando │ │ │ │ -0002a600: 6d50 6f69 6e74 4f6e 5261 7469 6f6e 616c mPointOnRational │ │ │ │ -0002a610: 5661 7269 6574 7928 4964 6561 6c29 3a0a Variety(Ideal):. │ │ │ │ -0002a620: 2020 2020 7261 6e64 6f6d 506f 696e 744f randomPointO │ │ │ │ -0002a630: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ -0002a640: 5f6c 7049 6465 616c 5f72 702c 202d 2d20 _lpIdeal_rp, -- │ │ │ │ -0002a650: 6669 6e64 2061 2072 616e 646f 6d20 706f find a random po │ │ │ │ -0002a660: 696e 7420 6f6e 2061 0a20 2020 2076 6172 int on a. var │ │ │ │ -0002a670: 6965 7479 2074 6861 7420 6361 6e20 6265 iety that can be │ │ │ │ -0002a680: 2064 6574 6563 7465 6420 746f 2062 6520 detected to be │ │ │ │ -0002a690: 7261 7469 6f6e 616c 0a2d 2d2d 2d2d 2d2d rational.------- │ │ │ │ -0002a6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a6e0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -0002a6f0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -0002a700: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -0002a710: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -0002a720: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -0002a730: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163 .25.11+ds/M2/Mac │ │ │ │ -0002a740: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -0002a750: 0a47 726f 6562 6e65 7253 7472 6174 612e .GroebnerStrata. │ │ │ │ -0002a760: 6d32 3a39 3339 3a30 2e0a 1f0a 4669 6c65 m2:939:0....File │ │ │ │ -0002a770: 3a20 4772 6f65 626e 6572 5374 7261 7461 : GroebnerStrata │ │ │ │ -0002a780: 2e69 6e66 6f2c 204e 6f64 653a 2072 616e .info, Node: ran │ │ │ │ -0002a790: 646f 6d50 6f69 6e74 734f 6e52 6174 696f domPointsOnRatio │ │ │ │ -0002a7a0: 6e61 6c56 6172 6965 7479 5f6c 7049 6465 nalVariety_lpIde │ │ │ │ -0002a7b0: 616c 5f63 6d5a 5a5f 7270 2c20 4e65 7874 al_cmZZ_rp, Next │ │ │ │ -0002a7c0: 3a20 736d 616c 6c65 724d 6f6e 6f6d 6961 : smallerMonomia │ │ │ │ -0002a7d0: 6c73 2c20 5072 6576 3a20 7261 6e64 6f6d ls, Prev: random │ │ │ │ -0002a7e0: 506f 696e 744f 6e52 6174 696f 6e61 6c56 PointOnRationalV │ │ │ │ -0002a7f0: 6172 6965 7479 5f6c 7049 6465 616c 5f72 ariety_lpIdeal_r │ │ │ │ -0002a800: 702c 2055 703a 2054 6f70 0a0a 7261 6e64 p, Up: Top..rand │ │ │ │ -0002a810: 6f6d 506f 696e 7473 4f6e 5261 7469 6f6e omPointsOnRation │ │ │ │ -0002a820: 616c 5661 7269 6574 7928 4964 6561 6c2c alVariety(Ideal, │ │ │ │ -0002a830: 5a5a 2920 2d2d 2066 696e 6420 7261 6e64 ZZ) -- find rand │ │ │ │ -0002a840: 6f6d 2070 6f69 6e74 7320 6f6e 2061 2076 om points on a v │ │ │ │ -0002a850: 6172 6965 7479 2074 6861 7420 6361 6e20 ariety that can │ │ │ │ -0002a860: 6265 2064 6574 6563 7465 6420 746f 2062 be detected to b │ │ │ │ -0002a870: 6520 7261 7469 6f6e 616c 0a2a 2a2a 2a2a e rational.***** │ │ │ │ -0002a880: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a8a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a8b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a8c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a8d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002a8e0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 *********.. * F │ │ │ │ -0002a8f0: 756e 6374 696f 6e3a 202a 6e6f 7465 2072 unction: *note r │ │ │ │ -0002a900: 616e 646f 6d50 6f69 6e74 734f 6e52 6174 andomPointsOnRat │ │ │ │ -0002a910: 696f 6e61 6c56 6172 6965 7479 3a0a 2020 ionalVariety:. │ │ │ │ -0002a920: 2020 7261 6e64 6f6d 506f 696e 7473 4f6e randomPointsOn │ │ │ │ -0002a930: 5261 7469 6f6e 616c 5661 7269 6574 795f RationalVariety_ │ │ │ │ -0002a940: 6c70 4964 6561 6c5f 636d 5a5a 5f72 702c lpIdeal_cmZZ_rp, │ │ │ │ -0002a950: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -0002a960: 2020 2020 2072 616e 646f 6d50 6f69 6e74 randomPoint │ │ │ │ -0002a970: 734f 6e52 6174 696f 6e61 6c56 6172 6965 sOnRationalVarie │ │ │ │ -0002a980: 7479 2849 2c20 6e29 0a20 2020 2020 2020 ty(I, n). │ │ │ │ -0002a990: 2072 616e 646f 6d50 6f69 6e74 4f6e 5261 randomPointOnRa │ │ │ │ -0002a9a0: 7469 6f6e 616c 5661 7269 6574 790a 2020 tionalVariety. │ │ │ │ -0002a9b0: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ -0002a9c0: 2a20 492c 2061 6e20 2a6e 6f74 6520 6964 * I, an *note id │ │ │ │ -0002a9d0: 6561 6c3a 2028 4d61 6361 756c 6179 3244 eal: (Macaulay2D │ │ │ │ -0002a9e0: 6f63 2949 6465 616c 2c2c 2041 6e20 6964 oc)Ideal,, An id │ │ │ │ -0002a9f0: 6561 6c20 696e 2061 2070 6f6c 796e 6f6d eal in a polynom │ │ │ │ -0002aa00: 6961 6c20 7269 6e67 0a20 2020 2020 2020 ial ring. │ │ │ │ -0002aa10: 2024 5324 206f 7665 7220 6120 6669 656c $S$ over a fiel │ │ │ │ -0002aa20: 642c 2077 6869 6368 2064 6566 696e 6573 d, which defines │ │ │ │ -0002aa30: 2061 2070 7269 6d65 2069 6465 616c 0a20 a prime ideal. │ │ │ │ -0002aa40: 2020 2020 202a 206e 2c20 616e 202a 6e6f * n, an *no │ │ │ │ -0002aa50: 7465 2069 6e74 6567 6572 3a20 284d 6163 te integer: (Mac │ │ │ │ -0002aa60: 6175 6c61 7932 446f 6329 5a5a 2c2c 2054 aulay2Doc)ZZ,, T │ │ │ │ -0002aa70: 6865 206e 756d 6265 7220 6f66 2070 6f69 he number of poi │ │ │ │ -0002aa80: 6e74 7320 746f 0a20 2020 2020 2020 2067 nts to. g │ │ │ │ -0002aa90: 656e 6572 6174 650a 2020 2a20 4f75 7470 enerate. * Outp │ │ │ │ -0002aaa0: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ -0002aab0: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ -0002aac0: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ -0002aad0: 4120 6c69 7374 206f 6620 246e 2420 6f6e A list of $n$ on │ │ │ │ -0002aae0: 6520 726f 7720 6d61 7472 6963 6573 206f e row matrices o │ │ │ │ -0002aaf0: 7665 720a 2020 2020 2020 2020 7468 6520 ver. the │ │ │ │ -0002ab00: 6261 7365 2066 6965 6c64 206f 6620 2453 base field of $S │ │ │ │ -0002ab10: 242c 2074 6861 7420 6172 6520 7261 6e64 $, that are rand │ │ │ │ -0002ab20: 6f6d 6c79 2063 686f 7365 6e20 706f 696e omly chosen poin │ │ │ │ -0002ab30: 7473 206f 6e20 2449 242e 2020 6e75 6c6c ts on $I$. null │ │ │ │ -0002ab40: 2069 730a 2020 2020 2020 2020 7265 7475 is. retu │ │ │ │ -0002ab50: 726e 6564 2069 6e20 7468 6520 6361 7365 rned in the case │ │ │ │ -0002ab60: 2077 6865 6e20 7468 6520 726f 7574 696e when the routin │ │ │ │ -0002ab70: 6520 6361 6e6e 6f74 2064 6574 6572 6d69 e cannot determi │ │ │ │ -0002ab80: 6e65 2069 6620 7468 6520 7661 7269 6574 ne if the variet │ │ │ │ -0002ab90: 790a 2020 2020 2020 2020 6973 2072 6174 y. is rat │ │ │ │ -0002aba0: 696f 6e61 6c20 616e 6420 6972 7265 6475 ional and irredu │ │ │ │ -0002abb0: 6369 626c 652e 0a0a 4465 7363 7269 7074 cible...Descript │ │ │ │ -0002abc0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -0002abd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002abe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ac00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ac10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002ac20: 0a7c 6931 203a 206b 6b20 3d20 5a5a 2f31 .|i1 : kk = ZZ/1 │ │ │ │ -0002ac30: 3031 3b20 2020 2020 2020 2020 2020 2020 01; │ │ │ │ -0002ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ac50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ac60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002a3a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a3b0: 2020 2020 207c 0a7c 2020 2020 2020 2b20 |.| + │ │ │ │ +0002a3c0: 3136 6420 2c20 6120 202b 2031 3762 2a63 16d , a + 17b*c │ │ │ │ +0002a3d0: 202d 2032 3063 2020 2d20 3461 2a64 202d - 20c - 4a*d - │ │ │ │ +0002a3e0: 2062 2a64 202d 2032 632a 6420 2b20 3436 b*d - 2c*d + 46 │ │ │ │ +0002a3f0: 6420 297d 2020 2020 2020 2020 2020 2020 d )} │ │ │ │ +0002a400: 2020 2020 207c 0a7c 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 207c 0a7c 6f31 3820 3a20 4c69 |.|o18 : Li │ │ │ │ +0002a460: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ +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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a4f0: 2d2d 2d2d 2d2b 0a0a 4974 2074 7572 6e73 -----+..It turns │ │ │ │ +0002a500: 206f 7574 2074 6861 7420 7468 6973 2069 out that this i │ │ │ │ +0002a510: 7320 7468 6520 6964 6561 6c20 6f66 2032 s the ideal of 2 │ │ │ │ +0002a520: 2073 6b65 7720 6c69 6e65 732c 206a 7573 skew lines, jus │ │ │ │ +0002a530: 7420 6e6f 7420 6465 6669 6e65 6420 6f76 t not defined ov │ │ │ │ +0002a540: 6572 2074 6869 730a 6669 656c 642e 0a0a er this.field... │ │ │ │ +0002a550: 4361 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54 Caveat.======..T │ │ │ │ +0002a560: 6869 7320 726f 7574 696e 6520 6578 7065 his routine expe │ │ │ │ +0002a570: 6374 7320 7468 6520 696e 7075 7420 746f cts the input to │ │ │ │ +0002a580: 2072 6570 7265 7365 6e74 2061 6e20 6972 represent an ir │ │ │ │ +0002a590: 7265 6475 6369 626c 6520 7661 7269 6574 reducible variet │ │ │ │ +0002a5a0: 790a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d y..See also.==== │ │ │ │ +0002a5b0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +0002a5c0: 7261 6e64 6f6d 506f 696e 7473 4f6e 5261 randomPointsOnRa │ │ │ │ +0002a5d0: 7469 6f6e 616c 5661 7269 6574 793a 0a20 tionalVariety:. │ │ │ │ +0002a5e0: 2020 2072 616e 646f 6d50 6f69 6e74 734f randomPointsO │ │ │ │ +0002a5f0: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ +0002a600: 5f6c 7049 6465 616c 5f63 6d5a 5a5f 7270 _lpIdeal_cmZZ_rp │ │ │ │ +0002a610: 2c20 2d2d 2066 696e 6420 7261 6e64 6f6d , -- find random │ │ │ │ +0002a620: 2070 6f69 6e74 7320 6f6e 2061 0a20 2020 points on a. │ │ │ │ +0002a630: 2076 6172 6965 7479 2074 6861 7420 6361 variety that ca │ │ │ │ +0002a640: 6e20 6265 2064 6574 6563 7465 6420 746f n be detected to │ │ │ │ +0002a650: 2062 6520 7261 7469 6f6e 616c 0a0a 5761 be rational..Wa │ │ │ │ +0002a660: 7973 2074 6f20 7573 6520 7468 6973 206d ys to use this m │ │ │ │ +0002a670: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d ethod:.========= │ │ │ │ +0002a680: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0002a690: 0a20 202a 202a 6e6f 7465 2072 616e 646f . * *note rando │ │ │ │ +0002a6a0: 6d50 6f69 6e74 4f6e 5261 7469 6f6e 616c mPointOnRational │ │ │ │ +0002a6b0: 5661 7269 6574 7928 4964 6561 6c29 3a0a Variety(Ideal):. │ │ │ │ +0002a6c0: 2020 2020 7261 6e64 6f6d 506f 696e 744f randomPointO │ │ │ │ +0002a6d0: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ +0002a6e0: 5f6c 7049 6465 616c 5f72 702c 202d 2d20 _lpIdeal_rp, -- │ │ │ │ +0002a6f0: 6669 6e64 2061 2072 616e 646f 6d20 706f find a random po │ │ │ │ +0002a700: 696e 7420 6f6e 2061 0a20 2020 2076 6172 int on a. var │ │ │ │ +0002a710: 6965 7479 2074 6861 7420 6361 6e20 6265 iety that can be │ │ │ │ +0002a720: 2064 6574 6563 7465 6420 746f 2062 6520 detected to be │ │ │ │ +0002a730: 7261 7469 6f6e 616c 0a2d 2d2d 2d2d 2d2d rational.------- │ │ │ │ +0002a740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a780: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0002a790: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0002a7a0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0002a7b0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0002a7c0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0002a7d0: 2e32 352e 3131 2b64 732f 4d32 2f4d 6163 .25.11+ds/M2/Mac │ │ │ │ +0002a7e0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0002a7f0: 0a47 726f 6562 6e65 7253 7472 6174 612e .GroebnerStrata. │ │ │ │ +0002a800: 6d32 3a39 3339 3a30 2e0a 1f0a 4669 6c65 m2:939:0....File │ │ │ │ +0002a810: 3a20 4772 6f65 626e 6572 5374 7261 7461 : GroebnerStrata │ │ │ │ +0002a820: 2e69 6e66 6f2c 204e 6f64 653a 2072 616e .info, Node: ran │ │ │ │ +0002a830: 646f 6d50 6f69 6e74 734f 6e52 6174 696f domPointsOnRatio │ │ │ │ +0002a840: 6e61 6c56 6172 6965 7479 5f6c 7049 6465 nalVariety_lpIde │ │ │ │ +0002a850: 616c 5f63 6d5a 5a5f 7270 2c20 4e65 7874 al_cmZZ_rp, Next │ │ │ │ +0002a860: 3a20 736d 616c 6c65 724d 6f6e 6f6d 6961 : smallerMonomia │ │ │ │ +0002a870: 6c73 2c20 5072 6576 3a20 7261 6e64 6f6d ls, Prev: random │ │ │ │ +0002a880: 506f 696e 744f 6e52 6174 696f 6e61 6c56 PointOnRationalV │ │ │ │ +0002a890: 6172 6965 7479 5f6c 7049 6465 616c 5f72 ariety_lpIdeal_r │ │ │ │ +0002a8a0: 702c 2055 703a 2054 6f70 0a0a 7261 6e64 p, Up: Top..rand │ │ │ │ +0002a8b0: 6f6d 506f 696e 7473 4f6e 5261 7469 6f6e omPointsOnRation │ │ │ │ +0002a8c0: 616c 5661 7269 6574 7928 4964 6561 6c2c alVariety(Ideal, │ │ │ │ +0002a8d0: 5a5a 2920 2d2d 2066 696e 6420 7261 6e64 ZZ) -- find rand │ │ │ │ +0002a8e0: 6f6d 2070 6f69 6e74 7320 6f6e 2061 2076 om points on a v │ │ │ │ +0002a8f0: 6172 6965 7479 2074 6861 7420 6361 6e20 ariety that can │ │ │ │ +0002a900: 6265 2064 6574 6563 7465 6420 746f 2062 be detected to b │ │ │ │ +0002a910: 6520 7261 7469 6f6e 616c 0a2a 2a2a 2a2a e rational.***** │ │ │ │ +0002a920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002a930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002a940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002a950: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002a960: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002a970: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002a980: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 *********.. * F │ │ │ │ +0002a990: 756e 6374 696f 6e3a 202a 6e6f 7465 2072 unction: *note r │ │ │ │ +0002a9a0: 616e 646f 6d50 6f69 6e74 734f 6e52 6174 andomPointsOnRat │ │ │ │ +0002a9b0: 696f 6e61 6c56 6172 6965 7479 3a0a 2020 ionalVariety:. │ │ │ │ +0002a9c0: 2020 7261 6e64 6f6d 506f 696e 7473 4f6e randomPointsOn │ │ │ │ +0002a9d0: 5261 7469 6f6e 616c 5661 7269 6574 795f RationalVariety_ │ │ │ │ +0002a9e0: 6c70 4964 6561 6c5f 636d 5a5a 5f72 702c lpIdeal_cmZZ_rp, │ │ │ │ +0002a9f0: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +0002aa00: 2020 2020 2072 616e 646f 6d50 6f69 6e74 randomPoint │ │ │ │ +0002aa10: 734f 6e52 6174 696f 6e61 6c56 6172 6965 sOnRationalVarie │ │ │ │ +0002aa20: 7479 2849 2c20 6e29 0a20 2020 2020 2020 ty(I, n). │ │ │ │ +0002aa30: 2072 616e 646f 6d50 6f69 6e74 4f6e 5261 randomPointOnRa │ │ │ │ +0002aa40: 7469 6f6e 616c 5661 7269 6574 790a 2020 tionalVariety. │ │ │ │ +0002aa50: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +0002aa60: 2a20 492c 2061 6e20 2a6e 6f74 6520 6964 * I, an *note id │ │ │ │ +0002aa70: 6561 6c3a 2028 4d61 6361 756c 6179 3244 eal: (Macaulay2D │ │ │ │ +0002aa80: 6f63 2949 6465 616c 2c2c 2041 6e20 6964 oc)Ideal,, An id │ │ │ │ +0002aa90: 6561 6c20 696e 2061 2070 6f6c 796e 6f6d eal in a polynom │ │ │ │ +0002aaa0: 6961 6c20 7269 6e67 0a20 2020 2020 2020 ial ring. │ │ │ │ +0002aab0: 2024 5324 206f 7665 7220 6120 6669 656c $S$ over a fiel │ │ │ │ +0002aac0: 642c 2077 6869 6368 2064 6566 696e 6573 d, which defines │ │ │ │ +0002aad0: 2061 2070 7269 6d65 2069 6465 616c 0a20 a prime ideal. │ │ │ │ +0002aae0: 2020 2020 202a 206e 2c20 616e 202a 6e6f * n, an *no │ │ │ │ +0002aaf0: 7465 2069 6e74 6567 6572 3a20 284d 6163 te integer: (Mac │ │ │ │ +0002ab00: 6175 6c61 7932 446f 6329 5a5a 2c2c 2054 aulay2Doc)ZZ,, T │ │ │ │ +0002ab10: 6865 206e 756d 6265 7220 6f66 2070 6f69 he number of poi │ │ │ │ +0002ab20: 6e74 7320 746f 0a20 2020 2020 2020 2067 nts to. g │ │ │ │ +0002ab30: 656e 6572 6174 650a 2020 2a20 4f75 7470 enerate. * Outp │ │ │ │ +0002ab40: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ +0002ab50: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ +0002ab60: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ +0002ab70: 4120 6c69 7374 206f 6620 246e 2420 6f6e A list of $n$ on │ │ │ │ +0002ab80: 6520 726f 7720 6d61 7472 6963 6573 206f e row matrices o │ │ │ │ +0002ab90: 7665 720a 2020 2020 2020 2020 7468 6520 ver. the │ │ │ │ +0002aba0: 6261 7365 2066 6965 6c64 206f 6620 2453 base field of $S │ │ │ │ +0002abb0: 242c 2074 6861 7420 6172 6520 7261 6e64 $, that are rand │ │ │ │ +0002abc0: 6f6d 6c79 2063 686f 7365 6e20 706f 696e omly chosen poin │ │ │ │ +0002abd0: 7473 206f 6e20 2449 242e 2020 6e75 6c6c ts on $I$. null │ │ │ │ +0002abe0: 2069 730a 2020 2020 2020 2020 7265 7475 is. retu │ │ │ │ +0002abf0: 726e 6564 2069 6e20 7468 6520 6361 7365 rned in the case │ │ │ │ +0002ac00: 2077 6865 6e20 7468 6520 726f 7574 696e when the routin │ │ │ │ +0002ac10: 6520 6361 6e6e 6f74 2064 6574 6572 6d69 e cannot determi │ │ │ │ +0002ac20: 6e65 2069 6620 7468 6520 7661 7269 6574 ne if the variet │ │ │ │ +0002ac30: 790a 2020 2020 2020 2020 6973 2072 6174 y. is rat │ │ │ │ +0002ac40: 696f 6e61 6c20 616e 6420 6972 7265 6475 ional and irredu │ │ │ │ +0002ac50: 6369 626c 652e 0a0a 4465 7363 7269 7074 cible...Descript │ │ │ │ +0002ac60: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ 0002ac70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002ac80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002acb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002acc0: 0a7c 6932 203a 2053 203d 206b 6b5b 612e .|i2 : S = kk[a. │ │ │ │ -0002acd0: 2e66 5d3b 2020 2020 2020 2020 2020 2020 .f]; │ │ │ │ +0002acc0: 0a7c 6931 203a 206b 6b20 3d20 5a5a 2f31 .|i1 : kk = ZZ/1 │ │ │ │ +0002acd0: 3031 3b20 2020 2020 2020 2020 2020 2020 01; │ │ │ │ 0002ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002acf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ad00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002ad10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002ad20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002ad60: 0a7c 6933 203a 2049 203d 206d 696e 6f72 .|i3 : I = minor │ │ │ │ -0002ad70: 7328 322c 2067 656e 6572 6963 5379 6d6d s(2, genericSymm │ │ │ │ -0002ad80: 6574 7269 634d 6174 7269 7828 532c 2033 etricMatrix(S, 3 │ │ │ │ -0002ad90: 2929 2020 2020 2020 2020 2020 2020 2020 )) │ │ │ │ +0002ad60: 0a7c 6932 203a 2053 203d 206b 6b5b 612e .|i2 : S = kk[a. │ │ │ │ +0002ad70: 2e66 5d3b 2020 2020 2020 2020 2020 2020 .f]; │ │ │ │ +0002ad80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ad90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ada0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002adb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002adf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002ae00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002ae10: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ae40: 2020 2020 3220 2020 2020 2020 2020 207c 2 | │ │ │ │ -0002ae50: 0a7c 6f33 203d 2069 6465 616c 2028 2d20 .|o3 = ideal (- │ │ │ │ -0002ae60: 6220 202b 2061 2a64 2c20 2d20 622a 6320 b + a*d, - b*c │ │ │ │ -0002ae70: 2b20 612a 652c 202d 2063 2a64 202b 2062 + a*e, - c*d + b │ │ │ │ -0002ae80: 2a65 2c20 2d20 622a 6320 2b20 612a 652c *e, - b*c + a*e, │ │ │ │ -0002ae90: 202d 2063 2020 2b20 612a 662c 202d 207c - c + a*f, - | │ │ │ │ -0002aea0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0002aeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002aed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0002aef0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002af00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002af10: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -0002af20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002af30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002af40: 0a7c 2020 2020 2063 2a65 202b 2062 2a66 .| c*e + b*f │ │ │ │ -0002af50: 2c20 2d20 632a 6420 2b20 622a 652c 202d , - c*d + b*e, - │ │ │ │ -0002af60: 2063 2a65 202b 2062 2a66 2c20 2d20 6520 c*e + b*f, - e │ │ │ │ -0002af70: 202b 2064 2a66 2920 2020 2020 2020 2020 + d*f) │ │ │ │ -0002af80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002adb0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002adc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002add0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ade0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002adf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002ae00: 0a7c 6933 203a 2049 203d 206d 696e 6f72 .|i3 : I = minor │ │ │ │ +0002ae10: 7328 322c 2067 656e 6572 6963 5379 6d6d s(2, genericSymm │ │ │ │ +0002ae20: 6574 7269 634d 6174 7269 7828 532c 2033 etricMatrix(S, 3 │ │ │ │ +0002ae30: 2929 2020 2020 2020 2020 2020 2020 2020 )) │ │ │ │ +0002ae40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002ae50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ae80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ae90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002aea0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002aeb0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002aec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002aed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002aee0: 2020 2020 3220 2020 2020 2020 2020 207c 2 | │ │ │ │ +0002aef0: 0a7c 6f33 203d 2069 6465 616c 2028 2d20 .|o3 = ideal (- │ │ │ │ +0002af00: 6220 202b 2061 2a64 2c20 2d20 622a 6320 b + a*d, - b*c │ │ │ │ +0002af10: 2b20 612a 652c 202d 2063 2a64 202b 2062 + a*e, - c*d + b │ │ │ │ +0002af20: 2a65 2c20 2d20 622a 6320 2b20 612a 652c *e, - b*c + a*e, │ │ │ │ +0002af30: 202d 2063 2020 2b20 612a 662c 202d 207c - c + a*f, - | │ │ │ │ +0002af40: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +0002af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ 0002af90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002afb0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002afd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002afe0: 0a7c 6f33 203a 2049 6465 616c 206f 6620 .|o3 : Ideal of │ │ │ │ -0002aff0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ -0002b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002afe0: 0a7c 2020 2020 2063 2a65 202b 2062 2a66 .| c*e + b*f │ │ │ │ +0002aff0: 2c20 2d20 632a 6420 2b20 622a 652c 202d , - c*d + b*e, - │ │ │ │ +0002b000: 2063 2a65 202b 2062 2a66 2c20 2d20 6520 c*e + b*f, - e │ │ │ │ +0002b010: 202b 2064 2a66 2920 2020 2020 2020 2020 + d*f) │ │ │ │ 0002b020: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b030: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002b080: 0a7c 6934 203a 2070 7473 203d 2072 616e .|i4 : pts = ran │ │ │ │ -0002b090: 646f 6d50 6f69 6e74 734f 6e52 6174 696f domPointsOnRatio │ │ │ │ -0002b0a0: 6e61 6c56 6172 6965 7479 2849 2c20 3429 nalVariety(I, 4) │ │ │ │ +0002b030: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b070: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b080: 0a7c 6f33 203a 2049 6465 616c 206f 6620 .|o3 : Ideal of │ │ │ │ +0002b090: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +0002b0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b0c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b0d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002b0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b110: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b120: 0a7c 6f34 203d 207b 7c20 2d32 3520 3230 .|o4 = {| -25 20 │ │ │ │ -0002b130: 202d 3330 202d 3136 2032 3420 2d33 3620 -30 -16 24 -36 │ │ │ │ -0002b140: 7c2c 207c 2031 3920 2d32 3920 3139 2032 |, | 19 -29 19 2 │ │ │ │ -0002b150: 3320 2d32 3920 3139 207c 2c20 7c20 2d34 3 -29 19 |, | -4 │ │ │ │ -0002b160: 3420 3436 202d 3820 3720 2d31 3020 207c 4 46 -8 7 -10 | │ │ │ │ -0002b170: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0002b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0002b1c0: 0a7c 2020 2020 202d 3239 207c 2c20 7c20 .| -29 |, | │ │ │ │ -0002b1d0: 3820 3431 202d 3234 2034 3620 2d32 3220 8 41 -24 46 -22 │ │ │ │ -0002b1e0: 2d32 3920 7c7d 2020 2020 2020 2020 2020 -29 |} │ │ │ │ -0002b1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b200: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b210: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b250: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b260: 0a7c 6f34 203a 204c 6973 7420 2020 2020 .|o4 : List │ │ │ │ -0002b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b0d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002b120: 0a7c 6934 203a 2070 7473 203d 2072 616e .|i4 : pts = ran │ │ │ │ +0002b130: 646f 6d50 6f69 6e74 734f 6e52 6174 696f domPointsOnRatio │ │ │ │ +0002b140: 6e61 6c56 6172 6965 7479 2849 2c20 3429 nalVariety(I, 4) │ │ │ │ +0002b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b160: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b170: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b1b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b1c0: 0a7c 6f34 203d 207b 7c20 2d32 3520 3230 .|o4 = {| -25 20 │ │ │ │ +0002b1d0: 202d 3330 202d 3136 2032 3420 2d33 3620 -30 -16 24 -36 │ │ │ │ +0002b1e0: 7c2c 207c 2031 3920 2d32 3920 3139 2032 |, | 19 -29 19 2 │ │ │ │ +0002b1f0: 3320 2d32 3920 3139 207c 2c20 7c20 2d34 3 -29 19 |, | -4 │ │ │ │ +0002b200: 3420 3436 202d 3820 3720 2d31 3020 207c 4 46 -8 7 -10 | │ │ │ │ +0002b210: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +0002b220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0002b260: 0a7c 2020 2020 202d 3239 207c 2c20 7c20 .| -29 |, | │ │ │ │ +0002b270: 3820 3431 202d 3234 2034 3620 2d32 3220 8 41 -24 46 -22 │ │ │ │ +0002b280: 2d32 3920 7c7d 2020 2020 2020 2020 2020 -29 |} │ │ │ │ 0002b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b2a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b2b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002b2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002b300: 0a7c 6935 203a 2066 6f72 2070 2069 6e20 .|i5 : for p in │ │ │ │ -0002b310: 7074 7320 6c69 7374 2073 7562 2849 2c20 pts list sub(I, │ │ │ │ -0002b320: 7029 203d 3d20 3020 2020 2020 2020 2020 p) == 0 │ │ │ │ +0002b2b0: 0a7c 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 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b300: 0a7c 6f34 203a 204c 6973 7420 2020 2020 .|o4 : List │ │ │ │ +0002b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b340: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b350: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b390: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b3a0: 0a7c 6f35 203d 207b 7472 7565 2c20 7472 .|o5 = {true, tr │ │ │ │ -0002b3b0: 7565 2c20 7472 7565 2c20 7472 7565 7d20 ue, true, true} │ │ │ │ -0002b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b350: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002b3a0: 0a7c 6935 203a 2066 6f72 2070 2069 6e20 .|i5 : for p in │ │ │ │ +0002b3b0: 7074 7320 6c69 7374 2073 7562 2849 2c20 pts list sub(I, │ │ │ │ +0002b3c0: 7029 203d 3d20 3020 2020 2020 2020 2020 p) == 0 │ │ │ │ 0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b3e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002b3f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b430: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b440: 0a7c 6f35 203a 204c 6973 7420 2020 2020 .|o5 : List │ │ │ │ -0002b450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b440: 0a7c 6f35 203d 207b 7472 7565 2c20 7472 .|o5 = {true, tr │ │ │ │ +0002b450: 7565 2c20 7472 7565 2c20 7472 7565 7d20 ue, true, true} │ │ │ │ 0002b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b480: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b490: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002b4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002b4e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002b530: 0a7c 6936 203a 2053 203d 206b 6b5b 612e .|i6 : S = kk[a. │ │ │ │ -0002b540: 2e64 5d3b 2020 2020 2020 2020 2020 2020 .d]; │ │ │ │ -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 207c | │ │ │ │ +0002b490: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b4d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b4e0: 0a7c 6f35 203a 204c 6973 7420 2020 2020 .|o5 : List │ │ │ │ +0002b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b520: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b530: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0002b580: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002b590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002b5d0: 0a7c 6937 203a 2046 203d 2067 726f 6562 .|i7 : F = groeb │ │ │ │ -0002b5e0: 6e65 7246 616d 696c 7920 6964 6561 6c22 nerFamily ideal" │ │ │ │ -0002b5f0: 6132 2c61 622c 6163 2c62 3222 2020 2020 a2,ab,ac,b2" │ │ │ │ +0002b5d0: 0a7c 6936 203a 2053 203d 206b 6b5b 612e .|i6 : S = kk[a. │ │ │ │ +0002b5e0: 2e64 5d3b 2020 2020 2020 2020 2020 2020 .d]; │ │ │ │ +0002b5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b610: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b620: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002b630: 2020 2020 2020 2020 2020 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 207c | │ │ │ │ -0002b670: 0a7c 2020 2020 2020 2020 2020 2020 2032 .| 2 │ │ │ │ -0002b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b690: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0002b6a0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +0002b620: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002b670: 0a7c 6937 203a 2046 203d 2067 726f 6562 .|i7 : F = groeb │ │ │ │ +0002b680: 6e65 7246 616d 696c 7920 6964 6561 6c22 nerFamily ideal" │ │ │ │ +0002b690: 6132 2c61 622c 6163 2c62 3222 2020 2020 a2,ab,ac,b2" │ │ │ │ +0002b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b6b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b6c0: 0a7c 6f37 203d 2069 6465 616c 2028 6120 .|o7 = ideal (a │ │ │ │ -0002b6d0: 202b 2074 2062 2a63 202b 2074 2061 2a64 + t b*c + t a*d │ │ │ │ -0002b6e0: 202b 2074 2063 2020 2b20 7420 622a 6420 + t c + t b*d │ │ │ │ -0002b6f0: 2b20 7420 632a 6420 2b20 7420 6420 2c20 + t c*d + t d , │ │ │ │ -0002b700: 612a 6220 2b20 7420 622a 6320 2b20 207c a*b + t b*c + | │ │ │ │ -0002b710: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002b720: 2020 2020 3120 2020 2020 2020 3320 2020 1 3 │ │ │ │ -0002b730: 2020 2020 3220 2020 2020 2034 2020 2020 2 4 │ │ │ │ -0002b740: 2020 2035 2020 2020 2020 2036 2020 2020 5 6 │ │ │ │ -0002b750: 2020 2020 2020 2037 2020 2020 2020 207c 7 | │ │ │ │ -0002b760: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0002b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0002b6c0: 0a7c 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 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b710: 0a7c 2020 2020 2020 2020 2020 2020 2032 .| 2 │ │ │ │ +0002b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b730: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +0002b740: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +0002b750: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b760: 0a7c 6f37 203d 2069 6465 616c 2028 6120 .|o7 = ideal (a │ │ │ │ +0002b770: 202b 2074 2062 2a63 202b 2074 2061 2a64 + t b*c + t a*d │ │ │ │ +0002b780: 202b 2074 2063 2020 2b20 7420 622a 6420 + t c + t b*d │ │ │ │ +0002b790: 2b20 7420 632a 6420 2b20 7420 6420 2c20 + t c*d + t d , │ │ │ │ +0002b7a0: 612a 6220 2b20 7420 622a 6320 2b20 207c a*b + t b*c + | │ │ │ │ 0002b7b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002b7c0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b7d0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -0002b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b7f0: 2020 2020 2020 2020 2020 2032 2020 207c 2 | │ │ │ │ -0002b800: 0a7c 2020 2020 2074 2061 2a64 202b 2074 .| t a*d + t │ │ │ │ -0002b810: 2063 2020 2b20 7420 2062 2a64 202b 2074 c + t b*d + t │ │ │ │ -0002b820: 2020 632a 6420 2b20 7420 2064 202c 2061 c*d + t d , a │ │ │ │ -0002b830: 2a63 202b 2074 2020 622a 6320 2b20 7420 *c + t b*c + t │ │ │ │ -0002b840: 2061 2a64 202b 2074 2020 6320 202b 207c a*d + t c + | │ │ │ │ -0002b850: 0a7c 2020 2020 2020 3920 2020 2020 2020 .| 9 │ │ │ │ -0002b860: 3820 2020 2020 2031 3020 2020 2020 2020 8 10 │ │ │ │ -0002b870: 3131 2020 2020 2020 2031 3220 2020 2020 11 12 │ │ │ │ -0002b880: 2020 2020 2020 3133 2020 2020 2020 2031 13 1 │ │ │ │ -0002b890: 3520 2020 2020 2020 3134 2020 2020 207c 5 14 | │ │ │ │ -0002b8a0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0002b8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0002b8f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002b900: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -0002b910: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b920: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -0002b930: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b940: 0a7c 2020 2020 2074 2020 622a 6420 2b20 .| t b*d + │ │ │ │ -0002b950: 7420 2063 2a64 202b 2074 2020 6420 2c20 t c*d + t d , │ │ │ │ -0002b960: 6220 202b 2074 2020 622a 6320 2b20 7420 b + t b*c + t │ │ │ │ -0002b970: 2061 2a64 202b 2074 2020 6320 202b 2074 a*d + t c + t │ │ │ │ -0002b980: 2020 622a 6420 2b20 7420 2063 2a64 207c b*d + t c*d | │ │ │ │ -0002b990: 0a7c 2020 2020 2020 3136 2020 2020 2020 .| 16 │ │ │ │ -0002b9a0: 2031 3720 2020 2020 2020 3138 2020 2020 17 18 │ │ │ │ -0002b9b0: 2020 2020 2020 3139 2020 2020 2020 2032 19 2 │ │ │ │ -0002b9c0: 3120 2020 2020 2020 3230 2020 2020 2020 1 20 │ │ │ │ -0002b9d0: 3232 2020 2020 2020 2032 3320 2020 207c 22 23 | │ │ │ │ -0002b9e0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0002b9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ba00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ba10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ba20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0002ba30: 0a7c 2020 2020 2020 2020 2020 2032 2020 .| 2 │ │ │ │ -0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002ba80: 0a7c 2020 2020 202b 2074 2020 6420 2920 .| + t d ) │ │ │ │ -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 207c | │ │ │ │ -0002bad0: 0a7c 2020 2020 2020 2020 3234 2020 2020 .| 24 │ │ │ │ +0002b7c0: 2020 2020 3120 2020 2020 2020 3320 2020 1 3 │ │ │ │ +0002b7d0: 2020 2020 3220 2020 2020 2034 2020 2020 2 4 │ │ │ │ +0002b7e0: 2020 2035 2020 2020 2020 2036 2020 2020 5 6 │ │ │ │ +0002b7f0: 2020 2020 2020 2037 2020 2020 2020 207c 7 | │ │ │ │ +0002b800: 0a7c 2020 2020 202d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0002b850: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002b860: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002b870: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +0002b880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b890: 2020 2020 2020 2020 2020 2032 2020 207c 2 | │ │ │ │ +0002b8a0: 0a7c 2020 2020 2074 2061 2a64 202b 2074 .| t a*d + t │ │ │ │ +0002b8b0: 2063 2020 2b20 7420 2062 2a64 202b 2074 c + t b*d + t │ │ │ │ +0002b8c0: 2020 632a 6420 2b20 7420 2064 202c 2061 c*d + t d , a │ │ │ │ +0002b8d0: 2a63 202b 2074 2020 622a 6320 2b20 7420 *c + t b*c + t │ │ │ │ +0002b8e0: 2061 2a64 202b 2074 2020 6320 202b 207c a*d + t c + | │ │ │ │ +0002b8f0: 0a7c 2020 2020 2020 3920 2020 2020 2020 .| 9 │ │ │ │ +0002b900: 3820 2020 2020 2031 3020 2020 2020 2020 8 10 │ │ │ │ +0002b910: 3131 2020 2020 2020 2031 3220 2020 2020 11 12 │ │ │ │ +0002b920: 2020 2020 2020 3133 2020 2020 2020 2031 13 1 │ │ │ │ +0002b930: 3520 2020 2020 2020 3134 2020 2020 207c 5 14 | │ │ │ │ +0002b940: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +0002b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0002b990: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002b9a0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +0002b9b0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002b9c0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0002b9d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b9e0: 0a7c 2020 2020 2074 2020 622a 6420 2b20 .| t b*d + │ │ │ │ +0002b9f0: 7420 2063 2a64 202b 2074 2020 6420 2c20 t c*d + t d , │ │ │ │ +0002ba00: 6220 202b 2074 2020 622a 6320 2b20 7420 b + t b*c + t │ │ │ │ +0002ba10: 2061 2a64 202b 2074 2020 6320 202b 2074 a*d + t c + t │ │ │ │ +0002ba20: 2020 622a 6420 2b20 7420 2063 2a64 207c b*d + t c*d | │ │ │ │ +0002ba30: 0a7c 2020 2020 2020 3136 2020 2020 2020 .| 16 │ │ │ │ +0002ba40: 2031 3720 2020 2020 2020 3138 2020 2020 17 18 │ │ │ │ +0002ba50: 2020 2020 2020 3139 2020 2020 2020 2032 19 2 │ │ │ │ +0002ba60: 3120 2020 2020 2020 3230 2020 2020 2020 1 20 │ │ │ │ +0002ba70: 3232 2020 2020 2020 2032 3320 2020 207c 22 23 | │ │ │ │ +0002ba80: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +0002ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002baa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0002bad0: 0a7c 2020 2020 2020 2020 2020 2032 2020 .| 2 │ │ │ │ 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 207c | │ │ │ │ -0002bb20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002bb20: 0a7c 2020 2020 202b 2074 2020 6420 2920 .| + t d ) │ │ │ │ 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 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bb70: 0a7c 6f37 203a 2049 6465 616c 206f 6620 .|o7 : Ideal of │ │ │ │ -0002bb80: 6b6b 5b74 202c 2074 202c 2074 2020 2c20 kk[t , t , t , │ │ │ │ -0002bb90: 7420 2c20 7420 2c20 7420 202c 2074 2020 t , t , t , t │ │ │ │ -0002bba0: 2c20 7420 202c 2074 202c 2074 202c 2074 , t , t , t , t │ │ │ │ -0002bbb0: 202c 2074 2020 2c20 7420 202c 2074 207c , t , t , t | │ │ │ │ +0002bb70: 0a7c 2020 2020 2020 2020 3234 2020 2020 .| 24 │ │ │ │ +0002bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bbb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002bbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002bbd0: 2020 2020 3620 2020 3520 2020 3132 2020 6 5 12 │ │ │ │ -0002bbe0: 2032 2020 2034 2020 2031 3120 2020 3138 2 4 11 18 │ │ │ │ -0002bbf0: 2020 2032 3420 2020 3120 2020 3320 2020 24 1 3 │ │ │ │ -0002bc00: 3820 2020 3130 2020 2031 3720 2020 327c 8 10 17 2| │ │ │ │ -0002bc10: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -0002bc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0002bc60: 0a7c 202c 2074 202c 2074 202c 2074 2020 .| , t , t , t │ │ │ │ -0002bc70: 2c20 7420 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -0002bc80: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ -0002bc90: 7420 205d 5b61 2e2e 645d 2020 2020 2020 t ][a..d] │ │ │ │ -0002bca0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bcb0: 0a7c 3320 2020 3720 2020 3920 2020 3134 .|3 7 9 14 │ │ │ │ -0002bcc0: 2020 2031 3620 2020 3230 2020 2032 3220 16 20 22 │ │ │ │ -0002bcd0: 2020 3133 2020 2031 3520 2020 3139 2020 13 15 19 │ │ │ │ -0002bce0: 2032 3120 2020 2020 2020 2020 2020 2020 21 │ │ │ │ -0002bcf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bd00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002bd50: 0a7c 6938 203a 204a 203d 2067 726f 6562 .|i8 : J = groeb │ │ │ │ -0002bd60: 6e65 7253 7472 6174 756d 2046 3b20 2020 nerStratum F; │ │ │ │ -0002bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 207c | │ │ │ │ +0002bc10: 0a7c 6f37 203a 2049 6465 616c 206f 6620 .|o7 : Ideal of │ │ │ │ +0002bc20: 6b6b 5b74 202c 2074 202c 2074 2020 2c20 kk[t , t , t , │ │ │ │ +0002bc30: 7420 2c20 7420 2c20 7420 202c 2074 2020 t , t , t , t │ │ │ │ +0002bc40: 2c20 7420 202c 2074 202c 2074 202c 2074 , t , t , t , t │ │ │ │ +0002bc50: 202c 2074 2020 2c20 7420 202c 2074 207c , t , t , t | │ │ │ │ +0002bc60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002bc70: 2020 2020 3620 2020 3520 2020 3132 2020 6 5 12 │ │ │ │ +0002bc80: 2032 2020 2034 2020 2031 3120 2020 3138 2 4 11 18 │ │ │ │ +0002bc90: 2020 2032 3420 2020 3120 2020 3320 2020 24 1 3 │ │ │ │ +0002bca0: 3820 2020 3130 2020 2031 3720 2020 327c 8 10 17 2| │ │ │ │ +0002bcb0: 0a7c 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0002bd00: 0a7c 202c 2074 202c 2074 202c 2074 2020 .| , t , t , t │ │ │ │ +0002bd10: 2c20 7420 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ +0002bd20: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ +0002bd30: 7420 205d 5b61 2e2e 645d 2020 2020 2020 t ][a..d] │ │ │ │ +0002bd40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002bd50: 0a7c 3320 2020 3720 2020 3920 2020 3134 .|3 7 9 14 │ │ │ │ +0002bd60: 2020 2031 3620 2020 3230 2020 2032 3220 16 20 22 │ │ │ │ +0002bd70: 2020 3133 2020 2031 3520 2020 3139 2020 13 15 19 │ │ │ │ +0002bd80: 2032 3120 2020 2020 2020 2020 2020 2020 21 │ │ │ │ 0002bd90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bda0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bde0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bdf0: 0a7c 6f38 203a 2049 6465 616c 206f 6620 .|o8 : Ideal of │ │ │ │ -0002be00: 6b6b 5b74 202c 2074 202c 2074 2020 2c20 kk[t , t , t , │ │ │ │ -0002be10: 7420 2c20 7420 2c20 7420 202c 2074 2020 t , t , t , t │ │ │ │ -0002be20: 2c20 7420 202c 2074 202c 2074 202c 2074 , t , t , t , t │ │ │ │ -0002be30: 202c 2074 2020 2c20 7420 202c 2020 207c , t , t , | │ │ │ │ +0002bda0: 0a2b 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 2d2d ---------------- │ │ │ │ +0002bde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002bdf0: 0a7c 6938 203a 204a 203d 2067 726f 6562 .|i8 : J = groeb │ │ │ │ +0002be00: 6e65 7253 7472 6174 756d 2046 3b20 2020 nerStratum F; │ │ │ │ +0002be10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002be40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002be50: 2020 2020 3620 2020 3520 2020 3132 2020 6 5 12 │ │ │ │ -0002be60: 2032 2020 2034 2020 2031 3120 2020 3138 2 4 11 18 │ │ │ │ -0002be70: 2020 2032 3420 2020 3120 2020 3320 2020 24 1 3 │ │ │ │ -0002be80: 3820 2020 3130 2020 2031 3720 2020 207c 8 10 17 | │ │ │ │ -0002be90: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -0002bea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002beb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0002bee0: 0a7c 7420 202c 2074 202c 2074 202c 2074 .|t , t , t , t │ │ │ │ -0002bef0: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ -0002bf00: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ -0002bf10: 2c20 7420 205d 2020 2020 2020 2020 2020 , t ] │ │ │ │ -0002bf20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bf30: 0a7c 2032 3320 2020 3720 2020 3920 2020 .| 23 7 9 │ │ │ │ -0002bf40: 3134 2020 2031 3620 2020 3230 2020 2032 14 16 20 2 │ │ │ │ -0002bf50: 3220 2020 3133 2020 2031 3520 2020 3139 2 13 15 19 │ │ │ │ -0002bf60: 2020 2032 3120 2020 2020 2020 2020 2020 21 │ │ │ │ -0002bf70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bf80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002bf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002bfd0: 0a7c 6939 203a 2063 6f6d 7073 4a20 3d20 .|i9 : compsJ = │ │ │ │ -0002bfe0: 6465 636f 6d70 6f73 6520 4a3b 2020 2020 decompose J; │ │ │ │ -0002bff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002be90: 0a7c 6f38 203a 2049 6465 616c 206f 6620 .|o8 : Ideal of │ │ │ │ +0002bea0: 6b6b 5b74 202c 2074 202c 2074 2020 2c20 kk[t , t , t , │ │ │ │ +0002beb0: 7420 2c20 7420 2c20 7420 202c 2074 2020 t , t , t , t │ │ │ │ +0002bec0: 2c20 7420 202c 2074 202c 2074 202c 2074 , t , t , t , t │ │ │ │ +0002bed0: 202c 2074 2020 2c20 7420 202c 2020 207c , t , t , | │ │ │ │ +0002bee0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002bef0: 2020 2020 3620 2020 3520 2020 3132 2020 6 5 12 │ │ │ │ +0002bf00: 2032 2020 2034 2020 2031 3120 2020 3138 2 4 11 18 │ │ │ │ +0002bf10: 2020 2032 3420 2020 3120 2020 3320 2020 24 1 3 │ │ │ │ +0002bf20: 3820 2020 3130 2020 2031 3720 2020 207c 8 10 17 | │ │ │ │ +0002bf30: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ +0002bf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bf60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0002bf80: 0a7c 7420 202c 2074 202c 2074 202c 2074 .|t , t , t , t │ │ │ │ +0002bf90: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ +0002bfa0: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ +0002bfb0: 2c20 7420 205d 2020 2020 2020 2020 2020 , t ] │ │ │ │ +0002bfc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002bfd0: 0a7c 2032 3320 2020 3720 2020 3920 2020 .| 23 7 9 │ │ │ │ +0002bfe0: 3134 2020 2031 3620 2020 3230 2020 2032 14 16 20 2 │ │ │ │ +0002bff0: 3220 2020 3133 2020 2031 3520 2020 3139 2 13 15 19 │ │ │ │ +0002c000: 2020 2032 3120 2020 2020 2020 2020 2020 21 │ │ │ │ 0002c010: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002c020: 0a2b 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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002c070: 0a7c 6931 3020 3a20 636f 6d70 734a 203d .|i10 : compsJ = │ │ │ │ -0002c080: 2063 6f6d 7073 4a2f 7472 696d 3b20 2020 compsJ/trim; │ │ │ │ +0002c070: 0a7c 6939 203a 2063 6f6d 7073 4a20 3d20 .|i9 : compsJ = │ │ │ │ +0002c080: 6465 636f 6d70 6f73 6520 4a3b 2020 2020 decompose J; │ │ │ │ 0002c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c0b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002c0c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002c110: 0a7c 6931 3120 3a20 2363 6f6d 7073 4a20 .|i11 : #compsJ │ │ │ │ -0002c120: 3d3d 2032 2020 2020 2020 2020 2020 2020 == 2 │ │ │ │ +0002c110: 0a7c 6931 3020 3a20 636f 6d70 734a 203d .|i10 : compsJ = │ │ │ │ +0002c120: 2063 6f6d 7073 4a2f 7472 696d 3b20 2020 compsJ/trim; │ │ │ │ 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 207c | │ │ │ │ -0002c160: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002c170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c1a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c1b0: 0a7c 6f31 3120 3d20 7472 7565 2020 2020 .|o11 = true │ │ │ │ -0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c160: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002c170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002c1b0: 0a7c 6931 3120 3a20 2363 6f6d 7073 4a20 .|i11 : #compsJ │ │ │ │ +0002c1c0: 3d3d 2032 2020 2020 2020 2020 2020 2020 == 2 │ │ │ │ 0002c1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c1f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c200: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002c210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002c250: 0a7c 6931 3220 3a20 636f 6d70 734a 2f64 .|i12 : compsJ/d │ │ │ │ -0002c260: 696d 2020 2020 2020 2020 2020 2020 2020 im │ │ │ │ +0002c200: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002c210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c240: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002c250: 0a7c 6f31 3120 3d20 7472 7565 2020 2020 .|o11 = true │ │ │ │ +0002c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c290: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c2a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002c2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c2f0: 0a7c 6f31 3220 3d20 7b31 312c 2038 7d20 .|o12 = {11, 8} │ │ │ │ -0002c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c2a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002c2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002c2f0: 0a7c 6931 3220 3a20 636f 6d70 734a 2f64 .|i12 : compsJ/d │ │ │ │ +0002c300: 696d 2020 2020 2020 2020 2020 2020 2020 im │ │ │ │ 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 207c | │ │ │ │ 0002c340: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c380: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c390: 0a7c 6f31 3220 3a20 4c69 7374 2020 2020 .|o12 : List │ │ │ │ +0002c390: 0a7c 6f31 3220 3d20 7b31 312c 2038 7d20 .|o12 = {11, 8} │ │ │ │ 0002c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c3d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c3e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0002c3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002c430: 0a0a 5468 6572 6520 6172 6520 3220 636f ..There are 2 co │ │ │ │ -0002c440: 6d70 6f6e 656e 7473 2e20 2057 6520 6174 mponents. We at │ │ │ │ -0002c450: 7465 6d70 7420 746f 2066 696e 6420 706f tempt to find po │ │ │ │ -0002c460: 696e 7473 206f 6e20 6561 6368 206f 6620 ints on each of │ │ │ │ -0002c470: 7468 6573 6520 7477 6f0a 636f 6d70 6f6e these two.compon │ │ │ │ -0002c480: 656e 7473 2e20 5765 2061 7265 2073 7563 ents. We are suc │ │ │ │ -0002c490: 6365 7373 6675 6c2e 2020 5468 6973 2069 cessful. This i │ │ │ │ -0002c4a0: 6e64 6963 6174 6573 2074 6861 7420 7468 ndicates that th │ │ │ │ -0002c4b0: 6520 636f 7272 6573 706f 6e64 696e 6720 e corresponding │ │ │ │ -0002c4c0: 7661 7269 6574 6965 730a 6172 6520 626f varieties.are bo │ │ │ │ -0002c4d0: 7468 2072 6174 696f 6e61 6c2e 2041 6c73 th rational. Als │ │ │ │ -0002c4e0: 6f2c 2069 6620 7765 2063 616e 2066 696e o, if we can fin │ │ │ │ -0002c4f0: 6420 6f6e 6520 706f 696e 742c 2077 6520 d one point, we │ │ │ │ -0002c500: 6361 6e20 6669 6e64 2061 7320 6d61 6e79 can find as many │ │ │ │ -0002c510: 2061 7320 7765 0a77 616e 742e 0a0a 2b2d as we.want...+- │ │ │ │ -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 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002c570: 3133 203a 206e 6574 4c69 7374 2072 616e 13 : netList ran │ │ │ │ -0002c580: 646f 6d50 6f69 6e74 734f 6e52 6174 696f domPointsOnRatio │ │ │ │ -0002c590: 6e61 6c56 6172 6965 7479 2863 6f6d 7073 nalVariety(comps │ │ │ │ -0002c5a0: 4a5f 302c 2031 3029 2020 2020 2020 2020 J_0, 10) │ │ │ │ -0002c5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002c5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c600: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002c610: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ -0002c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6f ------------|.|o │ │ │ │ -0002c660: 3133 203d 207c 7c20 3239 202d 3430 2031 13 = || 29 -40 1 │ │ │ │ -0002c670: 3520 2d34 3920 3320 2d31 3320 2d36 202d 5 -49 3 -13 -6 - │ │ │ │ -0002c680: 3339 2032 2033 3920 3437 2031 3520 3139 39 2 39 47 15 19 │ │ │ │ -0002c690: 202d 3437 202d 3436 202d 3339 202d 3136 -47 -46 -39 -16 │ │ │ │ -0002c6a0: 2033 3220 2d34 3320 3334 202d 7c0a 7c20 32 -43 34 -|.| │ │ │ │ +0002c3e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002c3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c420: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002c430: 0a7c 6f31 3220 3a20 4c69 7374 2020 2020 .|o12 : List │ │ │ │ +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 207c | │ │ │ │ +0002c480: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002c490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002c4d0: 0a0a 5468 6572 6520 6172 6520 3220 636f ..There are 2 co │ │ │ │ +0002c4e0: 6d70 6f6e 656e 7473 2e20 2057 6520 6174 mponents. We at │ │ │ │ +0002c4f0: 7465 6d70 7420 746f 2066 696e 6420 706f tempt to find po │ │ │ │ +0002c500: 696e 7473 206f 6e20 6561 6368 206f 6620 ints on each of │ │ │ │ +0002c510: 7468 6573 6520 7477 6f0a 636f 6d70 6f6e these two.compon │ │ │ │ +0002c520: 656e 7473 2e20 5765 2061 7265 2073 7563 ents. We are suc │ │ │ │ +0002c530: 6365 7373 6675 6c2e 2020 5468 6973 2069 cessful. This i │ │ │ │ +0002c540: 6e64 6963 6174 6573 2074 6861 7420 7468 ndicates that th │ │ │ │ +0002c550: 6520 636f 7272 6573 706f 6e64 696e 6720 e corresponding │ │ │ │ +0002c560: 7661 7269 6574 6965 730a 6172 6520 626f varieties.are bo │ │ │ │ +0002c570: 7468 2072 6174 696f 6e61 6c2e 2041 6c73 th rational. Als │ │ │ │ +0002c580: 6f2c 2069 6620 7765 2063 616e 2066 696e o, if we can fin │ │ │ │ +0002c590: 6420 6f6e 6520 706f 696e 742c 2077 6520 d one point, we │ │ │ │ +0002c5a0: 6361 6e20 6669 6e64 2061 7320 6d61 6e79 can find as many │ │ │ │ +0002c5b0: 2061 7320 7765 0a77 616e 742e 0a0a 2b2d as we.want...+- │ │ │ │ +0002c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002c610: 3133 203a 206e 6574 4c69 7374 2072 616e 13 : netList ran │ │ │ │ +0002c620: 646f 6d50 6f69 6e74 734f 6e52 6174 696f domPointsOnRatio │ │ │ │ +0002c630: 6e61 6c56 6172 6965 7479 2863 6f6d 7073 nalVariety(comps │ │ │ │ +0002c640: 4a5f 302c 2031 3029 2020 2020 2020 2020 J_0, 10) │ │ │ │ +0002c650: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0002c6b0: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ 0002c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ -0002c700: 2020 2020 207c 7c20 3337 202d 3720 2d32 || 37 -7 -2 │ │ │ │ -0002c710: 3420 3820 2d32 3620 3338 2039 202d 3331 4 8 -26 38 9 -31 │ │ │ │ -0002c720: 2032 3420 2d34 3720 2d33 3420 3132 2031 24 -47 -34 12 1 │ │ │ │ -0002c730: 3620 3232 202d 3232 2034 3520 2d32 3820 6 22 -22 45 -28 │ │ │ │ -0002c740: 3136 202d 3437 2032 202d 3438 7c0a 7c20 16 -47 2 -48|.| │ │ │ │ +0002c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6f ------------|.|o │ │ │ │ +0002c700: 3133 203d 207c 7c20 3133 2031 3520 3320 13 = || 13 15 3 │ │ │ │ +0002c710: 3336 2032 2034 3820 3434 202d 3335 202d 36 2 48 44 -35 - │ │ │ │ +0002c720: 3334 2033 3920 3520 2d33 3220 3334 2031 34 39 5 -32 34 1 │ │ │ │ +0002c730: 3920 2d34 3220 2d34 3720 2d31 3620 2d33 9 -42 -47 -16 -3 │ │ │ │ +0002c740: 3420 2d33 3920 2d31 3320 2d31 7c0a 7c20 4 -39 -13 -1|.| │ │ │ │ 0002c750: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ 0002c760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ -0002c7a0: 2020 2020 207c 7c20 3620 3120 2d33 3120 || 6 1 -31 │ │ │ │ -0002c7b0: 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0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0002e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 ------------+..C │ │ │ │ -0002e230: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 5468 aveat.======..Th │ │ │ │ -0002e240: 6973 2072 6f75 7469 6e65 2065 7870 6563 is routine expec │ │ │ │ -0002e250: 7473 2074 6865 2069 6e70 7574 2074 6f20 ts the input to │ │ │ │ -0002e260: 7265 7072 6573 656e 7420 616e 2069 7272 represent an irr │ │ │ │ -0002e270: 6564 7563 6962 6c65 2076 6172 6965 7479 educible variety │ │ │ │ -0002e280: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -0002e290: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2072 ===.. * *note r │ │ │ │ -0002e2a0: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ -0002e2b0: 6f6e 616c 5661 7269 6574 7928 4964 6561 onalVariety(Idea │ │ │ │ -0002e2c0: 6c29 3a0a 2020 2020 7261 6e64 6f6d 506f l):. randomPo │ │ │ │ -0002e2d0: 696e 744f 6e52 6174 696f 6e61 6c56 6172 intOnRationalVar │ │ │ │ -0002e2e0: 6965 7479 5f6c 7049 6465 616c 5f72 702c iety_lpIdeal_rp, │ │ │ │ -0002e2f0: 202d 2d20 6669 6e64 2061 2072 616e 646f -- find a rando │ │ │ │ -0002e300: 6d20 706f 696e 7420 6f6e 2061 0a20 2020 m point on a. │ │ │ │ -0002e310: 2076 6172 6965 7479 2074 6861 7420 6361 variety that ca │ │ │ │ -0002e320: 6e20 6265 2064 6574 6563 7465 6420 746f n be detected to │ │ │ │ -0002e330: 2062 6520 7261 7469 6f6e 616c 0a0a 5761 be rational..Wa │ │ │ │ -0002e340: 7973 2074 6f20 7573 6520 7468 6973 206d ys to use this m │ │ │ │ -0002e350: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d ethod:.========= │ │ │ │ -0002e360: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0002e370: 0a20 202a 202a 6e6f 7465 2072 616e 646f . * *note rando │ │ │ │ -0002e380: 6d50 6f69 6e74 734f 6e52 6174 696f 6e61 mPointsOnRationa │ │ │ │ -0002e390: 6c56 6172 6965 7479 2849 6465 616c 2c5a lVariety(Ideal,Z │ │ │ │ -0002e3a0: 5a29 3a0a 2020 2020 7261 6e64 6f6d 506f Z):. randomPo │ │ │ │ -0002e3b0: 696e 7473 4f6e 5261 7469 6f6e 616c 5661 intsOnRationalVa │ │ │ │ -0002e3c0: 7269 6574 795f 6c70 4964 6561 6c5f 636d riety_lpIdeal_cm │ │ │ │ -0002e3d0: 5a5a 5f72 702c 202d 2d20 6669 6e64 2072 ZZ_rp, -- find r │ │ │ │ -0002e3e0: 616e 646f 6d20 706f 696e 7473 206f 6e20 andom points on │ │ │ │ -0002e3f0: 610a 2020 2020 7661 7269 6574 7920 7468 a. variety th │ │ │ │ -0002e400: 6174 2063 616e 2062 6520 6465 7465 6374 at can be detect │ │ │ │ -0002e410: 6564 2074 6f20 6265 2072 6174 696f 6e61 ed to be rationa │ │ │ │ -0002e420: 6c0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d l.-------------- │ │ │ │ -0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e470: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0002e480: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0002e490: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0002e4a0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0002e4b0: 6361 756c 6179 322d 312e 3235 2e31 312b caulay2-1.25.11+ │ │ │ │ -0002e4c0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0002e4d0: 7061 636b 6167 6573 2f0a 4772 6f65 626e packages/.Groebn │ │ │ │ -0002e4e0: 6572 5374 7261 7461 2e6d 323a 3838 333a erStrata.m2:883: │ │ │ │ -0002e4f0: 302e 0a1f 0a46 696c 653a 2047 726f 6562 0....File: Groeb │ │ │ │ -0002e500: 6e65 7253 7472 6174 612e 696e 666f 2c20 nerStrata.info, │ │ │ │ -0002e510: 4e6f 6465 3a20 736d 616c 6c65 724d 6f6e Node: smallerMon │ │ │ │ -0002e520: 6f6d 6961 6c73 2c20 4e65 7874 3a20 7374 omials, Next: st │ │ │ │ -0002e530: 616e 6461 7264 4d6f 6e6f 6d69 616c 732c andardMonomials, │ │ │ │ -0002e540: 2050 7265 763a 2072 616e 646f 6d50 6f69 Prev: randomPoi │ │ │ │ -0002e550: 6e74 734f 6e52 6174 696f 6e61 6c56 6172 ntsOnRationalVar │ │ │ │ -0002e560: 6965 7479 5f6c 7049 6465 616c 5f63 6d5a iety_lpIdeal_cmZ │ │ │ │ -0002e570: 5a5f 7270 2c20 5570 3a20 546f 700a 0a73 Z_rp, Up: Top..s │ │ │ │ -0002e580: 6d61 6c6c 6572 4d6f 6e6f 6d69 616c 7320 mallerMonomials │ │ │ │ -0002e590: 2d2d 2072 6574 7572 6e73 2074 6865 2073 -- returns the s │ │ │ │ -0002e5a0: 7461 6e64 6172 6420 6d6f 6e6f 6d69 616c tandard monomial │ │ │ │ -0002e5b0: 7320 736d 616c 6c65 7220 6275 7420 6f66 s smaller but of │ │ │ │ -0002e5c0: 2074 6865 2073 616d 6520 6465 6772 6565 the same degree │ │ │ │ -0002e5d0: 2061 7320 6769 7665 6e20 6d6f 6e6f 6d69 as given monomi │ │ │ │ -0002e5e0: 616c 2873 290a 2a2a 2a2a 2a2a 2a2a 2a2a al(s).********** │ │ │ │ -0002e5f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e600: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e610: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e620: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002e640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -0002e650: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -0002e660: 2020 4c20 3d20 736d 616c 6c65 724d 6f6e L = smallerMon │ │ │ │ -0002e670: 6f6d 6961 6c73 204d 0a20 2020 2020 2020 omials M. │ │ │ │ -0002e680: 204c 203d 2073 6d61 6c6c 6572 4d6f 6e6f L = smallerMono │ │ │ │ -0002e690: 6d69 616c 7328 4d2c 206d 290a 2020 2a20 mials(M, m). * │ │ │ │ -0002e6a0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -0002e6b0: 4d2c 2061 6e20 2a6e 6f74 6520 6964 6561 M, an *note idea │ │ │ │ -0002e6c0: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ -0002e6d0: 2949 6465 616c 2c2c 2024 4d24 2073 686f )Ideal,, $M$ sho │ │ │ │ -0002e6e0: 756c 6420 6265 2061 206d 6f6e 6f6d 6961 uld be a monomia │ │ │ │ -0002e6f0: 6c20 6964 6561 6c0a 2020 2020 2020 2020 l ideal. │ │ │ │ -0002e700: 2861 6e20 6964 6561 6c20 6765 6e65 7261 (an ideal genera │ │ │ │ -0002e710: 7465 6420 6279 206d 6f6e 6f6d 6961 6c73 ted by monomials │ │ │ │ -0002e720: 290a 2020 2020 2020 2a20 6d2c 2061 202a ). * m, a * │ │ │ │ -0002e730: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ -0002e740: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -0002e750: 2952 696e 6745 6c65 6d65 6e74 2c2c 206f )RingElement,, o │ │ │ │ -0002e760: 7074 696f 6e61 6c2c 0a20 202a 204f 7574 ptional,. * Out │ │ │ │ -0002e770: 7075 7473 3a0a 2020 2020 2020 2a20 4c2c puts:. * L, │ │ │ │ -0002e780: 2061 202a 6e6f 7465 206c 6973 743a 2028 a *note list: ( │ │ │ │ -0002e790: 4d61 6361 756c 6179 3244 6f63 294c 6973 Macaulay2Doc)Lis │ │ │ │ -0002e7a0: 742c 2c20 6120 6c69 7374 206f 6620 6c69 t,, a list of li │ │ │ │ -0002e7b0: 7374 733a 2066 6f72 2065 6163 680a 2020 sts: for each. │ │ │ │ -0002e7c0: 2020 2020 2020 6765 6e65 7261 746f 7220 generator │ │ │ │ -0002e7d0: 246d 2420 6f66 2024 4d24 2c20 7468 6520 $m$ of $M$, the │ │ │ │ -0002e7e0: 6c69 7374 206f 6620 616c 6c20 6d6f 6e6f list of all mono │ │ │ │ -0002e7f0: 6d69 616c 7320 6f66 2074 6865 2073 616d mials of the sam │ │ │ │ -0002e800: 6520 6465 6772 6565 2061 730a 2020 2020 e degree as. │ │ │ │ -0002e810: 2020 2020 246d 242c 206e 6f74 2069 6e20 $m$, not in │ │ │ │ -0002e820: 7468 6520 6d6f 6e6f 6d69 616c 2069 6465 the monomial ide │ │ │ │ -0002e830: 616c 2061 6e64 2073 6d61 6c6c 6572 2074 al and smaller t │ │ │ │ -0002e840: 6861 6e20 7468 6174 2067 656e 6572 6174 han that generat │ │ │ │ -0002e850: 6f72 2069 6e20 7468 650a 2020 2020 2020 or in the. │ │ │ │ -0002e860: 2020 7465 726d 206f 7264 6572 206f 6620 term order of │ │ │ │ -0002e870: 7468 6520 616d 6269 656e 7420 7269 6e67 the ambient ring │ │ │ │ -0002e880: 2e20 2049 6620 696e 7374 6561 6420 246d . If instead $m │ │ │ │ -0002e890: 2420 6973 2067 6976 656e 2c20 7468 6520 $ is given, the │ │ │ │ -0002e8a0: 6c69 7374 206f 660a 2020 2020 2020 2020 list of. │ │ │ │ -0002e8b0: 7468 6520 7374 616e 6461 7264 206d 6f6e the standard mon │ │ │ │ -0002e8c0: 6f6d 6961 6c73 206f 6620 7468 6520 7361 omials of the sa │ │ │ │ -0002e8d0: 6d65 2064 6567 7265 652c 2073 6d61 6c6c me degree, small │ │ │ │ -0002e8e0: 6572 2074 6861 6e20 246d 242c 2069 730a er than $m$, is. │ │ │ │ -0002e8f0: 2020 2020 2020 2020 7265 7475 726e 6564 returned │ │ │ │ -0002e900: 2e0a 0a44 6573 6372 6970 7469 6f6e 0a3d ...Description.= │ │ │ │ -0002e910: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 496e 7075 ==========..Inpu │ │ │ │ -0002e920: 7474 696e 6720 616e 2069 6465 616c 2024 tting an ideal $ │ │ │ │ -0002e930: 4d24 2072 6574 7572 6e73 2074 6865 2073 M$ returns the s │ │ │ │ -0002e940: 6d61 6c6c 6572 206d 6f6e 6f6d 6961 6c73 maller monomials │ │ │ │ -0002e950: 206f 6620 6561 6368 206f 6620 7468 6520 of each of the │ │ │ │ -0002e960: 6769 7665 6e0a 6765 6e65 7261 746f 7273 given.generators │ │ │ │ -0002e970: 206f 6620 7468 6520 6964 6561 6c2e 0a0a of the ideal... │ │ │ │ -0002e980: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002e990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002e9d0: 7c69 3120 3a20 5220 3d20 5a5a 2f33 3230 |i1 : R = ZZ/320 │ │ │ │ -0002e9e0: 3033 5b61 2e2e 645d 3b20 2020 2020 2020 03[a..d]; │ │ │ │ -0002e9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002e1d0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ +0002e1e0: 3133 2034 2030 207c 2020 2020 2020 2020 13 4 0 | │ │ │ │ +0002e1f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e220: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ +0002e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e240: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e270: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 ------------+..C │ │ │ │ +0002e2d0: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 5468 aveat.======..Th │ │ │ │ +0002e2e0: 6973 2072 6f75 7469 6e65 2065 7870 6563 is routine expec │ │ │ │ +0002e2f0: 7473 2074 6865 2069 6e70 7574 2074 6f20 ts the input to │ │ │ │ +0002e300: 7265 7072 6573 656e 7420 616e 2069 7272 represent an irr │ │ │ │ +0002e310: 6564 7563 6962 6c65 2076 6172 6965 7479 educible variety │ │ │ │ +0002e320: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +0002e330: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2072 ===.. * *note r │ │ │ │ +0002e340: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ +0002e350: 6f6e 616c 5661 7269 6574 7928 4964 6561 onalVariety(Idea │ │ │ │ +0002e360: 6c29 3a0a 2020 2020 7261 6e64 6f6d 506f l):. randomPo │ │ │ │ +0002e370: 696e 744f 6e52 6174 696f 6e61 6c56 6172 intOnRationalVar │ │ │ │ +0002e380: 6965 7479 5f6c 7049 6465 616c 5f72 702c iety_lpIdeal_rp, │ │ │ │ +0002e390: 202d 2d20 6669 6e64 2061 2072 616e 646f -- find a rando │ │ │ │ +0002e3a0: 6d20 706f 696e 7420 6f6e 2061 0a20 2020 m point on a. │ │ │ │ +0002e3b0: 2076 6172 6965 7479 2074 6861 7420 6361 variety that ca │ │ │ │ +0002e3c0: 6e20 6265 2064 6574 6563 7465 6420 746f n be detected to │ │ │ │ +0002e3d0: 2062 6520 7261 7469 6f6e 616c 0a0a 5761 be rational..Wa │ │ │ │ +0002e3e0: 7973 2074 6f20 7573 6520 7468 6973 206d ys to use this m │ │ │ │ +0002e3f0: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d ethod:.========= │ │ │ │ +0002e400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0002e410: 0a20 202a 202a 6e6f 7465 2072 616e 646f . * *note rando │ │ │ │ +0002e420: 6d50 6f69 6e74 734f 6e52 6174 696f 6e61 mPointsOnRationa │ │ │ │ +0002e430: 6c56 6172 6965 7479 2849 6465 616c 2c5a lVariety(Ideal,Z │ │ │ │ +0002e440: 5a29 3a0a 2020 2020 7261 6e64 6f6d 506f Z):. randomPo │ │ │ │ +0002e450: 696e 7473 4f6e 5261 7469 6f6e 616c 5661 intsOnRationalVa │ │ │ │ +0002e460: 7269 6574 795f 6c70 4964 6561 6c5f 636d riety_lpIdeal_cm │ │ │ │ +0002e470: 5a5a 5f72 702c 202d 2d20 6669 6e64 2072 ZZ_rp, -- find r │ │ │ │ +0002e480: 616e 646f 6d20 706f 696e 7473 206f 6e20 andom points on │ │ │ │ +0002e490: 610a 2020 2020 7661 7269 6574 7920 7468 a. variety th │ │ │ │ +0002e4a0: 6174 2063 616e 2062 6520 6465 7465 6374 at can be detect │ │ │ │ +0002e4b0: 6564 2074 6f20 6265 2072 6174 696f 6e61 ed to be rationa │ │ │ │ +0002e4c0: 6c0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d l.-------------- │ │ │ │ +0002e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e510: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +0002e520: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +0002e530: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +0002e540: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +0002e550: 6361 756c 6179 322d 312e 3235 2e31 312b caulay2-1.25.11+ │ │ │ │ +0002e560: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +0002e570: 7061 636b 6167 6573 2f0a 4772 6f65 626e packages/.Groebn │ │ │ │ +0002e580: 6572 5374 7261 7461 2e6d 323a 3838 333a erStrata.m2:883: │ │ │ │ +0002e590: 302e 0a1f 0a46 696c 653a 2047 726f 6562 0....File: Groeb │ │ │ │ +0002e5a0: 6e65 7253 7472 6174 612e 696e 666f 2c20 nerStrata.info, │ │ │ │ +0002e5b0: 4e6f 6465 3a20 736d 616c 6c65 724d 6f6e Node: smallerMon │ │ │ │ +0002e5c0: 6f6d 6961 6c73 2c20 4e65 7874 3a20 7374 omials, Next: st │ │ │ │ +0002e5d0: 616e 6461 7264 4d6f 6e6f 6d69 616c 732c andardMonomials, │ │ │ │ +0002e5e0: 2050 7265 763a 2072 616e 646f 6d50 6f69 Prev: randomPoi │ │ │ │ +0002e5f0: 6e74 734f 6e52 6174 696f 6e61 6c56 6172 ntsOnRationalVar │ │ │ │ +0002e600: 6965 7479 5f6c 7049 6465 616c 5f63 6d5a iety_lpIdeal_cmZ │ │ │ │ +0002e610: 5a5f 7270 2c20 5570 3a20 546f 700a 0a73 Z_rp, Up: Top..s │ │ │ │ +0002e620: 6d61 6c6c 6572 4d6f 6e6f 6d69 616c 7320 mallerMonomials │ │ │ │ +0002e630: 2d2d 2072 6574 7572 6e73 2074 6865 2073 -- returns the s │ │ │ │ +0002e640: 7461 6e64 6172 6420 6d6f 6e6f 6d69 616c tandard monomial │ │ │ │ +0002e650: 7320 736d 616c 6c65 7220 6275 7420 6f66 s smaller but of │ │ │ │ +0002e660: 2074 6865 2073 616d 6520 6465 6772 6565 the same degree │ │ │ │ +0002e670: 2061 7320 6769 7665 6e20 6d6f 6e6f 6d69 as given monomi │ │ │ │ +0002e680: 616c 2873 290a 2a2a 2a2a 2a2a 2a2a 2a2a al(s).********** │ │ │ │ +0002e690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e6a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e6d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002e6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +0002e6f0: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +0002e700: 2020 4c20 3d20 736d 616c 6c65 724d 6f6e L = smallerMon │ │ │ │ +0002e710: 6f6d 6961 6c73 204d 0a20 2020 2020 2020 omials M. │ │ │ │ +0002e720: 204c 203d 2073 6d61 6c6c 6572 4d6f 6e6f L = smallerMono │ │ │ │ +0002e730: 6d69 616c 7328 4d2c 206d 290a 2020 2a20 mials(M, m). * │ │ │ │ +0002e740: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ +0002e750: 4d2c 2061 6e20 2a6e 6f74 6520 6964 6561 M, an *note idea │ │ │ │ +0002e760: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ +0002e770: 2949 6465 616c 2c2c 2024 4d24 2073 686f )Ideal,, $M$ sho │ │ │ │ +0002e780: 756c 6420 6265 2061 206d 6f6e 6f6d 6961 uld be a monomia │ │ │ │ +0002e790: 6c20 6964 6561 6c0a 2020 2020 2020 2020 l ideal. │ │ │ │ +0002e7a0: 2861 6e20 6964 6561 6c20 6765 6e65 7261 (an ideal genera │ │ │ │ +0002e7b0: 7465 6420 6279 206d 6f6e 6f6d 6961 6c73 ted by monomials │ │ │ │ +0002e7c0: 290a 2020 2020 2020 2a20 6d2c 2061 202a ). * m, a * │ │ │ │ +0002e7d0: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ +0002e7e0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +0002e7f0: 2952 696e 6745 6c65 6d65 6e74 2c2c 206f )RingElement,, o │ │ │ │ +0002e800: 7074 696f 6e61 6c2c 0a20 202a 204f 7574 ptional,. * Out │ │ │ │ +0002e810: 7075 7473 3a0a 2020 2020 2020 2a20 4c2c puts:. * L, │ │ │ │ +0002e820: 2061 202a 6e6f 7465 206c 6973 743a 2028 a *note list: ( │ │ │ │ +0002e830: 4d61 6361 756c 6179 3244 6f63 294c 6973 Macaulay2Doc)Lis │ │ │ │ +0002e840: 742c 2c20 6120 6c69 7374 206f 6620 6c69 t,, a list of li │ │ │ │ +0002e850: 7374 733a 2066 6f72 2065 6163 680a 2020 sts: for each. │ │ │ │ +0002e860: 2020 2020 2020 6765 6e65 7261 746f 7220 generator │ │ │ │ +0002e870: 246d 2420 6f66 2024 4d24 2c20 7468 6520 $m$ of $M$, the │ │ │ │ +0002e880: 6c69 7374 206f 6620 616c 6c20 6d6f 6e6f list of all mono │ │ │ │ +0002e890: 6d69 616c 7320 6f66 2074 6865 2073 616d mials of the sam │ │ │ │ +0002e8a0: 6520 6465 6772 6565 2061 730a 2020 2020 e degree as. │ │ │ │ +0002e8b0: 2020 2020 246d 242c 206e 6f74 2069 6e20 $m$, not in │ │ │ │ +0002e8c0: 7468 6520 6d6f 6e6f 6d69 616c 2069 6465 the monomial ide │ │ │ │ +0002e8d0: 616c 2061 6e64 2073 6d61 6c6c 6572 2074 al and smaller t │ │ │ │ +0002e8e0: 6861 6e20 7468 6174 2067 656e 6572 6174 han that generat │ │ │ │ +0002e8f0: 6f72 2069 6e20 7468 650a 2020 2020 2020 or in the. │ │ │ │ +0002e900: 2020 7465 726d 206f 7264 6572 206f 6620 term order of │ │ │ │ +0002e910: 7468 6520 616d 6269 656e 7420 7269 6e67 the ambient ring │ │ │ │ +0002e920: 2e20 2049 6620 696e 7374 6561 6420 246d . If instead $m │ │ │ │ +0002e930: 2420 6973 2067 6976 656e 2c20 7468 6520 $ is given, the │ │ │ │ +0002e940: 6c69 7374 206f 660a 2020 2020 2020 2020 list of. │ │ │ │ +0002e950: 7468 6520 7374 616e 6461 7264 206d 6f6e the standard mon │ │ │ │ +0002e960: 6f6d 6961 6c73 206f 6620 7468 6520 7361 omials of the sa │ │ │ │ +0002e970: 6d65 2064 6567 7265 652c 2073 6d61 6c6c me degree, small │ │ │ │ +0002e980: 6572 2074 6861 6e20 246d 242c 2069 730a er than $m$, is. │ │ │ │ +0002e990: 2020 2020 2020 2020 7265 7475 726e 6564 returned │ │ │ │ +0002e9a0: 2e0a 0a44 6573 6372 6970 7469 6f6e 0a3d ...Description.= │ │ │ │ +0002e9b0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 496e 7075 ==========..Inpu │ │ │ │ +0002e9c0: 7474 696e 6720 616e 2069 6465 616c 2024 tting an ideal $ │ │ │ │ +0002e9d0: 4d24 2072 6574 7572 6e73 2074 6865 2073 M$ returns the s │ │ │ │ +0002e9e0: 6d61 6c6c 6572 206d 6f6e 6f6d 6961 6c73 maller monomials │ │ │ │ +0002e9f0: 206f 6620 6561 6368 206f 6620 7468 6520 of each of the │ │ │ │ +0002ea00: 6769 7665 6e0a 6765 6e65 7261 746f 7273 given.generators │ │ │ │ +0002ea10: 206f 6620 7468 6520 6964 6561 6c2e 0a0a of the ideal... │ │ │ │ 0002ea20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0002ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ea40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ea60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002ea70: 7c69 3220 3a20 4d20 3d20 6964 6561 6c20 |i2 : M = ideal │ │ │ │ -0002ea80: 2861 5e32 2c20 625e 322c 2061 2a62 2a63 (a^2, b^2, a*b*c │ │ │ │ -0002ea90: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ +0002ea70: 7c69 3120 3a20 5220 3d20 5a5a 2f33 3230 |i1 : R = ZZ/320 │ │ │ │ +0002ea80: 3033 5b61 2e2e 645d 3b20 2020 2020 2020 03[a..d]; │ │ │ │ +0002ea90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eab0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002eac0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002eb10: 7c6f 3220 3a20 4964 6561 6c20 6f66 2052 |o2 : Ideal of R │ │ │ │ -0002eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eac0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002ead0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002eb10: 7c69 3220 3a20 4d20 3d20 6964 6561 6c20 |i2 : M = ideal │ │ │ │ +0002eb20: 2861 5e32 2c20 625e 322c 2061 2a62 2a63 (a^2, b^2, a*b*c │ │ │ │ +0002eb30: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ 0002eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eb50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002eb60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002eb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002ebb0: 7c69 3320 3a20 4c31 203d 2073 6d61 6c6c |i3 : L1 = small │ │ │ │ -0002ebc0: 6572 4d6f 6e6f 6d69 616c 7320 4d20 2020 erMonomials M │ │ │ │ +0002eb60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eba0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002ebb0: 7c6f 3220 3a20 4964 6561 6c20 6f66 2052 |o2 : Ideal of R │ │ │ │ +0002ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ebd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ebe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ebf0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ec00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ec50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002ec60: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -0002ec70: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -0002ec80: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0002ec00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002ec10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002ec50: 7c69 3320 3a20 4c31 203d 2073 6d61 6c6c |i3 : L1 = small │ │ │ │ +0002ec60: 6572 4d6f 6e6f 6d69 616c 7320 4d20 2020 erMonomials M │ │ │ │ +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 7c0a |. │ │ │ │ -0002eca0: 7c6f 3320 3d20 7b7b 612a 622c 2061 2a63 |o3 = {{a*b, a*c │ │ │ │ -0002ecb0: 2c20 622a 632c 2063 202c 2061 2a64 2c20 , b*c, c , a*d, │ │ │ │ -0002ecc0: 622a 642c 2063 2a64 2c20 6420 7d2c 207b b*d, c*d, d }, { │ │ │ │ -0002ecd0: 612a 632c 2062 2a63 2c20 6320 2c20 612a a*c, b*c, c , a* │ │ │ │ -0002ece0: 642c 2062 2a64 2c20 632a 642c 2020 7c0a d, b*d, c*d, |. │ │ │ │ -0002ecf0: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ -0002ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ed10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ed20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ed30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -0002ed40: 7c20 2020 2020 2032 2020 2020 2020 2032 | 2 2 │ │ │ │ -0002ed50: 2020 2020 2032 2020 2033 2020 2020 2020 2 3 │ │ │ │ -0002ed60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed70: 2020 3220 2020 2020 2032 2020 2020 2032 2 2 2 │ │ │ │ -0002ed80: 2020 2020 2032 2020 2033 2020 2020 7c0a 2 3 |. │ │ │ │ -0002ed90: 7c20 2020 2020 6420 7d2c 207b 612a 6320 | d }, {a*c │ │ │ │ -0002eda0: 2c20 622a 6320 2c20 6320 2c20 612a 622a , b*c , c , a*b* │ │ │ │ -0002edb0: 642c 2061 2a63 2a64 2c20 622a 632a 642c d, a*c*d, b*c*d, │ │ │ │ -0002edc0: 2063 2064 2c20 612a 6420 2c20 622a 6420 c d, a*d , b*d │ │ │ │ -0002edd0: 2c20 632a 6420 2c20 6420 7d7d 2020 7c0a , c*d , d }} |. │ │ │ │ -0002ede0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eca0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002ecb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ecc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ecd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ece0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002ecf0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002ed00: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0002ed10: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0002ed20: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0002ed30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002ed40: 7c6f 3320 3d20 7b7b 612a 622c 2061 2a63 |o3 = {{a*b, a*c │ │ │ │ +0002ed50: 2c20 622a 632c 2063 202c 2061 2a64 2c20 , b*c, c , a*d, │ │ │ │ +0002ed60: 622a 642c 2063 2a64 2c20 6420 7d2c 207b b*d, c*d, d }, { │ │ │ │ +0002ed70: 612a 632c 2062 2a63 2c20 6320 2c20 612a a*c, b*c, c , a* │ │ │ │ +0002ed80: 642c 2062 2a64 2c20 632a 642c 2020 7c0a d, b*d, c*d, |. │ │ │ │ +0002ed90: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ +0002eda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +0002ede0: 7c20 2020 2020 2032 2020 2020 2020 2032 | 2 2 │ │ │ │ +0002edf0: 2020 2020 2032 2020 2033 2020 2020 2020 2 3 │ │ │ │ 0002ee00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ee30: 7c6f 3320 3a20 4c69 7374 2020 2020 2020 |o3 : List │ │ │ │ -0002ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ee80: 2b2d 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 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002eed0: 7c69 3420 3a20 736d 616c 6c65 724d 6f6e |i4 : smallerMon │ │ │ │ -0002eee0: 6f6d 6961 6c73 284d 2c20 625e 3229 2020 omials(M, b^2) │ │ │ │ +0002ee10: 2020 3220 2020 2020 2032 2020 2020 2032 2 2 2 │ │ │ │ +0002ee20: 2020 2020 2032 2020 2033 2020 2020 7c0a 2 3 |. │ │ │ │ +0002ee30: 7c20 2020 2020 6420 7d2c 207b 612a 6320 | d }, {a*c │ │ │ │ +0002ee40: 2c20 622a 6320 2c20 6320 2c20 612a 622a , b*c , c , a*b* │ │ │ │ +0002ee50: 642c 2061 2a63 2a64 2c20 622a 632a 642c d, a*c*d, b*c*d, │ │ │ │ +0002ee60: 2063 2064 2c20 612a 6420 2c20 622a 6420 c d, a*d , b*d │ │ │ │ +0002ee70: 2c20 632a 6420 2c20 6420 7d7d 2020 7c0a , c*d , d }} |. │ │ │ │ +0002ee80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eec0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002eed0: 7c6f 3320 3a20 4c69 7374 2020 2020 2020 |o3 : List │ │ │ │ +0002eee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ef00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ef10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ef20: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002ef30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ef40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ef50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ef60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ef70: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002ef80: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002ef90: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +0002ef20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ef40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ef50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ef60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002ef70: 7c69 3420 3a20 736d 616c 6c65 724d 6f6e |i4 : smallerMon │ │ │ │ +0002ef80: 6f6d 6961 6c73 284d 2c20 625e 3229 2020 omials(M, b^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 7c0a |. │ │ │ │ -0002efc0: 7c6f 3420 3d20 7b61 2a63 2c20 622a 632c |o4 = {a*c, b*c, │ │ │ │ -0002efd0: 2063 202c 2061 2a64 2c20 622a 642c 2063 c , a*d, b*d, c │ │ │ │ -0002efe0: 2a64 2c20 6420 7d20 2020 2020 2020 2020 *d, d } │ │ │ │ +0002efc0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002efd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f000: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0002f010: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f020: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002f030: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 0002f040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f060: 7c6f 3420 3a20 4c69 7374 2020 2020 2020 |o4 : List │ │ │ │ -0002f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f060: 7c6f 3420 3d20 7b61 2a63 2c20 622a 632c |o4 = {a*c, b*c, │ │ │ │ +0002f070: 2063 202c 2061 2a64 2c20 622a 642c 2063 c , a*d, b*d, c │ │ │ │ +0002f080: 2a64 2c20 6420 7d20 2020 2020 2020 2020 *d, d } │ │ │ │ 0002f090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f0a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f0b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002f0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002f100: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -0002f110: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7461 ==.. * *note ta │ │ │ │ -0002f120: 696c 4d6f 6e6f 6d69 616c 733a 2074 6169 ilMonomials: tai │ │ │ │ -0002f130: 6c4d 6f6e 6f6d 6961 6c73 2c20 2d2d 2066 lMonomials, -- f │ │ │ │ -0002f140: 696e 6420 7461 696c 206d 6f6e 6f6d 6961 ind tail monomia │ │ │ │ -0002f150: 6c73 0a20 202a 202a 6e6f 7465 2073 7461 ls. * *note sta │ │ │ │ -0002f160: 6e64 6172 644d 6f6e 6f6d 6961 6c73 3a20 ndardMonomials: │ │ │ │ -0002f170: 7374 616e 6461 7264 4d6f 6e6f 6d69 616c standardMonomial │ │ │ │ -0002f180: 732c 202d 2d20 636f 6d70 7574 6573 2073 s, -- computes s │ │ │ │ -0002f190: 7461 6e64 6172 6420 6d6f 6e6f 6d69 616c tandard monomial │ │ │ │ -0002f1a0: 730a 0a57 6179 7320 746f 2075 7365 2073 s..Ways to use s │ │ │ │ -0002f1b0: 6d61 6c6c 6572 4d6f 6e6f 6d69 616c 733a mallerMonomials: │ │ │ │ -0002f1c0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0002f1d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -0002f1e0: 2020 2a20 2273 6d61 6c6c 6572 4d6f 6e6f * "smallerMono │ │ │ │ -0002f1f0: 6d69 616c 7328 4964 6561 6c29 220a 2020 mials(Ideal)". │ │ │ │ -0002f200: 2a20 2273 6d61 6c6c 6572 4d6f 6e6f 6d69 * "smallerMonomi │ │ │ │ -0002f210: 616c 7328 4964 6561 6c2c 5269 6e67 456c als(Ideal,RingEl │ │ │ │ -0002f220: 656d 656e 7429 220a 0a46 6f72 2074 6865 ement)"..For the │ │ │ │ -0002f230: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -0002f240: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -0002f250: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -0002f260: 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 616c smallerMonomial │ │ │ │ -0002f270: 733a 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 s: smallerMonomi │ │ │ │ -0002f280: 616c 732c 2069 7320 6120 2a6e 6f74 6520 als, is a *note │ │ │ │ -0002f290: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a method.function: │ │ │ │ -0002f2a0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -0002f2b0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ -0002f2c0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f310: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -0002f320: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -0002f330: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -0002f340: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -0002f350: 6175 6c61 7932 2d31 2e32 352e 3131 2b64 aulay2-1.25.11+d │ │ │ │ -0002f360: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -0002f370: 6163 6b61 6765 732f 0a47 726f 6562 6e65 ackages/.Groebne │ │ │ │ -0002f380: 7253 7472 6174 612e 6d32 3a35 3234 3a30 rStrata.m2:524:0 │ │ │ │ -0002f390: 2e0a 1f0a 4669 6c65 3a20 4772 6f65 626e ....File: Groebn │ │ │ │ -0002f3a0: 6572 5374 7261 7461 2e69 6e66 6f2c 204e erStrata.info, N │ │ │ │ -0002f3b0: 6f64 653a 2073 7461 6e64 6172 644d 6f6e ode: standardMon │ │ │ │ -0002f3c0: 6f6d 6961 6c73 2c20 4e65 7874 3a20 7461 omials, Next: ta │ │ │ │ -0002f3d0: 696c 4d6f 6e6f 6d69 616c 732c 2050 7265 ilMonomials, Pre │ │ │ │ -0002f3e0: 763a 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 v: smallerMonomi │ │ │ │ -0002f3f0: 616c 732c 2055 703a 2054 6f70 0a0a 7374 als, Up: Top..st │ │ │ │ -0002f400: 616e 6461 7264 4d6f 6e6f 6d69 616c 7320 andardMonomials │ │ │ │ -0002f410: 2d2d 2063 6f6d 7075 7465 7320 7374 616e -- computes stan │ │ │ │ -0002f420: 6461 7264 206d 6f6e 6f6d 6961 6c73 0a2a dard monomials.* │ │ │ │ -0002f430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002f440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002f450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -0002f460: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -0002f470: 2020 2020 204c 203d 2073 7461 6e64 6172 L = standar │ │ │ │ -0002f480: 644d 6f6e 6f6d 6961 6c73 204d 0a20 2020 dMonomials M. │ │ │ │ -0002f490: 2020 2020 204c 203d 2073 7461 6e64 6172 L = standar │ │ │ │ -0002f4a0: 644d 6f6e 6f6d 6961 6c73 2864 2c20 4d29 dMonomials(d, M) │ │ │ │ -0002f4b0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -0002f4c0: 2020 202a 204d 2c20 616e 202a 6e6f 7465 * M, an *note │ │ │ │ -0002f4d0: 2069 6465 616c 3a20 284d 6163 6175 6c61 ideal: (Macaula │ │ │ │ -0002f4e0: 7932 446f 6329 4964 6561 6c2c 2c20 4d20 y2Doc)Ideal,, M │ │ │ │ -0002f4f0: 7368 6f75 6c64 2062 6520 6120 6d6f 6e6f should be a mono │ │ │ │ -0002f500: 6d69 616c 2069 6465 616c 0a20 2020 2020 mial ideal. │ │ │ │ -0002f510: 202a 2064 2c20 6120 2a6e 6f74 6520 6c69 * d, a *note li │ │ │ │ -0002f520: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ -0002f530: 6329 4c69 7374 2c2c 2061 2064 6567 7265 c)List,, a degre │ │ │ │ -0002f540: 650a 2020 2a20 4f75 7470 7574 733a 0a20 e. * Outputs:. │ │ │ │ -0002f550: 2020 2020 202a 204c 2c20 6120 2a6e 6f74 * L, a *not │ │ │ │ -0002f560: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ -0002f570: 7932 446f 6329 4c69 7374 2c2c 204c 2069 y2Doc)List,, L i │ │ │ │ -0002f580: 7320 6120 6c69 7374 206f 6620 6c69 7374 s a list of list │ │ │ │ -0002f590: 7320 6f66 2073 7461 6e64 6172 640a 2020 s of standard. │ │ │ │ -0002f5a0: 2020 2020 2020 6d6f 6e6f 6d69 616c 7320 monomials │ │ │ │ -0002f5b0: 666f 7220 7468 6520 6964 6561 6c20 244d for the ideal $M │ │ │ │ -0002f5c0: 242c 206f 6e65 2066 6f72 2065 6163 6820 $, one for each │ │ │ │ -0002f5d0: 6765 6e65 7261 746f 7220 6f66 2024 4d24 generator of $M$ │ │ │ │ -0002f5e0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0002f5f0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 206d 6f6e =========..A mon │ │ │ │ -0002f600: 6f6d 6961 6c20 246d 2420 6973 2073 7461 omial $m$ is sta │ │ │ │ -0002f610: 6e64 6172 6420 7769 7468 2072 6573 7065 ndard with respe │ │ │ │ -0002f620: 6374 2074 6f20 6120 6d6f 6e6f 6d69 616c ct to a monomial │ │ │ │ -0002f630: 2069 6465 616c 2024 4d24 2061 6e64 2061 ideal $M$ and a │ │ │ │ -0002f640: 2067 656e 6572 6174 6f72 0a24 6724 206f generator.$g$ o │ │ │ │ -0002f650: 6620 244d 2420 6966 2024 6d24 2069 7320 f $M$ if $m$ is │ │ │ │ -0002f660: 6f66 2074 6865 2073 616d 6520 6465 6772 of the same degr │ │ │ │ -0002f670: 6565 2061 7320 2467 2420 6275 7420 6973 ee as $g$ but is │ │ │ │ -0002f680: 206e 6f74 2061 6e20 656c 656d 656e 7420 not an element │ │ │ │ -0002f690: 6f66 2024 4d24 2e0a 0a49 6e70 7574 7469 of $M$...Inputti │ │ │ │ -0002f6a0: 6e67 2061 6e20 6964 6561 6c20 244d 2420 ng an ideal $M$ │ │ │ │ -0002f6b0: 7265 7475 726e 7320 7468 6520 7374 616e returns the stan │ │ │ │ -0002f6c0: 6461 7264 206d 6f6e 6f6d 6961 6c73 206f dard monomials o │ │ │ │ -0002f6d0: 6620 6561 6368 206f 6620 7468 6520 6769 f each of the gi │ │ │ │ -0002f6e0: 7665 6e0a 6765 6e65 7261 746f 7273 206f ven.generators o │ │ │ │ -0002f6f0: 6620 7468 6520 6964 6561 6c2e 0a0a 2b2d f the ideal...+- │ │ │ │ -0002f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002f750: 3120 3a20 5220 3d20 5a5a 2f33 3230 3033 1 : R = ZZ/32003 │ │ │ │ -0002f760: 5b61 2e2e 645d 3b20 2020 2020 2020 2020 [a..d]; │ │ │ │ -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 7c0a 2b2d |.+- │ │ │ │ +0002f0b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f0f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002f100: 7c6f 3420 3a20 4c69 7374 2020 2020 2020 |o4 : List │ │ │ │ +0002f110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 7c0a |. │ │ │ │ +0002f150: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002f1a0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +0002f1b0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7461 ==.. * *note ta │ │ │ │ +0002f1c0: 696c 4d6f 6e6f 6d69 616c 733a 2074 6169 ilMonomials: tai │ │ │ │ +0002f1d0: 6c4d 6f6e 6f6d 6961 6c73 2c20 2d2d 2066 lMonomials, -- f │ │ │ │ +0002f1e0: 696e 6420 7461 696c 206d 6f6e 6f6d 6961 ind tail monomia │ │ │ │ +0002f1f0: 6c73 0a20 202a 202a 6e6f 7465 2073 7461 ls. * *note sta │ │ │ │ +0002f200: 6e64 6172 644d 6f6e 6f6d 6961 6c73 3a20 ndardMonomials: │ │ │ │ +0002f210: 7374 616e 6461 7264 4d6f 6e6f 6d69 616c standardMonomial │ │ │ │ +0002f220: 732c 202d 2d20 636f 6d70 7574 6573 2073 s, -- computes s │ │ │ │ +0002f230: 7461 6e64 6172 6420 6d6f 6e6f 6d69 616c tandard monomial │ │ │ │ +0002f240: 730a 0a57 6179 7320 746f 2075 7365 2073 s..Ways to use s │ │ │ │ +0002f250: 6d61 6c6c 6572 4d6f 6e6f 6d69 616c 733a mallerMonomials: │ │ │ │ +0002f260: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +0002f270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0002f280: 2020 2a20 2273 6d61 6c6c 6572 4d6f 6e6f * "smallerMono │ │ │ │ +0002f290: 6d69 616c 7328 4964 6561 6c29 220a 2020 mials(Ideal)". │ │ │ │ +0002f2a0: 2a20 2273 6d61 6c6c 6572 4d6f 6e6f 6d69 * "smallerMonomi │ │ │ │ +0002f2b0: 616c 7328 4964 6561 6c2c 5269 6e67 456c als(Ideal,RingEl │ │ │ │ +0002f2c0: 656d 656e 7429 220a 0a46 6f72 2074 6865 ement)"..For the │ │ │ │ +0002f2d0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +0002f2e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0002f2f0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ +0002f300: 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 616c smallerMonomial │ │ │ │ +0002f310: 733a 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 s: smallerMonomi │ │ │ │ +0002f320: 616c 732c 2069 7320 6120 2a6e 6f74 6520 als, is a *note │ │ │ │ +0002f330: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a method.function: │ │ │ │ +0002f340: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +0002f350: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +0002f360: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +0002f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0002f3c0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0002f3d0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0002f3e0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0002f3f0: 6175 6c61 7932 2d31 2e32 352e 3131 2b64 aulay2-1.25.11+d │ │ │ │ +0002f400: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0002f410: 6163 6b61 6765 732f 0a47 726f 6562 6e65 ackages/.Groebne │ │ │ │ +0002f420: 7253 7472 6174 612e 6d32 3a35 3234 3a30 rStrata.m2:524:0 │ │ │ │ +0002f430: 2e0a 1f0a 4669 6c65 3a20 4772 6f65 626e ....File: Groebn │ │ │ │ +0002f440: 6572 5374 7261 7461 2e69 6e66 6f2c 204e erStrata.info, N │ │ │ │ +0002f450: 6f64 653a 2073 7461 6e64 6172 644d 6f6e ode: standardMon │ │ │ │ +0002f460: 6f6d 6961 6c73 2c20 4e65 7874 3a20 7461 omials, Next: ta │ │ │ │ +0002f470: 696c 4d6f 6e6f 6d69 616c 732c 2050 7265 ilMonomials, Pre │ │ │ │ +0002f480: 763a 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 v: smallerMonomi │ │ │ │ +0002f490: 616c 732c 2055 703a 2054 6f70 0a0a 7374 als, Up: Top..st │ │ │ │ +0002f4a0: 616e 6461 7264 4d6f 6e6f 6d69 616c 7320 andardMonomials │ │ │ │ +0002f4b0: 2d2d 2063 6f6d 7075 7465 7320 7374 616e -- computes stan │ │ │ │ +0002f4c0: 6461 7264 206d 6f6e 6f6d 6961 6c73 0a2a dard monomials.* │ │ │ │ +0002f4d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002f4e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002f4f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +0002f500: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +0002f510: 2020 2020 204c 203d 2073 7461 6e64 6172 L = standar │ │ │ │ +0002f520: 644d 6f6e 6f6d 6961 6c73 204d 0a20 2020 dMonomials M. │ │ │ │ +0002f530: 2020 2020 204c 203d 2073 7461 6e64 6172 L = standar │ │ │ │ +0002f540: 644d 6f6e 6f6d 6961 6c73 2864 2c20 4d29 dMonomials(d, M) │ │ │ │ +0002f550: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +0002f560: 2020 202a 204d 2c20 616e 202a 6e6f 7465 * M, an *note │ │ │ │ +0002f570: 2069 6465 616c 3a20 284d 6163 6175 6c61 ideal: (Macaula │ │ │ │ +0002f580: 7932 446f 6329 4964 6561 6c2c 2c20 4d20 y2Doc)Ideal,, M │ │ │ │ +0002f590: 7368 6f75 6c64 2062 6520 6120 6d6f 6e6f should be a mono │ │ │ │ +0002f5a0: 6d69 616c 2069 6465 616c 0a20 2020 2020 mial ideal. │ │ │ │ +0002f5b0: 202a 2064 2c20 6120 2a6e 6f74 6520 6c69 * d, a *note li │ │ │ │ +0002f5c0: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ +0002f5d0: 6329 4c69 7374 2c2c 2061 2064 6567 7265 c)List,, a degre │ │ │ │ +0002f5e0: 650a 2020 2a20 4f75 7470 7574 733a 0a20 e. * Outputs:. │ │ │ │ +0002f5f0: 2020 2020 202a 204c 2c20 6120 2a6e 6f74 * L, a *not │ │ │ │ +0002f600: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +0002f610: 7932 446f 6329 4c69 7374 2c2c 204c 2069 y2Doc)List,, L i │ │ │ │ +0002f620: 7320 6120 6c69 7374 206f 6620 6c69 7374 s a list of list │ │ │ │ +0002f630: 7320 6f66 2073 7461 6e64 6172 640a 2020 s of standard. │ │ │ │ +0002f640: 2020 2020 2020 6d6f 6e6f 6d69 616c 7320 monomials │ │ │ │ +0002f650: 666f 7220 7468 6520 6964 6561 6c20 244d for the ideal $M │ │ │ │ +0002f660: 242c 206f 6e65 2066 6f72 2065 6163 6820 $, one for each │ │ │ │ +0002f670: 6765 6e65 7261 746f 7220 6f66 2024 4d24 generator of $M$ │ │ │ │ +0002f680: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0002f690: 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 206d 6f6e =========..A mon │ │ │ │ +0002f6a0: 6f6d 6961 6c20 246d 2420 6973 2073 7461 omial $m$ is sta │ │ │ │ +0002f6b0: 6e64 6172 6420 7769 7468 2072 6573 7065 ndard with respe │ │ │ │ +0002f6c0: 6374 2074 6f20 6120 6d6f 6e6f 6d69 616c ct to a monomial │ │ │ │ +0002f6d0: 2069 6465 616c 2024 4d24 2061 6e64 2061 ideal $M$ and a │ │ │ │ +0002f6e0: 2067 656e 6572 6174 6f72 0a24 6724 206f generator.$g$ o │ │ │ │ +0002f6f0: 6620 244d 2420 6966 2024 6d24 2069 7320 f $M$ if $m$ is │ │ │ │ +0002f700: 6f66 2074 6865 2073 616d 6520 6465 6772 of the same degr │ │ │ │ +0002f710: 6565 2061 7320 2467 2420 6275 7420 6973 ee as $g$ but is │ │ │ │ +0002f720: 206e 6f74 2061 6e20 656c 656d 656e 7420 not an element │ │ │ │ +0002f730: 6f66 2024 4d24 2e0a 0a49 6e70 7574 7469 of $M$...Inputti │ │ │ │ +0002f740: 6e67 2061 6e20 6964 6561 6c20 244d 2420 ng an ideal $M$ │ │ │ │ +0002f750: 7265 7475 726e 7320 7468 6520 7374 616e returns the stan │ │ │ │ +0002f760: 6461 7264 206d 6f6e 6f6d 6961 6c73 206f dard monomials o │ │ │ │ +0002f770: 6620 6561 6368 206f 6620 7468 6520 6769 f each of the gi │ │ │ │ +0002f780: 7665 6e0a 6765 6e65 7261 746f 7273 206f ven.generators o │ │ │ │ +0002f790: 6620 7468 6520 6964 6561 6c2e 0a0a 2b2d f the ideal...+- │ │ │ │ 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002f7f0: 3220 3a20 4d20 3d20 6964 6561 6c20 2861 2 : M = ideal (a │ │ │ │ -0002f800: 5e32 2c20 612a 622c 2062 5e33 2c20 612a ^2, a*b, b^3, a* │ │ │ │ -0002f810: 6329 3b20 2020 2020 2020 2020 2020 2020 c); │ │ │ │ +0002f7f0: 3120 3a20 5220 3d20 5a5a 2f33 3230 3033 1 : R = ZZ/32003 │ │ │ │ +0002f800: 5b61 2e2e 645d 3b20 2020 2020 2020 2020 [a..d]; │ │ │ │ +0002f810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f830: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f880: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002f890: 3220 3a20 4964 6561 6c20 6f66 2052 2020 2 : Ideal of R │ │ │ │ -0002f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f830: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002f890: 3220 3a20 4d20 3d20 6964 6561 6c20 2861 2 : M = ideal (a │ │ │ │ +0002f8a0: 5e32 2c20 612a 622c 2062 5e33 2c20 612a ^2, a*b, b^3, a* │ │ │ │ +0002f8b0: 6329 3b20 2020 2020 2020 2020 2020 2020 c); │ │ │ │ 0002f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0002f8e0: 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 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002f930: 3320 3a20 4c31 203d 2073 7461 6e64 6172 3 : L1 = standar │ │ │ │ -0002f940: 644d 6f6e 6f6d 6961 6c73 204d 2020 2020 dMonomials M │ │ │ │ +0002f8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f920: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0002f930: 3220 3a20 4964 6561 6c20 6f66 2052 2020 2 : Ideal of R │ │ │ │ +0002f940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f970: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f9c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002f9d0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -0002f9e0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002f9f0: 2020 2032 2020 2020 2032 2020 2020 2020 2 2 │ │ │ │ -0002fa00: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002fa10: 2020 2020 2032 2020 2020 2020 7c0a 7c6f 2 |.|o │ │ │ │ -0002fa20: 3320 3d20 7b7b 6220 2c20 622a 632c 2063 3 = {{b , b*c, c │ │ │ │ -0002fa30: 202c 2061 2a64 2c20 622a 642c 2063 2a64 , a*d, b*d, c*d │ │ │ │ -0002fa40: 2c20 6420 7d2c 207b 6220 2c20 622a 632c , d }, {b , b*c, │ │ │ │ -0002fa50: 2063 202c 2061 2a64 2c20 622a 642c 2063 c , a*d, b*d, c │ │ │ │ -0002fa60: 2a64 2c20 6420 7d2c 2020 2020 7c0a 7c20 *d, d }, |.| │ │ │ │ -0002fa70: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ -0002fa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fa90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002faa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ -0002fac0: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ -0002fad0: 2033 2020 2032 2020 2020 2020 2020 2020 3 2 │ │ │ │ -0002fae0: 2032 2020 2020 2020 3220 2020 2020 3220 2 2 2 │ │ │ │ -0002faf0: 2020 2020 3220 2020 3320 2020 2020 3220 2 3 2 │ │ │ │ -0002fb00: 2020 2020 2020 2032 2020 2020 7c0a 7c20 2 |.| │ │ │ │ -0002fb10: 2020 2020 7b62 2063 2c20 622a 6320 2c20 {b c, b*c , │ │ │ │ -0002fb20: 6320 2c20 6220 642c 2062 2a63 2a64 2c20 c , b d, b*c*d, │ │ │ │ -0002fb30: 6320 642c 2061 2a64 202c 2062 2a64 202c c d, a*d , b*d , │ │ │ │ -0002fb40: 2063 2a64 202c 2064 207d 2c20 7b62 202c c*d , d }, {b , │ │ │ │ -0002fb50: 2062 2a63 2c20 6320 2c20 2020 7c0a 7c20 b*c, c , |.| │ │ │ │ -0002fb60: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ -0002fb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ -0002fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fbc0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0002fbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fbf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002fc00: 2020 2020 612a 642c 2062 2a64 2c20 632a a*d, b*d, c* │ │ │ │ -0002fc10: 642c 2064 207d 7d20 2020 2020 2020 2020 d, d }} │ │ │ │ -0002fc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002f970: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002f9d0: 3320 3a20 4c31 203d 2073 7461 6e64 6172 3 : L1 = standar │ │ │ │ +0002f9e0: 644d 6f6e 6f6d 6961 6c73 204d 2020 2020 dMonomials M │ │ │ │ +0002f9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fa10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002fa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fa60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002fa70: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002fa80: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002fa90: 2020 2032 2020 2020 2032 2020 2020 2020 2 2 │ │ │ │ +0002faa0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002fab0: 2020 2020 2032 2020 2020 2020 7c0a 7c6f 2 |.|o │ │ │ │ +0002fac0: 3320 3d20 7b7b 6220 2c20 622a 632c 2063 3 = {{b , b*c, c │ │ │ │ +0002fad0: 202c 2061 2a64 2c20 622a 642c 2063 2a64 , a*d, b*d, c*d │ │ │ │ +0002fae0: 2c20 6420 7d2c 207b 6220 2c20 622a 632c , d }, {b , b*c, │ │ │ │ +0002faf0: 2063 202c 2061 2a64 2c20 622a 642c 2063 c , a*d, b*d, c │ │ │ │ +0002fb00: 2a64 2c20 6420 7d2c 2020 2020 7c0a 7c20 *d, d }, |.| │ │ │ │ +0002fb10: 2020 2020 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 7c0a 7c20 ------------|.| │ │ │ │ +0002fb60: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ +0002fb70: 2033 2020 2032 2020 2020 2020 2020 2020 3 2 │ │ │ │ +0002fb80: 2032 2020 2020 2020 3220 2020 2020 3220 2 2 2 │ │ │ │ +0002fb90: 2020 2020 3220 2020 3320 2020 2020 3220 2 3 2 │ │ │ │ +0002fba0: 2020 2020 2020 2032 2020 2020 7c0a 7c20 2 |.| │ │ │ │ +0002fbb0: 2020 2020 7b62 2063 2c20 622a 6320 2c20 {b c, b*c , │ │ │ │ +0002fbc0: 6320 2c20 6220 642c 2062 2a63 2a64 2c20 c , b d, b*c*d, │ │ │ │ +0002fbd0: 6320 642c 2061 2a64 202c 2062 2a64 202c c d, a*d , b*d , │ │ │ │ +0002fbe0: 2063 2a64 202c 2064 207d 2c20 7b62 202c c*d , d }, {b , │ │ │ │ +0002fbf0: 2062 2a63 2c20 6320 2c20 2020 7c0a 7c20 b*c, c , |.| │ │ │ │ +0002fc00: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ +0002fc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ 0002fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fc60: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 0002fc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002fca0: 3320 3a20 4c69 7374 2020 2020 2020 2020 3 : List │ │ │ │ -0002fcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fc90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002fca0: 2020 2020 612a 642c 2062 2a64 2c20 632a a*d, b*d, c* │ │ │ │ +0002fcb0: 642c 2064 207d 7d20 2020 2020 2020 2020 d, d }} │ │ │ │ 0002fcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fce0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0002fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002fd40: 3420 3a20 7374 616e 6461 7264 4d6f 6e6f 4 : standardMono │ │ │ │ -0002fd50: 6d69 616c 7328 7b33 7d2c 204d 2920 2020 mials({3}, M) │ │ │ │ +0002fce0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002fcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fd30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0002fd40: 3320 3a20 4c69 7374 2020 2020 2020 2020 3 : List │ │ │ │ +0002fd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fd80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002fd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fdd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002fde0: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ -0002fdf0: 2033 2020 2032 2020 2020 2020 2020 2020 3 2 │ │ │ │ -0002fe00: 2032 2020 2020 2020 3220 2020 2020 3220 2 2 2 │ │ │ │ -0002fe10: 2020 2020 3220 2020 3320 2020 2020 2020 2 3 │ │ │ │ -0002fe20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002fe30: 3420 3d20 7b62 2063 2c20 622a 6320 2c20 4 = {b c, b*c , │ │ │ │ -0002fe40: 6320 2c20 6220 642c 2062 2a63 2a64 2c20 c , b d, b*c*d, │ │ │ │ -0002fe50: 6320 642c 2061 2a64 202c 2062 2a64 202c c d, a*d , b*d , │ │ │ │ -0002fe60: 2063 2a64 202c 2064 207d 2020 2020 2020 c*d , d } │ │ │ │ +0002fd80: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002fd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002fde0: 3420 3a20 7374 616e 6461 7264 4d6f 6e6f 4 : standardMono │ │ │ │ +0002fdf0: 6d69 616c 7328 7b33 7d2c 204d 2920 2020 mials({3}, M) │ │ │ │ +0002fe00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fe10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fe20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fe40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fe50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fe60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fe70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002fe80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fe90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002feb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fe80: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ +0002fe90: 2033 2020 2032 2020 2020 2020 2020 2020 3 2 │ │ │ │ +0002fea0: 2032 2020 2020 2020 3220 2020 2020 3220 2 2 2 │ │ │ │ +0002feb0: 2020 2020 3220 2020 3320 2020 2020 2020 2 3 │ │ │ │ 0002fec0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002fed0: 3420 3a20 4c69 7374 2020 2020 2020 2020 4 : List │ │ │ │ -0002fee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ff00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ff10: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0002ff20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ff30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ff60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 ------------+..I │ │ │ │ -0002ff70: 6e70 7574 7469 6e67 2061 6e20 696e 7465 nputting an inte │ │ │ │ -0002ff80: 6765 7220 2464 2420 286f 7220 6465 6772 ger $d$ (or degr │ │ │ │ -0002ff90: 6565 2024 6424 2920 616e 6420 616e 2069 ee $d$) and an i │ │ │ │ -0002ffa0: 6465 616c 2067 6976 6573 2074 6865 2073 deal gives the s │ │ │ │ -0002ffb0: 7461 6e64 6172 640a 6d6f 6e6f 6d69 616c tandard.monomial │ │ │ │ -0002ffc0: 7320 666f 7220 7468 6520 7370 6563 6966 s for the specif │ │ │ │ -0002ffd0: 6965 6420 6964 6561 6c20 696e 2064 6567 ied ideal in deg │ │ │ │ -0002ffe0: 7265 6520 2464 242e 0a0a 2b2d 2d2d 2d2d ree $d$...+----- │ │ │ │ +0002fed0: 3420 3d20 7b62 2063 2c20 622a 6320 2c20 4 = {b c, b*c , │ │ │ │ +0002fee0: 6320 2c20 6220 642c 2062 2a63 2a64 2c20 c , b d, b*c*d, │ │ │ │ +0002fef0: 6320 642c 2061 2a64 202c 2062 2a64 202c c d, a*d , b*d , │ │ │ │ +0002ff00: 2063 2a64 202c 2064 207d 2020 2020 2020 c*d , d } │ │ │ │ +0002ff10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002ff20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ff30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ff40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ff50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ff60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0002ff70: 3420 3a20 4c69 7374 2020 2020 2020 2020 4 : List │ │ │ │ +0002ff80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ff90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ffa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ffb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002ffc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ffd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ffe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030010: 2b0a 7c69 3520 3a20 7374 616e 6461 7264 +.|i5 : standard │ │ │ │ -00030020: 4d6f 6e6f 6d69 616c 7328 322c 204d 2920 Monomials(2, M) │ │ │ │ -00030030: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00030040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030060: 7c0a 7c20 2020 2020 2020 3220 2020 2020 |.| 2 │ │ │ │ -00030070: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00030080: 2020 2020 2020 3220 7c0a 7c6f 3520 3d20 2 |.|o5 = │ │ │ │ -00030090: 7b62 202c 2062 2a63 2c20 6320 2c20 612a {b , b*c, c , a* │ │ │ │ -000300a0: 642c 2062 2a64 2c20 632a 642c 2064 207d d, b*d, c*d, d } │ │ │ │ -000300b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000300c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000300d0: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20 |.|o5 : │ │ │ │ -000300e0: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ +00030000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 ------------+..I │ │ │ │ +00030010: 6e70 7574 7469 6e67 2061 6e20 696e 7465 nputting an inte │ │ │ │ +00030020: 6765 7220 2464 2420 286f 7220 6465 6772 ger $d$ (or degr │ │ │ │ +00030030: 6565 2024 6424 2920 616e 6420 616e 2069 ee $d$) and an i │ │ │ │ +00030040: 6465 616c 2067 6976 6573 2074 6865 2073 deal gives the s │ │ │ │ +00030050: 7461 6e64 6172 640a 6d6f 6e6f 6d69 616c tandard.monomial │ │ │ │ +00030060: 7320 666f 7220 7468 6520 7370 6563 6966 s for the specif │ │ │ │ +00030070: 6965 6420 6964 6561 6c20 696e 2064 6567 ied ideal in deg │ │ │ │ +00030080: 7265 6520 2464 242e 0a0a 2b2d 2d2d 2d2d ree $d$...+----- │ │ │ │ +00030090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000300a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000300b0: 2b0a 7c69 3520 3a20 7374 616e 6461 7264 +.|i5 : standard │ │ │ │ +000300c0: 4d6f 6e6f 6d69 616c 7328 322c 204d 2920 Monomials(2, M) │ │ │ │ +000300d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000300e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000300f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030100: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00030110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030120: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -00030130: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00030140: 2a20 2a6e 6f74 6520 7461 696c 4d6f 6e6f * *note tailMono │ │ │ │ -00030150: 6d69 616c 733a 2074 6169 6c4d 6f6e 6f6d mials: tailMonom │ │ │ │ -00030160: 6961 6c73 2c20 2d2d 2066 696e 6420 7461 ials, -- find ta │ │ │ │ -00030170: 696c 206d 6f6e 6f6d 6961 6c73 0a20 202a il monomials. * │ │ │ │ -00030180: 202a 6e6f 7465 2073 6d61 6c6c 6572 4d6f *note smallerMo │ │ │ │ -00030190: 6e6f 6d69 616c 733a 2073 6d61 6c6c 6572 nomials: smaller │ │ │ │ -000301a0: 4d6f 6e6f 6d69 616c 732c 202d 2d20 7265 Monomials, -- re │ │ │ │ -000301b0: 7475 726e 7320 7468 6520 7374 616e 6461 turns the standa │ │ │ │ -000301c0: 7264 206d 6f6e 6f6d 6961 6c73 0a20 2020 rd monomials. │ │ │ │ -000301d0: 2073 6d61 6c6c 6572 2062 7574 206f 6620 smaller but of │ │ │ │ -000301e0: 7468 6520 7361 6d65 2064 6567 7265 6520 the same degree │ │ │ │ -000301f0: 6173 2067 6976 656e 206d 6f6e 6f6d 6961 as given monomia │ │ │ │ -00030200: 6c28 7329 0a0a 5761 7973 2074 6f20 7573 l(s)..Ways to us │ │ │ │ -00030210: 6520 7374 616e 6461 7264 4d6f 6e6f 6d69 e standardMonomi │ │ │ │ -00030220: 616c 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d als:.=========== │ │ │ │ -00030230: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00030240: 3d3d 3d0a 0a20 202a 2022 7374 616e 6461 ===.. * "standa │ │ │ │ -00030250: 7264 4d6f 6e6f 6d69 616c 7328 4964 6561 rdMonomials(Idea │ │ │ │ -00030260: 6c29 220a 2020 2a20 2273 7461 6e64 6172 l)". * "standar │ │ │ │ -00030270: 644d 6f6e 6f6d 6961 6c73 284c 6973 742c dMonomials(List, │ │ │ │ -00030280: 4964 6561 6c29 220a 2020 2a20 2273 7461 Ideal)". * "sta │ │ │ │ -00030290: 6e64 6172 644d 6f6e 6f6d 6961 6c73 285a ndardMonomials(Z │ │ │ │ -000302a0: 5a2c 4964 6561 6c29 220a 0a46 6f72 2074 Z,Ideal)"..For t │ │ │ │ -000302b0: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -000302c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000302d0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -000302e0: 7465 2073 7461 6e64 6172 644d 6f6e 6f6d te standardMonom │ │ │ │ -000302f0: 6961 6c73 3a20 7374 616e 6461 7264 4d6f ials: standardMo │ │ │ │ -00030300: 6e6f 6d69 616c 732c 2069 7320 6120 2a6e nomials, is a *n │ │ │ │ -00030310: 6f74 6520 6d65 7468 6f64 0a66 756e 6374 ote method.funct │ │ │ │ -00030320: 696f 6e3a 2028 4d61 6361 756c 6179 3244 ion: (Macaulay2D │ │ │ │ -00030330: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00030340: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ -00030350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030390: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -000303a0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -000303b0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -000303c0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -000303d0: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ -000303e0: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61 11+ds/M2/Macaula │ │ │ │ -000303f0: 7932 2f70 6163 6b61 6765 732f 0a47 726f y2/packages/.Gro │ │ │ │ -00030400: 6562 6e65 7253 7472 6174 612e 6d32 3a34 ebnerStrata.m2:4 │ │ │ │ -00030410: 3838 3a30 2e0a 1f0a 4669 6c65 3a20 4772 88:0....File: Gr │ │ │ │ -00030420: 6f65 626e 6572 5374 7261 7461 2e69 6e66 oebnerStrata.inf │ │ │ │ -00030430: 6f2c 204e 6f64 653a 2074 6169 6c4d 6f6e o, Node: tailMon │ │ │ │ -00030440: 6f6d 6961 6c73 2c20 5072 6576 3a20 7374 omials, Prev: st │ │ │ │ -00030450: 616e 6461 7264 4d6f 6e6f 6d69 616c 732c andardMonomials, │ │ │ │ -00030460: 2055 703a 2054 6f70 0a0a 7461 696c 4d6f Up: Top..tailMo │ │ │ │ -00030470: 6e6f 6d69 616c 7320 2d2d 2066 696e 6420 nomials -- find │ │ │ │ -00030480: 7461 696c 206d 6f6e 6f6d 6961 6c73 0a2a tail monomials.* │ │ │ │ -00030490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000304a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000304b0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -000304c0: 0a20 2020 2020 2020 204c 203d 2074 6169 . L = tai │ │ │ │ -000304d0: 6c4d 6f6e 6f6d 6961 6c73 204d 0a20 2020 lMonomials M. │ │ │ │ -000304e0: 2020 2020 204c 203d 2074 6169 6c4d 6f6e L = tailMon │ │ │ │ -000304f0: 6f6d 6961 6c73 284d 2c20 6d29 0a20 202a omials(M, m). * │ │ │ │ -00030500: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00030510: 204d 2c20 616e 202a 6e6f 7465 2069 6465 M, an *note ide │ │ │ │ -00030520: 616c 3a20 284d 6163 6175 6c61 7932 446f al: (Macaulay2Do │ │ │ │ -00030530: 6329 4964 6561 6c2c 2c20 244d 2420 7368 c)Ideal,, $M$ sh │ │ │ │ -00030540: 6f75 6c64 2062 6520 6120 6d6f 6e6f 6d69 ould be a monomi │ │ │ │ -00030550: 616c 2069 6465 616c 0a20 2020 2020 2020 al ideal. │ │ │ │ -00030560: 2028 616e 2069 6465 616c 2067 656e 6572 (an ideal gener │ │ │ │ -00030570: 6174 6564 2062 7920 6d6f 6e6f 6d69 616c ated by monomial │ │ │ │ -00030580: 7329 0a20 2020 2020 202a 206d 2c20 6120 s). * m, a │ │ │ │ -00030590: 2a6e 6f74 6520 7269 6e67 2065 6c65 6d65 *note ring eleme │ │ │ │ -000305a0: 6e74 3a20 284d 6163 6175 6c61 7932 446f nt: (Macaulay2Do │ │ │ │ -000305b0: 6329 5269 6e67 456c 656d 656e 742c 2c20 c)RingElement,, │ │ │ │ -000305c0: 6f70 7469 6f6e 616c 2c20 6f6e 6c79 0a20 optional, only. │ │ │ │ -000305d0: 2020 2020 2020 2072 6574 7572 6e20 6120 return a │ │ │ │ -000305e0: 7369 6e67 6c65 206c 6973 7420 6f66 2074 single list of t │ │ │ │ -000305f0: 6865 2074 6169 6c20 6d6f 6e6f 6d69 616c he tail monomial │ │ │ │ -00030600: 7320 666f 7220 7468 6973 206d 6f6e 6f6d s for this monom │ │ │ │ -00030610: 6961 6c0a 2020 2a20 2a6e 6f74 6520 4f70 ial. * *note Op │ │ │ │ -00030620: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ -00030630: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ -00030640: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ -00030650: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ -00030660: 732c 3a0a 2020 2020 2020 2a20 416c 6c53 s,:. * AllS │ │ │ │ -00030670: 7461 6e64 6172 6420 3d3e 2061 202a 6e6f tandard => a *no │ │ │ │ -00030680: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ -00030690: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -000306a0: 426f 6f6c 6561 6e2c 2c20 6465 6661 756c Boolean,, defaul │ │ │ │ -000306b0: 740a 2020 2020 2020 2020 7661 6c75 6520 t. value │ │ │ │ -000306c0: 6661 6c73 652c 2077 6869 6368 206d 6f6e false, which mon │ │ │ │ -000306d0: 6f6d 6961 6c73 2073 686f 756c 6420 6265 omials should be │ │ │ │ -000306e0: 2063 6f6e 7369 6465 7265 6420 7461 696c considered tail │ │ │ │ -000306f0: 206d 6f6e 6f6d 6961 6c73 206f 6620 610a monomials of a. │ │ │ │ -00030700: 2020 2020 2020 2020 6d6f 6e6f 6d69 616c monomial │ │ │ │ -00030710: 2024 6d24 3a20 6569 7468 6572 2061 6c6c $m$: either all │ │ │ │ -00030720: 2073 7461 6e64 6172 6420 6d6f 6e6f 6d69 standard monomi │ │ │ │ -00030730: 616c 7320 6f66 2061 2067 6976 656e 2064 als of a given d │ │ │ │ -00030740: 6567 7265 652c 206f 7220 616c 6c0a 2020 egree, or all. │ │ │ │ -00030750: 2020 2020 2020 6d6f 6e6f 6d69 616c 7320 monomials │ │ │ │ -00030760: 736d 616c 6c65 7220 7468 616e 2024 6d24 smaller than $m$ │ │ │ │ -00030770: 2069 6e20 7468 6520 6769 7665 6e20 7465 in the given te │ │ │ │ -00030780: 726d 206f 7264 6572 2028 6275 7420 7374 rm order (but st │ │ │ │ -00030790: 696c 6c20 6f66 2074 6865 0a20 2020 2020 ill of the. │ │ │ │ -000307a0: 2020 2073 616d 6520 6465 6772 6565 290a same degree). │ │ │ │ -000307b0: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -000307c0: 2020 202a 204c 2c20 6120 2a6e 6f74 6520 * L, a *note │ │ │ │ -000307d0: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ -000307e0: 446f 6329 4c69 7374 2c2c 2061 206c 6973 Doc)List,, a lis │ │ │ │ -000307f0: 7420 6f66 206c 6973 7473 3a20 666f 7220 t of lists: for │ │ │ │ -00030800: 6561 6368 0a20 2020 2020 2020 2067 656e each. gen │ │ │ │ -00030810: 6572 6174 6f72 2024 6d24 206f 6620 244d erator $m$ of $M │ │ │ │ -00030820: 242c 2074 6865 206c 6973 7420 6f66 2061 $, the list of a │ │ │ │ -00030830: 6c6c 2074 6169 6c20 6d6f 6e6f 6d69 616c ll tail monomial │ │ │ │ -00030840: 7320 4966 2069 6e73 7465 6164 2024 6d24 s If instead $m$ │ │ │ │ -00030850: 2069 730a 2020 2020 2020 2020 6769 7665 is. give │ │ │ │ -00030860: 6e2c 2074 6865 206c 6973 7420 6f66 2074 n, the list of t │ │ │ │ -00030870: 6865 2074 6169 6c20 6d6f 6e6f 6d69 616c he tail monomial │ │ │ │ -00030880: 7320 6f66 2024 6d24 2069 7320 7265 7475 s of $m$ is retu │ │ │ │ -00030890: 726e 6564 0a0a 4465 7363 7269 7074 696f rned..Descriptio │ │ │ │ -000308a0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 n.===========..I │ │ │ │ -000308b0: 6e70 7574 7469 6e67 2061 6e20 6964 6561 nputting an idea │ │ │ │ -000308c0: 6c20 244d 2420 6765 6e65 7261 7465 6420 l $M$ generated │ │ │ │ -000308d0: 6279 206d 6f6e 6f6d 6961 6c73 2072 6574 by monomials ret │ │ │ │ -000308e0: 7572 6e73 2061 206c 6973 7420 6f66 206c urns a list of l │ │ │ │ -000308f0: 6973 7473 206f 6620 7461 696c 0a6d 6f6e ists of tail.mon │ │ │ │ -00030900: 6f6d 6961 6c73 2066 6f72 2065 6163 6820 omials for each │ │ │ │ -00030910: 6765 6e65 7261 746f 7220 6f66 2024 4d24 generator of $M$ │ │ │ │ -00030920: 2028 696e 2074 6865 2073 616d 6520 6f72 (in the same or │ │ │ │ -00030930: 6465 7229 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d der)...+-------- │ │ │ │ -00030940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030980: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d -----+.|i1 : R = │ │ │ │ -00030990: 205a 5a2f 3332 3030 335b 612e 2e64 5d3b ZZ/32003[a..d]; │ │ │ │ -000309a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00030100: 7c0a 7c20 2020 2020 2020 3220 2020 2020 |.| 2 │ │ │ │ +00030110: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00030120: 2020 2020 2020 3220 7c0a 7c6f 3520 3d20 2 |.|o5 = │ │ │ │ +00030130: 7b62 202c 2062 2a63 2c20 6320 2c20 612a {b , b*c, c , a* │ │ │ │ +00030140: 642c 2062 2a64 2c20 632a 642c 2064 207d d, b*d, c*d, d } │ │ │ │ +00030150: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00030160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030170: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20 |.|o5 : │ │ │ │ +00030180: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ +00030190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000301a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +000301b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000301c0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ +000301d0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +000301e0: 2a20 2a6e 6f74 6520 7461 696c 4d6f 6e6f * *note tailMono │ │ │ │ +000301f0: 6d69 616c 733a 2074 6169 6c4d 6f6e 6f6d mials: tailMonom │ │ │ │ +00030200: 6961 6c73 2c20 2d2d 2066 696e 6420 7461 ials, -- find ta │ │ │ │ +00030210: 696c 206d 6f6e 6f6d 6961 6c73 0a20 202a il monomials. * │ │ │ │ +00030220: 202a 6e6f 7465 2073 6d61 6c6c 6572 4d6f *note smallerMo │ │ │ │ +00030230: 6e6f 6d69 616c 733a 2073 6d61 6c6c 6572 nomials: smaller │ │ │ │ +00030240: 4d6f 6e6f 6d69 616c 732c 202d 2d20 7265 Monomials, -- re │ │ │ │ +00030250: 7475 726e 7320 7468 6520 7374 616e 6461 turns the standa │ │ │ │ +00030260: 7264 206d 6f6e 6f6d 6961 6c73 0a20 2020 rd monomials. │ │ │ │ +00030270: 2073 6d61 6c6c 6572 2062 7574 206f 6620 smaller but of │ │ │ │ +00030280: 7468 6520 7361 6d65 2064 6567 7265 6520 the same degree │ │ │ │ +00030290: 6173 2067 6976 656e 206d 6f6e 6f6d 6961 as given monomia │ │ │ │ +000302a0: 6c28 7329 0a0a 5761 7973 2074 6f20 7573 l(s)..Ways to us │ │ │ │ +000302b0: 6520 7374 616e 6461 7264 4d6f 6e6f 6d69 e standardMonomi │ │ │ │ +000302c0: 616c 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d als:.=========== │ │ │ │ +000302d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000302e0: 3d3d 3d0a 0a20 202a 2022 7374 616e 6461 ===.. * "standa │ │ │ │ +000302f0: 7264 4d6f 6e6f 6d69 616c 7328 4964 6561 rdMonomials(Idea │ │ │ │ +00030300: 6c29 220a 2020 2a20 2273 7461 6e64 6172 l)". * "standar │ │ │ │ +00030310: 644d 6f6e 6f6d 6961 6c73 284c 6973 742c dMonomials(List, │ │ │ │ +00030320: 4964 6561 6c29 220a 2020 2a20 2273 7461 Ideal)". * "sta │ │ │ │ +00030330: 6e64 6172 644d 6f6e 6f6d 6961 6c73 285a ndardMonomials(Z │ │ │ │ +00030340: 5a2c 4964 6561 6c29 220a 0a46 6f72 2074 Z,Ideal)"..For t │ │ │ │ +00030350: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +00030360: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00030370: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ +00030380: 7465 2073 7461 6e64 6172 644d 6f6e 6f6d te standardMonom │ │ │ │ +00030390: 6961 6c73 3a20 7374 616e 6461 7264 4d6f ials: standardMo │ │ │ │ +000303a0: 6e6f 6d69 616c 732c 2069 7320 6120 2a6e nomials, is a *n │ │ │ │ +000303b0: 6f74 6520 6d65 7468 6f64 0a66 756e 6374 ote method.funct │ │ │ │ +000303c0: 696f 6e3a 2028 4d61 6361 756c 6179 3244 ion: (Macaulay2D │ │ │ │ +000303d0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +000303e0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +000303f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030430: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +00030440: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +00030450: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +00030460: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +00030470: 2f6d 6163 6175 6c61 7932 2d31 2e32 352e /macaulay2-1.25. │ │ │ │ +00030480: 3131 2b64 732f 4d32 2f4d 6163 6175 6c61 11+ds/M2/Macaula │ │ │ │ +00030490: 7932 2f70 6163 6b61 6765 732f 0a47 726f y2/packages/.Gro │ │ │ │ +000304a0: 6562 6e65 7253 7472 6174 612e 6d32 3a34 ebnerStrata.m2:4 │ │ │ │ +000304b0: 3838 3a30 2e0a 1f0a 4669 6c65 3a20 4772 88:0....File: Gr │ │ │ │ +000304c0: 6f65 626e 6572 5374 7261 7461 2e69 6e66 oebnerStrata.inf │ │ │ │ +000304d0: 6f2c 204e 6f64 653a 2074 6169 6c4d 6f6e o, Node: tailMon │ │ │ │ +000304e0: 6f6d 6961 6c73 2c20 5072 6576 3a20 7374 omials, Prev: st │ │ │ │ +000304f0: 616e 6461 7264 4d6f 6e6f 6d69 616c 732c andardMonomials, │ │ │ │ +00030500: 2055 703a 2054 6f70 0a0a 7461 696c 4d6f Up: Top..tailMo │ │ │ │ +00030510: 6e6f 6d69 616c 7320 2d2d 2066 696e 6420 nomials -- find │ │ │ │ +00030520: 7461 696c 206d 6f6e 6f6d 6961 6c73 0a2a tail monomials.* │ │ │ │ +00030530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00030540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00030550: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00030560: 0a20 2020 2020 2020 204c 203d 2074 6169 . L = tai │ │ │ │ +00030570: 6c4d 6f6e 6f6d 6961 6c73 204d 0a20 2020 lMonomials M. │ │ │ │ +00030580: 2020 2020 204c 203d 2074 6169 6c4d 6f6e L = tailMon │ │ │ │ +00030590: 6f6d 6961 6c73 284d 2c20 6d29 0a20 202a omials(M, m). * │ │ │ │ +000305a0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +000305b0: 204d 2c20 616e 202a 6e6f 7465 2069 6465 M, an *note ide │ │ │ │ +000305c0: 616c 3a20 284d 6163 6175 6c61 7932 446f al: (Macaulay2Do │ │ │ │ +000305d0: 6329 4964 6561 6c2c 2c20 244d 2420 7368 c)Ideal,, $M$ sh │ │ │ │ +000305e0: 6f75 6c64 2062 6520 6120 6d6f 6e6f 6d69 ould be a monomi │ │ │ │ +000305f0: 616c 2069 6465 616c 0a20 2020 2020 2020 al ideal. │ │ │ │ +00030600: 2028 616e 2069 6465 616c 2067 656e 6572 (an ideal gener │ │ │ │ +00030610: 6174 6564 2062 7920 6d6f 6e6f 6d69 616c ated by monomial │ │ │ │ +00030620: 7329 0a20 2020 2020 202a 206d 2c20 6120 s). * m, a │ │ │ │ +00030630: 2a6e 6f74 6520 7269 6e67 2065 6c65 6d65 *note ring eleme │ │ │ │ +00030640: 6e74 3a20 284d 6163 6175 6c61 7932 446f nt: (Macaulay2Do │ │ │ │ +00030650: 6329 5269 6e67 456c 656d 656e 742c 2c20 c)RingElement,, │ │ │ │ +00030660: 6f70 7469 6f6e 616c 2c20 6f6e 6c79 0a20 optional, only. │ │ │ │ +00030670: 2020 2020 2020 2072 6574 7572 6e20 6120 return a │ │ │ │ +00030680: 7369 6e67 6c65 206c 6973 7420 6f66 2074 single list of t │ │ │ │ +00030690: 6865 2074 6169 6c20 6d6f 6e6f 6d69 616c he tail monomial │ │ │ │ +000306a0: 7320 666f 7220 7468 6973 206d 6f6e 6f6d s for this monom │ │ │ │ +000306b0: 6961 6c0a 2020 2a20 2a6e 6f74 6520 4f70 ial. * *note Op │ │ │ │ +000306c0: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ +000306d0: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ +000306e0: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ +000306f0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ +00030700: 732c 3a0a 2020 2020 2020 2a20 416c 6c53 s,:. * AllS │ │ │ │ +00030710: 7461 6e64 6172 6420 3d3e 2061 202a 6e6f tandard => a *no │ │ │ │ +00030720: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ +00030730: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00030740: 426f 6f6c 6561 6e2c 2c20 6465 6661 756c Boolean,, defaul │ │ │ │ +00030750: 740a 2020 2020 2020 2020 7661 6c75 6520 t. value │ │ │ │ +00030760: 6661 6c73 652c 2077 6869 6368 206d 6f6e false, which mon │ │ │ │ +00030770: 6f6d 6961 6c73 2073 686f 756c 6420 6265 omials should be │ │ │ │ +00030780: 2063 6f6e 7369 6465 7265 6420 7461 696c considered tail │ │ │ │ +00030790: 206d 6f6e 6f6d 6961 6c73 206f 6620 610a monomials of a. │ │ │ │ +000307a0: 2020 2020 2020 2020 6d6f 6e6f 6d69 616c monomial │ │ │ │ +000307b0: 2024 6d24 3a20 6569 7468 6572 2061 6c6c $m$: either all │ │ │ │ +000307c0: 2073 7461 6e64 6172 6420 6d6f 6e6f 6d69 standard monomi │ │ │ │ +000307d0: 616c 7320 6f66 2061 2067 6976 656e 2064 als of a given d │ │ │ │ +000307e0: 6567 7265 652c 206f 7220 616c 6c0a 2020 egree, or all. │ │ │ │ +000307f0: 2020 2020 2020 6d6f 6e6f 6d69 616c 7320 monomials │ │ │ │ +00030800: 736d 616c 6c65 7220 7468 616e 2024 6d24 smaller than $m$ │ │ │ │ +00030810: 2069 6e20 7468 6520 6769 7665 6e20 7465 in the given te │ │ │ │ +00030820: 726d 206f 7264 6572 2028 6275 7420 7374 rm order (but st │ │ │ │ +00030830: 696c 6c20 6f66 2074 6865 0a20 2020 2020 ill of the. │ │ │ │ +00030840: 2020 2073 616d 6520 6465 6772 6565 290a same degree). │ │ │ │ +00030850: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +00030860: 2020 202a 204c 2c20 6120 2a6e 6f74 6520 * L, a *note │ │ │ │ +00030870: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ +00030880: 446f 6329 4c69 7374 2c2c 2061 206c 6973 Doc)List,, a lis │ │ │ │ +00030890: 7420 6f66 206c 6973 7473 3a20 666f 7220 t of lists: for │ │ │ │ +000308a0: 6561 6368 0a20 2020 2020 2020 2067 656e each. gen │ │ │ │ +000308b0: 6572 6174 6f72 2024 6d24 206f 6620 244d erator $m$ of $M │ │ │ │ +000308c0: 242c 2074 6865 206c 6973 7420 6f66 2061 $, the list of a │ │ │ │ +000308d0: 6c6c 2074 6169 6c20 6d6f 6e6f 6d69 616c ll tail monomial │ │ │ │ +000308e0: 7320 4966 2069 6e73 7465 6164 2024 6d24 s If instead $m$ │ │ │ │ +000308f0: 2069 730a 2020 2020 2020 2020 6769 7665 is. give │ │ │ │ +00030900: 6e2c 2074 6865 206c 6973 7420 6f66 2074 n, the list of t │ │ │ │ +00030910: 6865 2074 6169 6c20 6d6f 6e6f 6d69 616c he tail monomial │ │ │ │ +00030920: 7320 6f66 2024 6d24 2069 7320 7265 7475 s of $m$ is retu │ │ │ │ +00030930: 726e 6564 0a0a 4465 7363 7269 7074 696f rned..Descriptio │ │ │ │ +00030940: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 n.===========..I │ │ │ │ +00030950: 6e70 7574 7469 6e67 2061 6e20 6964 6561 nputting an idea │ │ │ │ +00030960: 6c20 244d 2420 6765 6e65 7261 7465 6420 l $M$ generated │ │ │ │ +00030970: 6279 206d 6f6e 6f6d 6961 6c73 2072 6574 by monomials ret │ │ │ │ +00030980: 7572 6e73 2061 206c 6973 7420 6f66 206c urns a list of l │ │ │ │ +00030990: 6973 7473 206f 6620 7461 696c 0a6d 6f6e ists of tail.mon │ │ │ │ +000309a0: 6f6d 6961 6c73 2066 6f72 2065 6163 6820 omials for each │ │ │ │ +000309b0: 6765 6e65 7261 746f 7220 6f66 2024 4d24 generator of $M$ │ │ │ │ +000309c0: 2028 696e 2074 6865 2073 616d 6520 6f72 (in the same or │ │ │ │ +000309d0: 6465 7229 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d der)...+-------- │ │ │ │ 000309e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000309f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030a20: 2d2d 2d2d 2d2b 0a7c 6932 203a 204d 203d -----+.|i2 : M = │ │ │ │ -00030a30: 2069 6465 616c 2028 615e 322c 2062 5e32 ideal (a^2, b^2 │ │ │ │ -00030a40: 2c20 612a 622a 6329 3b20 2020 2020 2020 , a*b*c); │ │ │ │ +00030a20: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d -----+.|i1 : R = │ │ │ │ +00030a30: 205a 5a2f 3332 3030 335b 612e 2e64 5d3b ZZ/32003[a..d]; │ │ │ │ +00030a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00030a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ac0: 2020 2020 207c 0a7c 6f32 203a 2049 6465 |.|o2 : Ide │ │ │ │ -00030ad0: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ -00030ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030a70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00030a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030ac0: 2d2d 2d2d 2d2b 0a7c 6932 203a 204d 203d -----+.|i2 : M = │ │ │ │ +00030ad0: 2069 6465 616c 2028 615e 322c 2062 5e32 ideal (a^2, b^2 │ │ │ │ +00030ae0: 2c20 612a 622a 6329 3b20 2020 2020 2020 , a*b*c); │ │ │ │ 00030af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00030b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030b60: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 6169 -----+.|i3 : tai │ │ │ │ -00030b70: 6c4d 6f6e 6f6d 6961 6c73 204d 2020 2020 lMonomials M │ │ │ │ +00030b10: 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 │ │ │ │ +00030b60: 2020 2020 207c 0a7c 6f32 203a 2049 6465 |.|o2 : Ide │ │ │ │ +00030b70: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ 00030b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030bb0: 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 │ │ │ │ -00030c00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00030c10: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +00030bb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00030bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030c00: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 6169 -----+.|i3 : tai │ │ │ │ +00030c10: 6c4d 6f6e 6f6d 6961 6c73 204d 2020 2020 lMonomials M │ │ │ │ 00030c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c30: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00030c40: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00030c50: 2020 2020 207c 0a7c 6f33 203d 207b 7b61 |.|o3 = {{a │ │ │ │ -00030c60: 2a62 2c20 612a 632c 2062 2a63 2c20 6320 *b, a*c, b*c, c │ │ │ │ -00030c70: 2c20 612a 642c 2062 2a64 2c20 632a 642c , a*d, b*d, c*d, │ │ │ │ -00030c80: 2064 207d 2c20 7b61 2a63 2c20 622a 632c d }, {a*c, b*c, │ │ │ │ -00030c90: 2063 202c 2061 2a64 2c20 622a 642c 2063 c , a*d, b*d, c │ │ │ │ -00030ca0: 2a64 2c20 207c 0a7c 2020 2020 202d 2d2d *d, |.| --- │ │ │ │ -00030cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030cf0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3220 -----|.| 2 │ │ │ │ -00030d00: 2020 2020 2020 3220 2020 2020 3220 2020 2 2 │ │ │ │ -00030d10: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -00030d20: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00030d30: 3220 2020 2020 3220 2020 2020 3220 2020 2 2 2 │ │ │ │ -00030d40: 3320 2020 207c 0a7c 2020 2020 2064 207d 3 |.| d } │ │ │ │ -00030d50: 2c20 7b61 2a63 202c 2062 2a63 202c 2063 , {a*c , b*c , c │ │ │ │ -00030d60: 202c 2061 2a62 2a64 2c20 612a 632a 642c , a*b*d, a*c*d, │ │ │ │ -00030d70: 2062 2a63 2a64 2c20 6320 642c 2061 2a64 b*c*d, c d, a*d │ │ │ │ -00030d80: 202c 2062 2a64 202c 2063 2a64 202c 2064 , b*d , c*d , d │ │ │ │ -00030d90: 207d 7d20 207c 0a7c 2020 2020 2020 2020 }} |.| │ │ │ │ -00030da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030de0: 2020 2020 207c 0a7c 6f33 203a 204c 6973 |.|o3 : Lis │ │ │ │ -00030df0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00030e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00030e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030e70: 2d2d 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2 │ │ │ │ +00030ce0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00030cf0: 2020 2020 207c 0a7c 6f33 203d 207b 7b61 |.|o3 = {{a │ │ │ │ +00030d00: 2a62 2c20 612a 632c 2062 2a63 2c20 6320 *b, a*c, b*c, c │ │ │ │ +00030d10: 2c20 612a 642c 2062 2a64 2c20 632a 642c , a*d, b*d, c*d, │ │ │ │ +00030d20: 2064 207d 2c20 7b61 2a63 2c20 622a 632c d }, {a*c, b*c, │ │ │ │ +00030d30: 2063 202c 2061 2a64 2c20 622a 642c 2063 c , a*d, b*d, c │ │ │ │ +00030d40: 2a64 2c20 207c 0a7c 2020 2020 202d 2d2d *d, |.| --- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030d90: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3220 -----|.| 2 │ │ │ │ +00030da0: 2020 2020 2020 3220 2020 2020 3220 2020 2 2 │ │ │ │ +00030db0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00030dc0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00030dd0: 3220 2020 2020 3220 2020 2020 3220 2020 2 2 2 │ │ │ │ +00030de0: 3320 2020 207c 0a7c 2020 2020 2064 207d 3 |.| d } │ │ │ │ +00030df0: 2c20 7b61 2a63 202c 2062 2a63 202c 2063 , {a*c , b*c , c │ │ │ │ +00030e00: 202c 2061 2a62 2a64 2c20 612a 632a 642c , a*b*d, a*c*d, │ │ │ │ +00030e10: 2062 2a63 2a64 2c20 6320 642c 2061 2a64 b*c*d, c d, a*d │ │ │ │ +00030e20: 202c 2062 2a64 202c 2063 2a64 202c 2064 , b*d , c*d , d │ │ │ │ +00030e30: 207d 7d20 207c 0a7c 2020 2020 2020 2020 }} |.| │ │ │ │ +00030e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030e80: 2020 2020 207c 0a7c 6f33 203a 204c 6973 |.|o3 : Lis │ │ │ │ +00030e90: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00030ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ed0: 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 │ │ │ │ -00030f20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00030f30: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00030f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f50: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00030f60: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00030f70: 2020 2020 207c 0a7c 6f34 203d 207b 7b61 |.|o4 = {{a │ │ │ │ -00030f80: 2a62 2c20 612a 632c 2062 2a63 2c20 6320 *b, a*c, b*c, c │ │ │ │ -00030f90: 2c20 612a 642c 2062 2a64 2c20 632a 642c , a*d, b*d, c*d, │ │ │ │ -00030fa0: 2064 207d 2c20 7b61 2a62 2c20 612a 632c d }, {a*b, a*c, │ │ │ │ -00030fb0: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ -00030fc0: 2a64 2c20 207c 0a7c 2020 2020 202d 2d2d *d, |.| --- │ │ │ │ -00030fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031010: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -00031020: 2020 2032 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ -00031030: 2032 2020 2033 2020 2020 2020 2020 2020 2 3 │ │ │ │ -00031040: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00031050: 2020 2020 2032 2020 2020 2032 2020 2020 2 2 │ │ │ │ -00031060: 2032 2020 207c 0a7c 2020 2020 2063 2a64 2 |.| c*d │ │ │ │ -00031070: 2c20 6420 7d2c 207b 612a 6320 2c20 622a , d }, {a*c , b* │ │ │ │ -00031080: 6320 2c20 6320 2c20 612a 622a 642c 2061 c , c , a*b*d, a │ │ │ │ -00031090: 2a63 2a64 2c20 622a 632a 642c 2063 2064 *c*d, b*c*d, c d │ │ │ │ -000310a0: 2c20 612a 6420 2c20 622a 6420 2c20 632a , a*d , b*d , c* │ │ │ │ -000310b0: 6420 2c20 207c 0a7c 2020 2020 202d 2d2d d , |.| --- │ │ │ │ -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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031100: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3320 -----|.| 3 │ │ │ │ -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 2020 2020 2020 2020 2020 │ │ │ │ -00031150: 2020 2020 207c 0a7c 2020 2020 2064 207d |.| d } │ │ │ │ -00031160: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ -00031170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00030ed0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00030ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030f20: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 6169 -----+.|i4 : tai │ │ │ │ +00030f30: 6c4d 6f6e 6f6d 6961 6c73 284d 2c20 416c lMonomials(M, Al │ │ │ │ +00030f40: 6c53 7461 6e64 6172 6420 3d3e 2074 7275 lStandard => tru │ │ │ │ +00030f50: 6529 2020 2020 2020 2020 2020 2020 2020 e) │ │ │ │ +00030f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030f70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00030f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00030fd0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +00030fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030ff0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00031000: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00031010: 2020 2020 207c 0a7c 6f34 203d 207b 7b61 |.|o4 = {{a │ │ │ │ +00031020: 2a62 2c20 612a 632c 2062 2a63 2c20 6320 *b, a*c, b*c, c │ │ │ │ +00031030: 2c20 612a 642c 2062 2a64 2c20 632a 642c , a*d, b*d, c*d, │ │ │ │ +00031040: 2064 207d 2c20 7b61 2a62 2c20 612a 632c d }, {a*b, a*c, │ │ │ │ +00031050: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ +00031060: 2a64 2c20 207c 0a7c 2020 2020 202d 2d2d *d, |.| --- │ │ │ │ +00031070: 2d2d 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207c 0a7c 2020 2020 202d 2d2d d , |.| --- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000311a0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3320 -----|.| 3 │ │ │ │ 000311b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000311c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000311d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000311e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311f0: 2020 2020 207c 0a7c 6f34 203a 204c 6973 |.|o4 : Lis │ │ │ │ -00031200: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +000311f0: 2020 2020 207c 0a7c 2020 2020 2064 207d |.| d } │ │ │ │ +00031200: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ 00031210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031240: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00031250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031290: 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 6169 -----+.|i5 : tai │ │ │ │ -000312a0: 6c4d 6f6e 6f6d 6961 6c73 284d 2c20 625e lMonomials(M, b^ │ │ │ │ -000312b0: 3229 2020 2020 2020 2020 2020 2020 2020 2) │ │ │ │ +00031240: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00031250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031290: 2020 2020 207c 0a7c 6f34 203a 204c 6973 |.|o4 : Lis │ │ │ │ +000312a0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +000312b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000312c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000312d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000312e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000312f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031330: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00031340: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00031350: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +000312e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000312f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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programmer. │ │ │ │ +00031910: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00031920: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00031930: 6e6f 7465 2074 6169 6c4d 6f6e 6f6d 6961 note tailMonomia │ │ │ │ +00031940: 6c73 3a20 7461 696c 4d6f 6e6f 6d69 616c ls: tailMonomial │ │ │ │ +00031950: 732c 2069 7320 6120 2a6e 6f74 6520 6d65 s, is a *note me │ │ │ │ +00031960: 7468 6f64 2066 756e 6374 696f 6e20 7769 thod function wi │ │ │ │ +00031970: 7468 0a6f 7074 696f 6e73 3a20 284d 6163 th.options: (Mac │ │ │ │ +00031980: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ +00031990: 4675 6e63 7469 6f6e 5769 7468 4f70 7469 FunctionWithOpti │ │ │ │ +000319a0: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ons,...--------- │ │ │ │ +000319b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000319c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000319d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- 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6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ +00031b90: 5f6c 7049 6465 616c 5f72 707f 3134 3937 _lpIdeal_rp.1497 │ │ │ │ +00031ba0: 3733 0a4e 6f64 653a 2072 616e 646f 6d50 73.Node: randomP │ │ │ │ +00031bb0: 6f69 6e74 734f 6e52 6174 696f 6e61 6c56 ointsOnRationalV │ │ │ │ +00031bc0: 6172 6965 7479 5f6c 7049 6465 616c 5f63 ariety_lpIdeal_c │ │ │ │ +00031bd0: 6d5a 5a5f 7270 7f31 3734 3039 300a 4e6f mZZ_rp.174090.No │ │ │ │ +00031be0: 6465 3a20 736d 616c 6c65 724d 6f6e 6f6d de: smallerMonom │ │ │ │ +00031bf0: 6961 6c73 7f31 3839 3834 330a 4e6f 6465 ials.189843.Node │ │ │ │ +00031c00: 3a20 7374 616e 6461 7264 4d6f 6e6f 6d69 : standardMonomi │ │ │ │ +00031c10: 616c 737f 3139 3335 3836 0a4e 6f64 653a als.193586.Node: │ │ │ │ +00031c20: 2074 6169 6c4d 6f6e 6f6d 6961 6c73 7f31 tailMonomials.1 │ │ │ │ +00031c30: 3937 3831 340a 1f0a 456e 6420 5461 6720 97814...End Tag │ │ │ │ +00031c40: 5461 626c 650a Table. │ │ ├── ./usr/share/info/GroebnerWalk.info.gz │ │ │ ├── GroebnerWalk.info │ │ │ │ @@ -207,16 +207,16 @@ │ │ │ │ 00000ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00000d10: 7c69 3520 3a20 656c 6170 7365 6454 696d |i5 : elapsedTim │ │ │ │ 00000d20: 6520 6762 2049 3220 2020 2020 2020 2020 e gb I2 │ │ │ │ 00000d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00000d50: 2020 2020 2020 7c0a 7c20 2d2d 2033 2e31 |.| -- 3.1 │ │ │ │ -00000d60: 3439 3637 7320 656c 6170 7365 6420 2020 4967s elapsed │ │ │ │ +00000d50: 2020 2020 2020 7c0a 7c20 2d2d 2032 2e32 |.| -- 2.2 │ │ │ │ +00000d60: 3532 3831 7320 656c 6170 7365 6420 2020 5281s elapsed │ │ │ │ 00000d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000d90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00000da0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00000db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -249,15 +249,15 @@ │ │ │ │ 00000f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000fa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2065 -------+.|i6 : e │ │ │ │ 00000fb0: 6c61 7073 6564 5469 6d65 2067 726f 6562 lapsedTime groeb │ │ │ │ 00000fc0: 6e65 7257 616c 6b28 6762 2049 312c 2052 nerWalk(gb I1, R │ │ │ │ 00000fd0: 3229 2020 2020 2020 2020 2020 2020 2020 2) │ │ │ │ 00000fe0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00000ff0: 7c20 2d2d 2032 2e31 3532 3239 7320 656c | -- 2.15229s el │ │ │ │ +00000ff0: 7c20 2d2d 2031 2e37 3030 3132 7320 656c | -- 1.70012s el │ │ │ │ 00001000: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00001010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001030: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00001040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/HolonomicSystems.info.gz │ │ │ ├── HolonomicSystems.info │ │ │ │ @@ -4008,36 +4008,36 @@ │ │ │ │ 0000fa70: 7272 656e 7420 636f 6566 6669 6369 656e rrent coefficien │ │ │ │ 0000fa80: 7420 7269 6e67 206f 7220 2020 2020 2020 t ring or │ │ │ │ 0000fa90: 207c 0a7c 436f 6e76 6572 7469 6e67 2074 |.|Converting t │ │ │ │ 0000faa0: 6f20 4e61 6976 6520 616c 676f 7269 7468 o Naive algorith │ │ │ │ 0000fab0: 6d2e 2020 2020 2020 2020 2020 2020 2020 m. │ │ │ │ 0000fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fae0: 207c 0a7c 202d 2d20 2e30 3030 3030 3530 |.| -- .0000050 │ │ │ │ -0000faf0: 3873 2065 6c61 7073 6564 2020 2020 2020 8s elapsed │ │ │ │ +0000fae0: 207c 0a7c 202d 2d20 2e30 3030 3030 3535 |.| -- .0000055 │ │ │ │ +0000faf0: 3938 7320 656c 6170 7365 6420 2020 2020 98s elapsed │ │ │ │ 0000fb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fb30: 207c 0a7c 202d 2d20 2e30 3030 3030 3330 |.| -- .0000030 │ │ │ │ -0000fb40: 3535 7320 656c 6170 7365 6420 2020 2020 55s elapsed │ │ │ │ +0000fb30: 207c 0a7c 202d 2d20 2e30 3030 3030 3731 |.| -- .0000071 │ │ │ │ +0000fb40: 3232 7320 656c 6170 7365 6420 2020 2020 22s elapsed │ │ │ │ 0000fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fb80: 207c 0a7c 202d 2d20 2e30 3030 3030 3536 |.| -- .0000056 │ │ │ │ -0000fb90: 3173 2065 6c61 7073 6564 2020 2020 2020 1s elapsed │ │ │ │ +0000fb80: 207c 0a7c 202d 2d20 2e30 3030 3030 3637 |.| -- .0000067 │ │ │ │ +0000fb90: 3339 7320 656c 6170 7365 6420 2020 2020 39s elapsed │ │ │ │ 0000fba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fbd0: 207c 0a7c 202d 2d20 2e30 3030 3030 3337 |.| -- .0000037 │ │ │ │ -0000fbe0: 3237 7320 656c 6170 7365 6420 2020 2020 27s elapsed │ │ │ │ +0000fbd0: 207c 0a7c 202d 2d20 2e30 3030 3030 3631 |.| -- .0000061 │ │ │ │ +0000fbe0: 3973 2065 6c61 7073 6564 2020 2020 2020 9s elapsed │ │ │ │ 0000fbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fc20: 207c 0a7c 202d 2d20 2e30 3030 3030 3334 |.| -- .0000034 │ │ │ │ -0000fc30: 3736 7320 656c 6170 7365 6420 2020 2020 76s elapsed │ │ │ │ +0000fc20: 207c 0a7c 202d 2d20 2e30 3030 3030 3832 |.| -- .0000082 │ │ │ │ +0000fc30: 3732 7320 656c 6170 7365 6420 2020 2020 72s elapsed │ │ │ │ 0000fc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000fc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5558,15 +5558,15 @@ │ │ │ │ 00015b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00015b70: 3320 3a20 736f 6c76 6546 726f 6265 6e69 3 : solveFrobeni │ │ │ │ 00015b80: 7573 4964 6561 6c20 4920 2020 2020 2020 usIdeal I │ │ │ │ 00015b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00015bc0: 2d2d 202e 3030 3030 3034 3532 3973 2065 -- .000004529s e │ │ │ │ +00015bc0: 2d2d 202e 3030 3030 3036 3535 3973 2065 -- .000006559s e │ │ │ │ 00015bd0: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00015be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015c00: 2020 2020 2020 2020 2020 2020 7c0a 7c57 |.|W │ │ │ │ 00015c10: 6172 6e69 6e67 3a20 2046 3420 416c 676f arning: F4 Algo │ │ │ │ 00015c20: 7269 7468 6d20 6e6f 7420 6176 6169 6c61 rithm not availa │ │ │ │ 00015c30: 626c 6520 6f76 6572 2063 7572 7265 6e74 ble over current │ │ │ │ @@ -5678,15 +5678,15 @@ │ │ │ │ 000162d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000162e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 000162f0: 3520 3a20 736f 6c76 6546 726f 6265 6e69 5 : solveFrobeni │ │ │ │ 00016300: 7573 4964 6561 6c28 492c 2057 2920 2020 usIdeal(I, W) │ │ │ │ 00016310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016330: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00016340: 2d2d 202e 3030 3030 3034 3436 3973 2065 -- .000004469s e │ │ │ │ +00016340: 2d2d 202e 3030 3030 3036 3931 3273 2065 -- .000006912s e │ │ │ │ 00016350: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00016360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016380: 2020 2020 2020 2020 2020 2020 7c0a 7c57 |.|W │ │ │ │ 00016390: 6172 6e69 6e67 3a20 2046 3420 416c 676f arning: F4 Algo │ │ │ │ 000163a0: 7269 7468 6d20 6e6f 7420 6176 6169 6c61 rithm not availa │ │ │ │ 000163b0: 626c 6520 6f76 6572 2063 7572 7265 6e74 ble over current │ │ ├── ./usr/share/info/HomotopyLieAlgebra.info.gz │ │ │ ├── HomotopyLieAlgebra.info │ │ │ │ @@ -2134,351 +2134,351 @@ │ │ │ │ 00008550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008560: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00008570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000085a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000085b0: 2020 7c0a 7c6f 3135 203d 207b 287b 5420 |.|o15 = {({T │ │ │ │ -000085c0: 2c20 5420 7d2c 2054 2054 2020 2b20 5420 , T }, T T + T │ │ │ │ -000085d0: 5420 202d 207a 2a54 2020 202b 2079 2a54 T - z*T + y*T │ │ │ │ -000085e0: 2020 292c 2028 7b54 202c 2054 207d 2c20 ), ({T , T }, │ │ │ │ -000085f0: 2d20 5420 5420 202b 2079 2a54 2020 292c - T T + y*T ), │ │ │ │ -00008600: 2020 7c0a 7c20 2020 2020 2020 2020 2033 |.| 3 │ │ │ │ -00008610: 2020 2037 2020 2020 3420 3620 2020 2033 7 4 6 3 │ │ │ │ -00008620: 2037 2020 2020 2020 3131 2020 2020 2020 7 11 │ │ │ │ -00008630: 3133 2020 2020 2020 3220 2020 3920 2020 13 2 9 │ │ │ │ -00008640: 2020 2032 2039 2020 2020 2020 3136 2020 2 9 16 │ │ │ │ +000085c0: 2c20 5420 7d2c 202d 2054 2054 2020 2d20 , T }, - T T - │ │ │ │ +000085d0: 5420 5420 202d 207a 2a54 2020 202b 2078 T T - z*T + x │ │ │ │ +000085e0: 2a54 2020 292c 2028 7b54 202c 2054 207d *T ), ({T , T } │ │ │ │ +000085f0: 2c20 2d20 5420 5420 202b 207a 2a54 2020 , - T T + z*T │ │ │ │ +00008600: 292c 7c0a 7c20 2020 2020 2020 2020 2031 ),|.| 1 │ │ │ │ +00008610: 2020 2039 2020 2020 2020 3520 3620 2020 9 5 6 │ │ │ │ +00008620: 2031 2039 2020 2020 2020 3134 2020 2020 1 9 14 │ │ │ │ +00008630: 2020 3137 2020 2020 2020 3420 2020 3720 17 4 7 │ │ │ │ +00008640: 2020 2020 2034 2037 2020 2020 2020 3133 4 7 13 │ │ │ │ 00008650: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00008660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000086a0: 2d2d 7c0a 7c20 2020 2020 2028 7b54 202c --|.| ({T , │ │ │ │ -000086b0: 2054 207d 2c20 2d20 5420 5420 202d 2054 T }, - T T 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3130 2020 2020 3420 3620 2020 2031 10 4 6 1 │ │ │ │ -00008800: 2031 3020 2020 2020 2032 3020 2020 2020 10 20 │ │ │ │ -00008810: 2035 2020 2039 2020 2020 2020 3520 3920 5 9 5 9 │ │ │ │ -00008820: 2020 2020 2031 3620 2020 2020 2033 2020 16 3 │ │ │ │ +00008790: 2d2d 7c0a 7c20 2020 2020 2054 207d 2c20 --|.| T }, │ │ │ │ +000087a0: 5420 5420 202d 207a 2a54 2020 202b 2078 T T - z*T + x │ │ │ │ +000087b0: 2a54 2020 292c 2028 7b54 202c 2054 207d *T ), ({T , T } │ │ │ │ +000087c0: 2c20 2d20 5420 5420 202d 2054 2054 2020 , - T T - T T │ │ │ │ +000087d0: 2d20 7a2a 5420 2020 2b20 782a 5420 2029 - z*T + x*T ) │ │ │ │ +000087e0: 2c20 7c0a 7c20 2020 2020 2020 3720 2020 , |.| 7 │ │ │ │ +000087f0: 2033 2037 2020 2020 2020 3131 2020 2020 3 7 11 │ │ │ │ +00008800: 2020 3132 2020 2020 2020 3520 2020 3620 12 5 6 │ │ │ │ +00008810: 2020 2020 2035 2036 2020 2020 3120 3920 5 6 1 9 │ │ │ │ +00008820: 2020 2020 2031 3420 2020 2020 2031 3720 14 17 │ │ │ │ 00008830: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00008840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00008880: 2d2d 7c0a 7c20 2020 2020 2054 207d 2c20 --|.| T }, │ │ │ │ -00008890: 5420 5420 202d 207a 2a54 2020 202b 2078 T T - z*T + x │ │ │ │ -000088a0: 2a54 2020 292c 2028 7b54 202c 2054 207d *T ), ({T , T } │ │ │ │ -000088b0: 2c20 2d20 5420 5420 202d 2054 2054 2020 , - T T - T T │ │ │ │ -000088c0: 2d20 7a2a 5420 2020 2b20 782a 5420 2029 - z*T + x*T ) │ │ │ │ -000088d0: 2c20 7c0a 7c20 2020 2020 2020 3720 2020 , |.| 7 │ │ │ │ -000088e0: 2033 2037 2020 2020 2020 3131 2020 2020 3 7 11 │ │ │ │ -000088f0: 2020 3132 2020 2020 2020 3520 2020 3620 12 5 6 │ │ │ │ -00008900: 2020 2020 2035 2036 2020 2020 3120 3920 5 6 1 9 │ │ │ │ -00008910: 2020 2020 2031 3420 2020 2020 2031 3720 14 17 │ │ │ │ +00008880: 2d2d 7c0a 7c20 2020 2020 2028 7b54 202c --|.| ({T , │ │ │ │ +00008890: 2054 207d 2c20 5420 5420 202b 2054 2054 T }, T T + T T │ │ │ │ +000088a0: 2020 2d20 7a2a 5420 2020 2b20 782a 5420 - z*T + x*T │ │ │ │ +000088b0: 2029 2c20 287b 5420 2c20 5420 7d2c 202d ), ({T , T }, - │ │ │ │ +000088c0: 2054 2054 2020 2d20 5420 5420 202b 2020 T T - T T + │ │ │ │ +000088d0: 2020 7c0a 7c20 2020 2020 2020 2020 3520 |.| 5 │ │ │ │ +000088e0: 2020 3820 2020 2035 2038 2020 2020 3320 8 5 8 3 │ │ │ │ +000088f0: 3920 2020 2020 2031 3520 2020 2020 2031 9 15 1 │ │ │ │ +00008900: 3620 2020 2020 2034 2020 2038 2020 2020 6 4 8 │ │ │ │ +00008910: 2020 3220 3720 2020 2034 2038 2020 2020 2 7 4 8 │ │ │ │ 00008920: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00008930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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5420 202d 207a 2a54 T }, T T - z*T │ │ │ │ -00008a90: 2020 202b 2079 2a54 2020 292c 2028 7b54 + y*T ), ({T │ │ │ │ -00008aa0: 202c 2054 207d 2c20 2d20 5420 5420 202d , T }, - T T - │ │ │ │ -00008ab0: 2020 7c0a 7c20 2020 2020 2020 2020 3132 |.| 12 │ │ │ │ -00008ac0: 2020 2020 2020 3134 2020 2020 2020 3320 14 3 │ │ │ │ -00008ad0: 2020 3920 2020 2033 2039 2020 2020 2020 9 3 9 │ │ │ │ -00008ae0: 3135 2020 2020 2020 3137 2020 2020 2020 15 17 │ │ │ │ -00008af0: 3320 2020 3620 2020 2020 2033 2036 2020 3 6 3 6 │ │ │ │ +00008a60: 2d2d 7c0a 7c20 2020 2020 2054 2054 2020 --|.| T T │ │ │ │ +00008a70: 2d20 5420 5420 202b 207a 2a54 2020 202b - T T + z*T + │ │ │ │ +00008a80: 2078 2a54 2020 292c 2028 7b54 202c 2054 x*T ), ({T , T │ │ │ │ +00008a90: 207d 2c20 2d20 5420 5420 202b 2078 2a54 }, - T T + x*T │ │ │ │ +00008aa0: 2020 292c 2028 7b54 202c 2054 207d 2c20 ), ({T , T }, │ │ │ │ +00008ab0: 2d20 7c0a 7c20 2020 2020 2020 3520 3720 - |.| 5 7 │ │ │ │ +00008ac0: 2020 2031 2038 2020 2020 2020 3132 2020 1 8 12 │ │ │ │ +00008ad0: 2020 2020 3134 2020 2020 2020 3120 2020 14 1 │ │ │ │ +00008ae0: 3720 2020 2020 2031 2037 2020 2020 2020 7 1 7 │ │ │ │ +00008af0: 3133 2020 2020 2020 3520 2020 3920 2020 13 5 9 │ │ │ │ 00008b00: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00008b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008b50: 2d2d 7c0a 7c20 2020 2020 2054 2054 2020 --|.| T T │ │ │ │ -00008b60: 2d20 5420 5420 202b 207a 2a54 2020 202b - T T + z*T + │ │ │ │ -00008b70: 2078 2a54 2020 292c 2028 7b54 202c 2054 x*T ), ({T , T │ │ │ │ -00008b80: 207d 2c20 2d20 5420 5420 202b 2078 2a54 }, - T T + x*T │ │ │ │ -00008b90: 2020 292c 2028 7b54 202c 2054 207d 2c20 ), ({T , T }, │ │ │ │ -00008ba0: 2d20 7c0a 7c20 2020 2020 2020 3520 3720 - |.| 5 7 │ │ │ │ -00008bb0: 2020 2031 2038 2020 2020 2020 3132 2020 1 8 12 │ │ │ │ -00008bc0: 2020 2020 3134 2020 2020 2020 3120 2020 14 1 │ │ │ │ -00008bd0: 3720 2020 2020 2031 2037 2020 2020 2020 7 1 7 │ │ │ │ -00008be0: 3133 2020 2020 2020 3520 2020 3920 2020 13 5 9 │ │ │ │ +00008b60: 2d20 5420 5420 202b 2079 2a54 2020 292c - T T + y*T ), │ │ │ │ +00008b70: 2028 7b54 202c 2054 2020 7d2c 202d 2054 ({T , T }, - T │ │ │ │ +00008b80: 2054 2020 2b20 5420 5420 2020 2d20 7a2a T + T T - z* │ │ │ │ +00008b90: 5420 2020 2b20 782a 5420 2029 2c20 2020 T + x*T ), │ │ │ │ +00008ba0: 2020 7c0a 7c20 2020 2020 2020 3220 3820 |.| 2 8 │ │ │ │ +00008bb0: 2020 2035 2039 2020 2020 2020 3135 2020 5 9 15 │ │ │ │ +00008bc0: 2020 2020 3320 2020 3130 2020 2020 2020 3 10 │ │ │ │ +00008bd0: 3420 3820 2020 2033 2031 3020 2020 2020 4 8 3 10 │ │ │ │ +00008be0: 2031 3820 2020 2020 2031 3920 2020 2020 18 19 │ │ │ │ 00008bf0: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00008c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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({T , │ │ │ │ +00008c50: 2054 207d 2c20 2d20 5420 5420 202d 2054 T }, - T T - T │ │ │ │ +00008c60: 2054 2020 2b20 792a 5420 2020 2b20 7a2a T + y*T + z* │ │ │ │ +00008c70: 5420 2029 2c20 287b 5420 2c20 5420 7d2c T ), ({T , T }, │ │ │ │ +00008c80: 202d 2054 2054 2020 2b20 5420 5420 2020 - T T + T T │ │ │ │ +00008c90: 2d20 7c0a 7c20 2020 2020 2020 2020 3220 - |.| 2 │ │ │ │ +00008ca0: 2020 3720 2020 2020 2032 2037 2020 2020 7 2 7 │ │ │ │ +00008cb0: 3420 3820 2020 2020 2031 3220 2020 2020 4 8 12 │ │ │ │ +00008cc0: 2031 3420 2020 2020 2034 2020 2038 2020 14 4 8 │ │ │ │ +00008cd0: 2020 2020 3420 3820 2020 2033 2031 3020 4 8 3 10 │ │ │ │ 00008ce0: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00008cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00008d30: 2d2d 7c0a 7c20 2020 2020 2028 7b54 202c --|.| ({T , │ │ │ │ -00008d40: 2054 207d 2c20 2d20 5420 5420 202d 2054 T }, - T T - T │ │ │ │ -00008d50: 2054 2020 2b20 792a 5420 2020 2b20 7a2a T + y*T + z* │ │ │ │ -00008d60: 5420 2029 2c20 287b 5420 2c20 5420 7d2c T ), ({T , T }, │ │ │ │ -00008d70: 202d 2054 2054 2020 2b20 5420 5420 2020 - T T + T T │ │ │ │ -00008d80: 2d20 7c0a 7c20 2020 2020 2020 2020 3220 - |.| 2 │ │ │ │ -00008d90: 2020 3720 2020 2020 2032 2037 2020 2020 7 2 7 │ │ │ │ -00008da0: 3420 3820 2020 2020 2031 3220 2020 2020 4 8 12 │ │ │ │ -00008db0: 2031 3420 2020 2020 2034 2020 2038 2020 14 4 8 │ │ │ │ -00008dc0: 2020 2020 3420 3820 2020 2033 2031 3020 4 8 3 10 │ │ │ │ +00008d30: 2d2d 7c0a 7c20 2020 2020 207a 2a54 2020 --|.| z*T │ │ │ │ +00008d40: 202b 2078 2a54 2020 292c 2028 7b54 202c + x*T ), ({T , │ │ │ │ +00008d50: 2054 2020 7d2c 202d 2054 2054 2020 2d20 T }, - T T - │ │ │ │ +00008d60: 5420 5420 2020 2b20 792a 5420 2029 2c20 T T + y*T ), │ │ │ │ +00008d70: 287b 5420 2c20 5420 207d 2c20 2d20 2020 ({T , T }, - │ │ │ │ +00008d80: 2020 7c0a 7c20 2020 2020 2020 2020 3138 |.| 18 │ │ │ │ +00008d90: 2020 2020 2020 3139 2020 2020 2020 3220 19 2 │ │ │ │ +00008da0: 2020 3130 2020 2020 2020 3520 3820 2020 10 5 8 │ │ │ │ +00008db0: 2032 2031 3020 2020 2020 2031 3920 2020 2 10 19 │ │ │ │ +00008dc0: 2020 2034 2020 2031 3020 2020 2020 2020 4 10 │ │ │ │ 00008dd0: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00008de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00008e20: 2d2d 7c0a 7c20 2020 2020 207a 2a54 2020 --|.| z*T │ │ │ │ +00008e20: 2d2d 7c0a 7c20 2020 2020 2054 2054 2020 --|.| T T │ │ │ │ 00008e30: 202b 2078 2a54 2020 292c 2028 7b54 202c + x*T ), ({T , │ │ │ │ -00008e40: 2054 2020 7d2c 202d 2054 2054 2020 2d20 T }, - T T - │ │ │ │ -00008e50: 5420 5420 2020 2b20 792a 5420 2029 2c20 T T + y*T ), │ │ │ │ -00008e60: 287b 5420 2c20 5420 207d 2c20 2d20 2020 ({T , T }, - │ │ │ │ -00008e70: 2020 7c0a 7c20 2020 2020 2020 2020 3138 |.| 18 │ │ │ │ -00008e80: 2020 2020 2020 3139 2020 2020 2020 3220 19 2 │ │ │ │ -00008e90: 2020 3130 2020 2020 2020 3520 3820 2020 10 5 8 │ │ │ │ -00008ea0: 2032 2031 3020 2020 2020 2031 3920 2020 2 10 19 │ │ │ │ -00008eb0: 2020 2034 2020 2031 3020 2020 2020 2020 4 10 │ │ │ │ +00008e40: 2054 207d 2c20 2d20 5420 5420 202d 2054 T }, - T T - T │ │ │ │ +00008e50: 2054 2020 202b 2079 2a54 2020 292c 2028 T + y*T ), ( │ │ │ │ +00008e60: 7b54 202c 2054 207d 2c20 5420 5420 202b {T , T }, T T + │ │ │ │ +00008e70: 2020 7c0a 7c20 2020 2020 2020 3420 3130 |.| 4 10 │ │ │ │ +00008e80: 2020 2020 2020 3138 2020 2020 2020 3520 18 5 │ │ │ │ +00008e90: 2020 3820 2020 2020 2035 2038 2020 2020 8 5 8 │ │ │ │ +00008ea0: 3220 3130 2020 2020 2020 3139 2020 2020 2 10 19 │ │ │ │ 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2d2d 2d2d ---------------- │ │ │ │ 00008fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009000: 2d2d 7c0a 7c20 2020 2020 2078 2a54 2020 --|.| x*T │ │ │ │ -00009010: 202d 207a 2a54 2020 292c 2028 7b54 202c - z*T ), ({T , │ │ │ │ -00009020: 2054 207d 2c20 2d20 5420 5420 202d 2054 T }, - T T - T │ │ │ │ -00009030: 2054 2020 2d20 5420 5420 202b 207a 2a54 T - T T + z*T │ │ │ │ -00009040: 2020 202b 2078 2a54 2020 292c 2028 7b54 + x*T ), ({T │ │ │ │ -00009050: 202c 7c0a 7c20 2020 2020 2020 2020 3135 ,|.| 15 │ │ │ │ -00009060: 2020 2020 2020 3137 2020 2020 2020 3120 17 1 │ │ │ │ -00009070: 2020 3820 2020 2020 2033 2036 2020 2020 8 3 6 │ │ │ │ -00009080: 3520 3720 2020 2031 2038 2020 2020 2020 5 7 1 8 │ │ │ │ -00009090: 3132 2020 2020 2020 3134 2020 2020 2020 12 14 │ │ │ │ -000090a0: 3420 7c0a 7c20 2020 2020 202d 2d2d 2d2d 4 |.| ----- │ │ │ │ +00009000: 2d2d 7c0a 7c20 2020 2020 2054 207d 2c20 --|.| T }, │ │ │ │ +00009010: 5420 5420 202d 2054 2054 2020 202d 207a T T - T T - z │ │ │ │ +00009020: 2a54 2020 202b 207a 2a54 2020 292c 2028 *T + z*T ), ( │ │ │ │ +00009030: 7b54 202c 2054 207d 2c20 2d20 5420 5420 {T , T }, - T T │ │ │ │ +00009040: 202d 2054 2054 2020 2b20 792a 5420 2029 - T T + y*T ) │ │ │ │ +00009050: 2c20 7c0a 7c20 2020 2020 2020 3920 2020 , |.| 9 │ │ │ │ +00009060: 2034 2039 2020 2020 3520 3130 2020 2020 4 9 5 10 │ │ │ │ +00009070: 2020 3137 2020 2020 2020 3139 2020 2020 17 19 │ │ │ │ +00009080: 2020 3220 2020 3820 2020 2020 2032 2038 2 8 2 8 │ │ │ │ +00009090: 2020 2020 3520 3920 2020 2020 2031 3520 5 9 15 │ │ │ │ +000090a0: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 000090b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000090c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000090d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000090e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000090f0: 2d2d 7c0a 7c20 2020 2020 2054 207d 2c20 --|.| T }, │ │ │ │ -00009100: 5420 5420 202d 2054 2054 2020 202d 207a T T - T T - z │ │ │ │ -00009110: 2a54 2020 202b 207a 2a54 2020 292c 2028 *T + z*T ), ( │ │ │ │ -00009120: 7b54 202c 2054 207d 2c20 2d20 5420 5420 {T , T }, - T T │ │ │ │ -00009130: 202d 2054 2054 2020 2b20 792a 5420 2029 - T T + y*T ) │ │ │ │ -00009140: 2c20 7c0a 7c20 2020 2020 2020 3920 2020 , |.| 9 │ │ │ │ -00009150: 2034 2039 2020 2020 3520 3130 2020 2020 4 9 5 10 │ │ │ │ -00009160: 2020 3137 2020 2020 2020 3139 2020 2020 17 19 │ │ │ │ -00009170: 2020 3220 2020 3820 2020 2020 2032 2038 2 8 2 8 │ │ │ │ -00009180: 2020 2020 3520 3920 2020 2020 2031 3520 5 9 15 │ │ │ │ +000090f0: 2d2d 7c0a 7c20 2020 2020 2028 7b54 202c --|.| ({T , │ │ │ │ +00009100: 2054 207d 2c20 2d20 5420 5420 202d 2054 T }, - T T - T │ │ │ │ +00009110: 2054 2020 2d20 5420 5420 202b 207a 2a54 T - T T + z*T │ │ │ │ +00009120: 2020 202b 2078 2a54 2020 292c 2028 7b54 + x*T ), ({T │ │ │ │ +00009130: 202c 2054 207d 2c20 5420 5420 202b 2020 , T }, T T + │ │ │ │ +00009140: 2020 7c0a 7c20 2020 2020 2020 2020 3520 |.| 5 │ │ │ │ +00009150: 2020 3720 2020 2020 2033 2036 2020 2020 7 3 6 │ │ │ │ +00009160: 3520 3720 2020 2031 2038 2020 2020 2020 5 7 1 8 │ │ │ │ +00009170: 3132 2020 2020 2020 3134 2020 2020 2020 12 14 │ │ │ │ +00009180: 3420 2020 3920 2020 2032 2036 2020 2020 4 9 2 6 │ │ │ │ 00009190: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 000091a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000091b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000091c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000091d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000091e0: 2d2d 7c0a 7c20 2020 2020 2028 7b54 202c --|.| ({T , │ │ │ │ -000091f0: 2054 207d 2c20 2d20 5420 5420 202d 2054 T }, - T T - T │ │ │ │ -00009200: 2054 2020 2d20 5420 5420 202b 207a 2a54 T - T T + z*T │ │ │ │ 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z*T + y*T ) │ │ │ │ +00009500: 2c20 7c0a 7c20 2020 2020 2020 3520 3820 , |.| 5 8 │ │ │ │ +00009510: 2020 2033 2039 2020 2020 2020 3135 2020 3 9 15 │ │ │ │ +00009520: 2020 2020 3136 2020 2020 2020 3420 2020 16 4 │ │ │ │ +00009530: 3620 2020 2034 2036 2020 2020 3320 3720 6 4 6 3 7 │ │ │ │ +00009540: 2020 2020 2031 3120 2020 2020 2031 3320 11 13 │ │ │ │ 00009550: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00009560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000095a0: 2d2d 7c0a 7c20 2020 2020 2054 2054 2020 --|.| T T │ │ │ │ -000095b0: 2b20 5420 5420 202d 207a 2a54 2020 202b + T T - z*T + │ │ │ │ -000095c0: 2078 2a54 2020 292c 2028 7b54 202c 2054 x*T ), ({T , T │ │ │ │ -000095d0: 207d 2c20 5420 5420 202b 2054 2054 2020 }, T T + T T │ │ │ │ -000095e0: 2d20 7a2a 5420 2020 2b20 792a 5420 2029 - z*T + y*T ) │ │ │ │ -000095f0: 2c20 7c0a 7c20 2020 2020 2020 3520 3820 , |.| 5 8 │ │ │ │ -00009600: 2020 2033 2039 2020 2020 2020 3135 2020 3 9 15 │ │ │ │ -00009610: 2020 2020 3136 2020 2020 2020 3420 2020 16 4 │ │ │ │ -00009620: 3620 2020 2034 2036 2020 2020 3320 3720 6 4 6 3 7 │ │ │ │ -00009630: 2020 2020 2031 3120 2020 2020 2031 3320 11 13 │ │ │ │ +000095a0: 2d2d 7c0a 7c20 2020 2020 2028 7b54 202c --|.| ({T , │ │ │ │ +000095b0: 2054 2020 7d2c 202d 2054 2054 2020 202b T }, - T T + │ │ │ │ +000095c0: 2079 2a54 2020 292c 2028 7b54 202c 2054 y*T ), ({T , T │ │ │ │ +000095d0: 207d 2c20 5420 5420 202d 2054 2054 2020 }, T T - T T │ │ │ │ +000095e0: 202b 2078 2a54 2020 292c 2028 7b54 202c + x*T ), ({T , │ │ │ │ +000095f0: 2020 7c0a 7c20 2020 2020 2020 2020 3520 |.| 5 │ │ │ │ +00009600: 2020 3130 2020 2020 2020 3520 3130 2020 10 5 10 │ │ │ │ +00009610: 2020 2020 3138 2020 2020 2020 3420 2020 18 4 │ │ │ │ +00009620: 3620 2020 2034 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│ │ │ │ 00009750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009780: 2d2d 7c0a 7c20 2020 2020 2054 207d 2c20 --|.| T }, │ │ │ │ -00009790: 2d20 5420 5420 202d 2054 2054 2020 2b20 - T T - T T + │ │ │ │ -000097a0: 782a 5420 2029 2c20 287b 5420 2c20 5420 x*T ), ({T , T │ │ │ │ -000097b0: 7d2c 2054 2054 2020 2b20 5420 5420 202b }, T T + T T + │ │ │ │ -000097c0: 2054 2054 2020 2b20 792a 5420 2020 2d20 T T + y*T - │ │ │ │ -000097d0: 2020 7c0a 7c20 2020 2020 2020 3720 2020 |.| 7 │ │ │ │ -000097e0: 2020 2031 2036 2020 2020 3420 3720 2020 1 6 4 7 │ │ │ │ -000097f0: 2020 2031 3120 2020 2020 2032 2020 2036 11 2 6 │ │ │ │ -00009800: 2020 2020 3220 3620 2020 2033 2038 2020 2 6 3 8 │ │ │ │ -00009810: 2020 3420 3920 2020 2020 2031 3420 2020 4 9 14 │ │ │ │ +00009780: 2d2d 7c0a 7c20 2020 2020 207a 2a54 2020 --|.| z*T │ │ │ │ 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207a 2a54 2020 292c 2028 7b54 202c - z*T ), ({T , │ │ │ │ +000098c0: 2020 7c0a 7c20 2020 2020 2020 3320 3820 |.| 3 8 │ │ │ │ +000098d0: 2020 2034 2039 2020 2020 2020 3134 2020 4 9 14 │ │ │ │ +000098e0: 2020 2020 3137 2020 2020 2020 3320 2020 17 3 │ │ │ │ +000098f0: 3620 2020 2033 2036 2020 2020 2020 3131 6 3 6 11 │ │ │ │ +00009900: 2020 2020 2020 3132 2020 2020 2020 3520 12 5 │ │ │ │ 00009910: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00009920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009960: 2d2d 7c0a 7c20 2020 2020 2054 2054 2020 --|.| T T │ │ │ │ -00009970: 2b20 5420 5420 202b 2079 2a54 2020 202d + T T + y*T - │ │ │ │ -00009980: 207a 2a54 2020 292c 2028 7b54 202c 2054 z*T ), ({T , T │ │ │ │ -00009990: 207d 2c20 5420 5420 202b 2079 2a54 2020 }, T T + y*T │ │ │ │ -000099a0: 202d 207a 2a54 2020 292c 2028 7b54 202c - z*T ), ({T , │ │ │ │ -000099b0: 2020 7c0a 7c20 2020 2020 2020 3320 3820 |.| 3 8 │ │ │ │ -000099c0: 2020 2034 2039 2020 2020 2020 3134 2020 4 9 14 │ │ │ │ -000099d0: 2020 2020 3137 2020 2020 2020 3320 2020 17 3 │ │ │ │ -000099e0: 3620 2020 2033 2036 2020 2020 2020 3131 6 3 6 11 │ │ │ │ -000099f0: 2020 2020 2020 3132 2020 2020 2020 3520 12 5 │ │ │ │ +00009960: 2d2d 7c0a 7c20 2020 2020 2054 207d 2c20 --|.| T }, │ │ │ │ +00009970: 2d20 5420 5420 202d 2054 2054 2020 202b - T T - T T + │ │ │ │ +00009980: 207a 2a54 2020 202b 207a 2a54 2020 292c z*T + z*T ), │ │ │ │ +00009990: 2028 7b54 202c 2054 207d 2c20 5420 5420 ({T , T }, T T │ │ │ │ +000099a0: 202b 2054 2054 2020 2d20 7a2a 5420 2020 + T T - z*T │ │ │ │ +000099b0: 2b20 7c0a 7c20 2020 2020 2020 3720 2020 + |.| 7 │ │ │ │ +000099c0: 2020 2035 2037 2020 2020 3420 3130 2020 5 7 4 10 │ │ │ │ +000099d0: 2020 2020 3132 2020 2020 2020 3230 2020 12 20 │ │ │ │ +000099e0: 2020 2020 3320 2020 3720 2020 2034 2036 3 7 4 6 │ │ │ │ +000099f0: 2020 2020 3320 3720 2020 2020 2031 3120 3 7 11 │ │ │ │ 00009a00: 2020 7c0a 7c20 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 00009a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009a50: 2d2d 7c0a 7c20 2020 2020 2054 207d 2c20 --|.| T }, │ │ │ │ -00009a60: 2d20 5420 5420 202d 2054 2054 2020 202b - T T - T T + │ │ │ │ -00009a70: 207a 2a54 2020 202b 207a 2a54 2020 297d z*T + z*T )} │ │ │ │ +00009a50: 2d2d 7c0a 7c20 2020 2020 2079 2a54 2020 --|.| y*T │ │ │ │ +00009a60: 292c 2028 7b54 202c 2054 207d 2c20 2d20 ), ({T , T }, - │ │ │ │ +00009a70: 5420 5420 202b 2079 2a54 2020 297d 2020 T T + y*T )} │ │ │ │ 00009a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009aa0: 2020 7c0a 7c20 2020 2020 2020 3720 2020 |.| 7 │ │ │ │ -00009ab0: 2020 2035 2037 2020 2020 3420 3130 2020 5 7 4 10 │ │ │ │ -00009ac0: 2020 2020 3132 2020 2020 2020 3230 2020 12 20 │ │ │ │ +00009aa0: 2020 7c0a 7c20 2020 2020 2020 2020 3133 |.| 13 │ │ │ │ +00009ab0: 2020 2020 2020 3220 2020 3920 2020 2020 2 9 │ │ │ │ +00009ac0: 2032 2039 2020 2020 2020 3136 2020 2020 2 9 16 │ │ │ │ 00009ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009af0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00009b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2498,15 +2498,15 @@ │ │ │ │ 00009c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009c30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00009c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00009c80: 2020 7c0a 7c6f 3136 203d 2031 2020 2020 |.|o16 = 1 │ │ │ │ +00009c80: 2020 7c0a 7c6f 3136 203d 202d 3120 2020 |.|o16 = -1 │ │ │ │ 00009c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009cd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00009ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/HyperplaneArrangements.info.gz │ │ │ ├── HyperplaneArrangements.info │ │ │ │ @@ -7210,16 +7210,16 @@ │ │ │ │ 0001c290: 3134 203a 2063 4127 2720 3d20 7472 696d 14 : cA'' = trim │ │ │ │ 0001c2a0: 2063 6f6e 6528 412c 2078 2920 2020 2020 cone(A, x) │ │ │ │ 0001c2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c2c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0001c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c300: 2020 7c0a 7c6f 3134 203d 207b 7820 2d20 |.|o14 = {x - │ │ │ │ -0001c310: 792c 2079 2c20 787d 2020 2020 2020 2020 y, y, x} │ │ │ │ +0001c300: 2020 7c0a 7c6f 3134 203d 207b 792c 2078 |.|o14 = {y, x │ │ │ │ +0001c310: 2c20 7820 2d20 797d 2020 2020 2020 2020 , x - y} │ │ │ │ 0001c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c330: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0001c340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c370: 2020 2020 2020 2020 7c0a 7c6f 3134 203a |.|o14 : │ │ │ │ 0001c380: 2048 7970 6572 706c 616e 6520 4172 7261 Hyperplane Arra │ │ │ │ @@ -19190,21 +19190,21 @@ │ │ │ │ 0004af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004af70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0004af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004afc0: 207c 0a7c 6f36 203d 207b 7820 2c20 7820 |.|o6 = {x , x │ │ │ │ -0004afd0: 2c20 7820 202b 2078 207d 2020 2020 2020 , x + x } │ │ │ │ +0004afc0: 207c 0a7c 6f36 203d 207b 7820 202b 2078 |.|o6 = {x + x │ │ │ │ +0004afd0: 202c 2078 202c 2078 207d 2020 2020 2020 , x , x } │ │ │ │ 0004afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b010: 207c 0a7c 2020 2020 2020 2032 2020 2031 |.| 2 1 │ │ │ │ -0004b020: 2020 2031 2020 2020 3220 2020 2020 2020 1 2 │ │ │ │ +0004b010: 207c 0a7c 2020 2020 2020 2031 2020 2020 |.| 1 │ │ │ │ +0004b020: 3220 2020 3220 2020 3120 2020 2020 2020 2 2 1 │ │ │ │ 0004b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b060: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0004b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/IntegralClosure.info.gz │ │ │ ├── IntegralClosure.info │ │ │ │ @@ -4491,17 +4491,17 @@ │ │ │ │ 000118a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000118b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ 000118c0: 7469 6d65 2052 2720 3d20 696e 7465 6772 time R' = integr │ │ │ │ 000118d0: 616c 436c 6f73 7572 6520 5220 2020 2020 alClosure R │ │ │ │ 000118e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000118f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011900: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00011910: 7365 6420 302e 3639 3837 3435 7320 2863 sed 0.698745s (c │ │ │ │ -00011920: 7075 293b 2030 2e33 3838 3835 7320 2874 pu); 0.38885s (t │ │ │ │ -00011930: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00011910: 7365 6420 302e 3932 3031 3134 7320 2863 sed 0.920114s (c │ │ │ │ +00011920: 7075 293b 2030 2e34 3439 3036 3773 2028 pu); 0.449067s ( │ │ │ │ +00011930: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00011940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011950: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00011960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000119a0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ @@ -4981,17 +4981,17 @@ │ │ │ │ 00013740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013750: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a --------+.|i10 : │ │ │ │ 00013760: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 00013770: 7261 6c43 6c6f 7375 7265 2852 2c20 5374 ralClosure(R, St │ │ │ │ 00013780: 7261 7465 6779 203d 3e20 5261 6469 6361 rategy => Radica │ │ │ │ 00013790: 6c29 2020 2020 2020 2020 2020 2020 2020 l) │ │ │ │ 000137a0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000137b0: 7365 6420 302e 3735 3439 3637 7320 2863 sed 0.754967s (c │ │ │ │ -000137c0: 7075 293b 2030 2e34 3034 3437 3973 2028 pu); 0.404479s ( │ │ │ │ -000137d0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000137b0: 7365 6420 302e 3939 3938 3273 2028 6370 sed 0.99982s (cp │ │ │ │ +000137c0: 7529 3b20 302e 3435 3231 3835 7320 2874 u); 0.452185s (t │ │ │ │ +000137d0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 000137e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000137f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00013800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013840: 2020 2020 2020 2020 7c0a 7c6f 3130 203d |.|o10 = │ │ │ │ @@ -5471,17 +5471,17 @@ │ │ │ │ 000155e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000155f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a --------+.|i16 : │ │ │ │ 00015600: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 00015610: 7261 6c43 6c6f 7375 7265 2852 2c20 5374 ralClosure(R, St │ │ │ │ 00015620: 7261 7465 6779 203d 3e20 416c 6c43 6f64 rategy => AllCod │ │ │ │ 00015630: 696d 656e 7369 6f6e 7329 2020 2020 2020 imensions) │ │ │ │ 00015640: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00015650: 7365 6420 302e 3832 3738 3433 7320 2863 sed 0.827843s (c │ │ │ │ -00015660: 7075 293b 2030 2e33 3938 3335 3973 2028 pu); 0.398359s ( │ │ │ │ -00015670: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00015650: 7365 6420 312e 3839 3938 7320 2863 7075 sed 1.8998s (cpu │ │ │ │ +00015660: 293b 2030 2e35 3635 3233 3273 2028 7468 ); 0.565232s (th │ │ │ │ +00015670: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00015680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015690: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000156a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000156b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000156c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000156d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000156e0: 2020 2020 2020 2020 7c0a 7c6f 3136 203d |.|o16 = │ │ │ │ @@ -5916,17 +5916,17 @@ │ │ │ │ 000171b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000171c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a --------+.|i21 : │ │ │ │ 000171d0: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 000171e0: 7261 6c43 6c6f 7375 7265 2852 2c20 5374 ralClosure(R, St │ │ │ │ 000171f0: 7261 7465 6779 203d 3e20 5369 6d70 6c69 rategy => Simpli │ │ │ │ 00017200: 6679 4672 6163 7469 6f6e 7329 2020 2020 fyFractions) │ │ │ │ 00017210: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00017220: 7365 6420 302e 3932 3430 3332 7320 2863 sed 0.924032s (c │ │ │ │ -00017230: 7075 293b 2030 2e34 3535 3235 3773 2028 pu); 0.455257s ( │ │ │ │ -00017240: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00017220: 7365 6420 312e 3235 3037 3273 2028 6370 sed 1.25072s (cp │ │ │ │ +00017230: 7529 3b20 302e 3533 3436 3931 7320 2874 u); 0.534691s (t │ │ │ │ +00017240: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00017250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017260: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00017270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000172a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000172b0: 2020 2020 2020 2020 7c0a 7c6f 3231 203d |.|o21 = │ │ │ │ @@ -6361,17 +6361,17 @@ │ │ │ │ 00018d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018d90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3236 203a --------+.|i26 : │ │ │ │ 00018da0: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 00018db0: 7261 6c43 6c6f 7375 7265 2028 522c 2053 ralClosure (R, S │ │ │ │ 00018dc0: 7472 6174 6567 7920 3d3e 2052 6164 6963 trategy => Radic │ │ │ │ 00018dd0: 616c 436f 6469 6d31 2920 2020 2020 2020 alCodim1) │ │ │ │ 00018de0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00018df0: 7365 6420 312e 3630 3339 3673 2028 6370 sed 1.60396s (cp │ │ │ │ -00018e00: 7529 3b20 302e 3733 3536 3734 7320 2874 u); 0.735674s (t │ │ │ │ -00018e10: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00018df0: 7365 6420 322e 3331 3337 7320 2863 7075 sed 2.3137s (cpu │ │ │ │ +00018e00: 293b 2030 2e39 3133 3934 3573 2028 7468 ); 0.913945s (th │ │ │ │ +00018e10: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00018e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00018e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e80: 2020 2020 2020 2020 7c0a 7c6f 3236 203d |.|o26 = │ │ │ │ @@ -6806,16 +6806,16 @@ │ │ │ │ 0001a950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a960: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3331 203a --------+.|i31 : │ │ │ │ 0001a970: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 0001a980: 7261 6c43 6c6f 7375 7265 2028 522c 2053 ralClosure (R, S │ │ │ │ 0001a990: 7472 6174 6567 7920 3d3e 2056 6173 636f trategy => Vasco │ │ │ │ 0001a9a0: 6e63 656c 6f73 2920 2020 2020 2020 2020 ncelos) │ │ │ │ 0001a9b0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -0001a9c0: 7365 6420 302e 3434 3332 3731 7320 2863 sed 0.443271s (c │ │ │ │ -0001a9d0: 7075 293b 2030 2e33 3137 3636 3873 2028 pu); 0.317668s ( │ │ │ │ +0001a9c0: 7365 6420 302e 3631 3538 3335 7320 2863 sed 0.615835s (c │ │ │ │ +0001a9d0: 7075 293b 2030 2e34 3135 3030 3673 2028 pu); 0.415006s ( │ │ │ │ 0001a9e0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7223,18 +7223,18 @@ │ │ │ │ 0001c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c390: 2d2d 2d2d 2b0a 7c69 3336 203a 2074 696d ----+.|i36 : tim │ │ │ │ 0001c3a0: 6520 5227 203d 2069 6e74 6567 7261 6c43 e R' = integralC │ │ │ │ 0001c3b0: 6c6f 7375 7265 2052 2020 2020 2020 2020 losure R │ │ │ │ 0001c3c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001c3d0: 0a7c 202d 2d20 7573 6564 2030 2e30 3434 .| -- used 0.044 │ │ │ │ -0001c3e0: 3638 3432 7320 2863 7075 293b 2030 2e30 6842s (cpu); 0.0 │ │ │ │ -0001c3f0: 3434 3638 3133 7320 2874 6872 6561 6429 446813s (thread) │ │ │ │ -0001c400: 3b20 3073 2028 6763 2920 7c0a 7c20 2020 ; 0s (gc) |.| │ │ │ │ +0001c3d0: 0a7c 202d 2d20 7573 6564 2030 2e30 3537 .| -- used 0.057 │ │ │ │ +0001c3e0: 3933 3773 2028 6370 7529 3b20 302e 3035 937s (cpu); 0.05 │ │ │ │ +0001c3f0: 3739 3332 3973 2028 7468 7265 6164 293b 79329s (thread); │ │ │ │ +0001c400: 2030 7320 2867 6329 2020 7c0a 7c20 2020 0s (gc) |.| │ │ │ │ 0001c410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c440: 2020 2020 207c 0a7c 6f33 3620 3d20 5227 |.|o36 = R' │ │ │ │ 0001c450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7413,16 +7413,16 @@ │ │ │ │ 0001cf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cf60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3431 203a --------+.|i41 : │ │ │ │ 0001cf70: 2074 696d 6520 5227 203d 2069 6e74 6567 time R' = integ │ │ │ │ 0001cf80: 7261 6c43 6c6f 7375 7265 2852 2c20 5374 ralClosure(R, St │ │ │ │ 0001cf90: 7261 7465 6779 203d 3e20 5261 6469 6361 rategy => Radica │ │ │ │ 0001cfa0: 6c29 207c 0a7c 202d 2d20 7573 6564 2030 l) |.| -- used 0 │ │ │ │ -0001cfb0: 2e30 3435 3835 3532 7320 2863 7075 293b .0458552s (cpu); │ │ │ │ -0001cfc0: 2030 2e30 3435 3835 3537 7320 2874 6872 0.0458557s (thr │ │ │ │ +0001cfb0: 2e30 3538 3634 3834 7320 2863 7075 293b .0586484s (cpu); │ │ │ │ +0001cfc0: 2030 2e30 3538 3634 3739 7320 2874 6872 0.0586479s (thr │ │ │ │ 0001cfd0: 6561 6429 3b20 3073 2028 6763 2920 7c0a ead); 0s (gc) |. │ │ │ │ 0001cfe0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0001cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d010: 2020 2020 2020 2020 207c 0a7c 6f34 3120 |.|o41 │ │ │ │ 0001d020: 3d20 5227 2020 2020 2020 2020 2020 2020 = R' │ │ │ │ 0001d030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7555,16 +7555,16 @@ │ │ │ │ 0001d820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d840: 2b0a 7c69 3436 203a 2074 696d 6520 5227 +.|i46 : time R' │ │ │ │ 0001d850: 203d 2069 6e74 6567 7261 6c43 6c6f 7375 = integralClosu │ │ │ │ 0001d860: 7265 2852 2c20 5374 7261 7465 6779 203d re(R, Strategy = │ │ │ │ 0001d870: 3e20 416c 6c43 6f64 696d 656e 7369 6f6e > AllCodimension │ │ │ │ 0001d880: 7329 7c0a 7c20 2d2d 2075 7365 6420 302e s)|.| -- used 0. │ │ │ │ -0001d890: 3036 3031 3535 3973 2028 6370 7529 3b20 0601559s (cpu); │ │ │ │ -0001d8a0: 302e 3036 3031 3535 3673 2028 7468 7265 0.0601556s (thre │ │ │ │ +0001d890: 3038 3033 3739 3373 2028 6370 7529 3b20 0803793s (cpu); │ │ │ │ +0001d8a0: 302e 3038 3033 3734 3473 2028 7468 7265 0.0803744s (thre │ │ │ │ 0001d8b0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 0001d8c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0001d8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d900: 2020 2020 2020 7c0a 7c6f 3436 203d 2052 |.|o46 = R │ │ │ │ 0001d910: 2720 2020 2020 2020 2020 2020 2020 2020 ' │ │ │ │ @@ -7698,16 +7698,16 @@ │ │ │ │ 0001e110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e130: 2d2d 2b0a 7c69 3531 203a 2074 696d 6520 --+.|i51 : time │ │ │ │ 0001e140: 5227 203d 2069 6e74 6567 7261 6c43 6c6f R' = integralClo │ │ │ │ 0001e150: 7375 7265 2028 522c 2053 7472 6174 6567 sure (R, Strateg │ │ │ │ 0001e160: 7920 3d3e 2052 6164 6963 616c 436f 6469 y => RadicalCodi │ │ │ │ 0001e170: 6d31 297c 0a7c 202d 2d20 7573 6564 2030 m1)|.| -- used 0 │ │ │ │ -0001e180: 2e30 3436 3037 3237 7320 2863 7075 293b .0460727s (cpu); │ │ │ │ -0001e190: 2030 2e30 3436 3036 3935 7320 2874 6872 0.0460695s (thr │ │ │ │ +0001e180: 2e30 3535 3132 3136 7320 2863 7075 293b .0551216s (cpu); │ │ │ │ +0001e190: 2030 2e30 3535 3131 3635 7320 2874 6872 0.0551165s (thr │ │ │ │ 0001e1a0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 0001e1b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1f0: 2020 2020 207c 0a7c 6f35 3120 3d20 5227 |.|o51 = R' │ │ │ │ 0001e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7841,17 +7841,17 @@ │ │ │ │ 0001ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0001ea30: 7c69 3536 203a 2074 696d 6520 5227 203d |i56 : time R' = │ │ │ │ 0001ea40: 2069 6e74 6567 7261 6c43 6c6f 7375 7265 integralClosure │ │ │ │ 0001ea50: 2028 522c 2053 7472 6174 6567 7920 3d3e (R, Strategy => │ │ │ │ 0001ea60: 2056 6173 636f 6e63 656c 6f73 297c 0a7c Vasconcelos)|.| │ │ │ │ -0001ea70: 202d 2d20 7573 6564 2030 2e30 3539 3035 -- used 0.05905 │ │ │ │ -0001ea80: 3636 7320 2863 7075 293b 2030 2e30 3539 66s (cpu); 0.059 │ │ │ │ -0001ea90: 3035 3632 7320 2874 6872 6561 6429 3b20 0562s (thread); │ │ │ │ +0001ea70: 202d 2d20 7573 6564 2030 2e30 3731 3735 -- used 0.07175 │ │ │ │ +0001ea80: 3231 7320 2863 7075 293b 2030 2e30 3731 21s (cpu); 0.071 │ │ │ │ +0001ea90: 3735 3333 7320 2874 6872 6561 6429 3b20 7533s (thread); │ │ │ │ 0001eaa0: 3073 2028 6763 2920 2020 2020 7c0a 7c20 0s (gc) |.| │ │ │ │ 0001eab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eae0: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ 0001eaf0: 3620 3d20 5227 2020 2020 2020 2020 2020 6 = R' │ │ │ │ 0001eb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8358,17 +8358,17 @@ │ │ │ │ 00020a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020a60: 2d2d 2d2d 2b0a 7c69 3637 203a 2074 696d ----+.|i67 : tim │ │ │ │ 00020a70: 6520 5227 203d 2069 6e74 6567 7261 6c43 e R' = integralC │ │ │ │ 00020a80: 6c6f 7375 7265 2852 2c20 5374 7261 7465 losure(R, Strate │ │ │ │ 00020a90: 6779 203d 3e20 5261 6469 6361 6c29 2020 gy => Radical) │ │ │ │ 00020aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ab0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00020ac0: 302e 3135 3530 3035 7320 2863 7075 293b 0.155005s (cpu); │ │ │ │ -00020ad0: 2030 2e30 3838 3537 3932 7320 2874 6872 0.0885792s (thr │ │ │ │ -00020ae0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00020ac0: 302e 3233 3535 3634 7320 2863 7075 293b 0.235564s (cpu); │ │ │ │ +00020ad0: 2030 2e31 3137 3733 3673 2028 7468 7265 0.117736s (thre │ │ │ │ +00020ae0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00020af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00020b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 2020 │ │ │ │ 00020b50: 2020 2020 7c0a 7c6f 3637 203d 2052 2720 |.|o67 = R' │ │ │ │ @@ -8853,17 +8853,17 @@ │ │ │ │ 00022940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022950: 2d2d 2d2d 2b0a 7c69 3738 203a 2074 696d ----+.|i78 : tim │ │ │ │ 00022960: 6520 5227 203d 2069 6e74 6567 7261 6c43 e R' = integralC │ │ │ │ 00022970: 6c6f 7375 7265 2852 2c20 5374 7261 7465 losure(R, Strate │ │ │ │ 00022980: 6779 203d 3e20 5261 6469 6361 6c29 2020 gy => Radical) │ │ │ │ 00022990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000229a0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000229b0: 302e 3430 3739 3773 2028 6370 7529 3b20 0.40797s (cpu); │ │ │ │ -000229c0: 302e 3333 3539 3735 7320 2874 6872 6561 0.335975s (threa │ │ │ │ -000229d0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +000229b0: 302e 3535 3133 3937 7320 2863 7075 293b 0.551397s (cpu); │ │ │ │ +000229c0: 2030 2e34 3239 3839 3273 2028 7468 7265 0.429892s (thre │ │ │ │ +000229d0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000229e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000229f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00022a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a40: 2020 2020 7c0a 7c6f 3738 203d 2052 2720 |.|o78 = R' │ │ │ │ @@ -8985,17 +8985,17 @@ │ │ │ │ 00023180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000231b0: 7c69 3832 203a 2074 696d 6520 5227 203d |i82 : time R' = │ │ │ │ 000231c0: 2069 6e74 6567 7261 6c43 6c6f 7375 7265 integralClosure │ │ │ │ 000231d0: 2852 2c20 5374 7261 7465 6779 203d 3e20 (R, Strategy => │ │ │ │ 000231e0: 416c 6c43 6f64 696d 656e 7369 6f6e 7329 AllCodimensions) │ │ │ │ -000231f0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3437 |.| -- used 0.47 │ │ │ │ -00023200: 3737 3133 7320 2863 7075 293b 2030 2e33 7713s (cpu); 0.3 │ │ │ │ -00023210: 3438 3733 3173 2028 7468 7265 6164 293b 48731s (thread); │ │ │ │ +000231f0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3638 |.| -- used 0.68 │ │ │ │ +00023200: 3530 3433 7320 2863 7075 293b 2030 2e34 5043s (cpu); 0.4 │ │ │ │ +00023210: 3538 3636 3373 2028 7468 7265 6164 293b 58663s (thread); │ │ │ │ 00023220: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00023230: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00023240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023270: 2020 2020 7c0a 7c6f 3832 203d 2052 2720 |.|o82 = R' │ │ │ │ 00023280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9118,42 +9118,42 @@ │ │ │ │ 000239d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000239e0: 7c69 3836 203a 2074 696d 6520 5227 203d |i86 : time R' = │ │ │ │ 000239f0: 2069 6e74 6567 7261 6c43 6c6f 7375 7265 integralClosure │ │ │ │ 00023a00: 2028 522c 2053 7472 6174 6567 7920 3d3e (R, Strategy => │ │ │ │ 00023a10: 2052 6164 6963 616c 436f 6469 6d31 2c20 RadicalCodim1, │ │ │ │ 00023a20: 5665 7262 6f73 6974 7920 3d3e 2020 7c0a Verbosity => |. │ │ │ │ 00023a30: 7c20 5b6a 6163 6f62 6961 6e20 7469 6d65 | [jacobian time │ │ │ │ -00023a40: 202e 3030 3035 3033 3133 3420 7365 6320 .000503134 sec │ │ │ │ -00023a50: 236d 696e 6f72 7320 345d 2020 2020 2020 #minors 4] │ │ │ │ +00023a40: 202e 3030 3130 3635 3631 2073 6563 2023 .00106561 sec # │ │ │ │ +00023a50: 6d69 6e6f 7273 2034 5d20 2020 2020 2020 minors 4] │ │ │ │ 00023a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00023a80: 7c69 6e74 6567 7261 6c20 636c 6f73 7572 |integral closur │ │ │ │ 00023a90: 6520 6e76 6172 7320 3420 6e75 6d67 656e e nvars 4 numgen │ │ │ │ 00023aa0: 7320 3120 6973 2053 3220 636f 6469 6d20 s 1 is S2 codim │ │ │ │ 00023ab0: 3120 636f 6469 6d4a 2032 2020 2020 2020 1 codimJ 2 │ │ │ │ 00023ac0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00023ad0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00023ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00023b20: 7c20 5b73 7465 7020 303a 2020 2074 696d | [step 0: tim │ │ │ │ -00023b30: 6520 2e32 3032 3736 3620 7365 6320 2023 e .202766 sec # │ │ │ │ +00023b30: 6520 2e32 3934 3736 3520 7365 6320 2023 e .294765 sec # │ │ │ │ 00023b40: 6672 6163 7469 6f6e 7320 365d 2020 2020 fractions 6] │ │ │ │ 00023b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00023b70: 7c20 5b73 7465 7020 313a 2020 2074 696d | [step 1: tim │ │ │ │ -00023b80: 6520 2e32 3038 3336 3220 7365 6320 2023 e .208362 sec # │ │ │ │ -00023b90: 6672 6163 7469 6f6e 7320 365d 2020 2020 fractions 6] │ │ │ │ +00023b80: 6520 2e33 3230 3234 2073 6563 2020 2366 e .32024 sec #f │ │ │ │ +00023b90: 7261 6374 696f 6e73 2036 5d20 2020 2020 ractions 6] │ │ │ │ 00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023bb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00023bc0: 7c20 2d2d 2075 7365 6420 302e 3431 3438 | -- used 0.4148 │ │ │ │ -00023bd0: 3631 7320 2863 7075 293b 2030 2e32 3931 61s (cpu); 0.291 │ │ │ │ -00023be0: 3338 7320 2874 6872 6561 6429 3b20 3073 38s (thread); 0s │ │ │ │ -00023bf0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00023bc0: 7c20 2d2d 2075 7365 6420 302e 3632 3233 | -- used 0.6223 │ │ │ │ +00023bd0: 3932 7320 2863 7075 293b 2030 2e33 3731 92s (cpu); 0.371 │ │ │ │ +00023be0: 3533 3973 2028 7468 7265 6164 293b 2030 539s (thread); 0 │ │ │ │ +00023bf0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00023c00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00023c10: 7c20 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 7c0a |. │ │ │ │ 00023c60: 7c6f 3836 203d 2052 2720 2020 2020 2020 |o86 = R' │ │ │ │ @@ -9297,40 +9297,40 @@ │ │ │ │ 00024500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024510: 2d2b 0a7c 6939 3020 3a20 7469 6d65 2052 -+.|i90 : time R │ │ │ │ 00024520: 2720 3d20 696e 7465 6772 616c 436c 6f73 ' = integralClos │ │ │ │ 00024530: 7572 6520 2852 2c20 5374 7261 7465 6779 ure (R, Strategy │ │ │ │ 00024540: 203d 3e20 5661 7363 6f6e 6365 6c6f 732c => Vasconcelos, │ │ │ │ 00024550: 2056 6572 626f 7369 7479 203d 3e20 3129 Verbosity => 1) │ │ │ │ 00024560: 7c0a 7c20 5b6a 6163 6f62 6961 6e20 7469 |.| [jacobian ti │ │ │ │ -00024570: 6d65 202e 3030 3035 3536 3732 3420 7365 me .000556724 se │ │ │ │ +00024570: 6d65 202e 3030 3036 3634 3735 3920 7365 me .000664759 se │ │ │ │ 00024580: 6320 236d 696e 6f72 7320 345d 2020 2020 c #minors 4] │ │ │ │ 00024590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000245a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 000245b0: 0a7c 696e 7465 6772 616c 2063 6c6f 7375 .|integral closu │ │ │ │ 000245c0: 7265 206e 7661 7273 2034 206e 756d 6765 re nvars 4 numge │ │ │ │ 000245d0: 6e73 2031 2069 7320 5332 2063 6f64 696d ns 1 is S2 codim │ │ │ │ 000245e0: 2031 2063 6f64 696d 4a20 3220 2020 2020 1 codimJ 2 │ │ │ │ 000245f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00024600: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00024610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024640: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00024650: 205b 7374 6570 2030 3a20 2020 7469 6d65 [step 0: time │ │ │ │ -00024660: 202e 3038 3938 3535 3720 7365 6320 2023 .0898557 sec # │ │ │ │ -00024670: 6672 6163 7469 6f6e 7320 365d 2020 2020 fractions 6] │ │ │ │ +00024660: 202e 3131 3038 3232 2073 6563 2020 2366 .110822 sec #f │ │ │ │ +00024670: 7261 6374 696f 6e73 2036 5d20 2020 2020 ractions 6] │ │ │ │ 00024680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024690: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000246a0: 5b73 7465 7020 313a 2020 2074 696d 6520 [step 1: time │ │ │ │ -000246b0: 2e33 3538 3135 3220 7365 6320 2023 6672 .358152 sec #fr │ │ │ │ +000246b0: 2e36 3434 3633 3520 7365 6320 2023 6672 .644635 sec #fr │ │ │ │ 000246c0: 6163 7469 6f6e 7320 365d 2020 2020 2020 actions 6] │ │ │ │ 000246d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000246e0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -000246f0: 2d20 7573 6564 2030 2e34 3531 3934 3973 - used 0.451949s │ │ │ │ -00024700: 2028 6370 7529 3b20 302e 3333 3733 3031 (cpu); 0.337301 │ │ │ │ +000246f0: 2d20 7573 6564 2030 2e37 3630 3034 3473 - used 0.760044s │ │ │ │ +00024700: 2028 6370 7529 3b20 302e 3530 3839 3735 (cpu); 0.508975 │ │ │ │ 00024710: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00024720: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00024730: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00024740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9474,41 +9474,41 @@ │ │ │ │ 00025010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025020: 2d2d 2d2d 2b0a 7c69 3934 203a 2074 696d ----+.|i94 : tim │ │ │ │ 00025030: 6520 5227 203d 2069 6e74 6567 7261 6c43 e R' = integralC │ │ │ │ 00025040: 6c6f 7375 7265 2028 522c 2053 7472 6174 losure (R, Strat │ │ │ │ 00025050: 6567 7920 3d3e 207b 5661 7363 6f6e 6365 egy => {Vasconce │ │ │ │ 00025060: 6c6f 732c 2020 2020 2020 2020 2020 2020 los, │ │ │ │ 00025070: 2020 2020 7c0a 7c20 5b6a 6163 6f62 6961 |.| [jacobia │ │ │ │ -00025080: 6e20 7469 6d65 202e 3030 3036 3031 3839 n time .00060189 │ │ │ │ -00025090: 3920 7365 6320 236d 696e 6f72 7320 315d 9 sec #minors 1] │ │ │ │ +00025080: 6e20 7469 6d65 202e 3030 3039 3135 3130 n time .00091510 │ │ │ │ +00025090: 3120 7365 6320 236d 696e 6f72 7320 315d 1 sec #minors 1] │ │ │ │ 000250a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000250b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000250c0: 2020 2020 7c0a 7c69 6e74 6567 7261 6c20 |.|integral │ │ │ │ 000250d0: 636c 6f73 7572 6520 6e76 6172 7320 3420 closure nvars 4 │ │ │ │ 000250e0: 6e75 6d67 656e 7320 3120 6973 2053 3220 numgens 1 is S2 │ │ │ │ 000250f0: 636f 6469 6d20 3120 636f 6469 6d4a 2032 codim 1 codimJ 2 │ │ │ │ 00025100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025110: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00025120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025160: 2020 2020 7c0a 7c20 5b73 7465 7020 303a |.| [step 0: │ │ │ │ -00025170: 2020 2074 696d 6520 2e31 3037 3833 3920 time .107839 │ │ │ │ +00025170: 2020 2074 696d 6520 2e31 3438 3031 3820 time .148018 │ │ │ │ 00025180: 7365 6320 2023 6672 6163 7469 6f6e 7320 sec #fractions │ │ │ │ 00025190: 365d 2020 2020 2020 2020 2020 2020 2020 6] │ │ │ │ 000251a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000251b0: 2020 2020 7c0a 7c20 5b73 7465 7020 313a |.| [step 1: │ │ │ │ -000251c0: 2020 2074 696d 6520 2e34 3037 3830 3120 time .407801 │ │ │ │ +000251c0: 2020 2074 696d 6520 2e36 3231 3636 3520 time .621665 │ │ │ │ 000251d0: 7365 6320 2023 6672 6163 7469 6f6e 7320 sec #fractions │ │ │ │ 000251e0: 365d 2020 2020 2020 2020 2020 2020 2020 6] │ │ │ │ 000251f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025200: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00025210: 302e 3531 3932 3139 7320 2863 7075 293b 0.519219s (cpu); │ │ │ │ -00025220: 2030 2e34 3033 3638 3573 2028 7468 7265 0.403685s (thre │ │ │ │ +00025210: 302e 3737 3437 3836 7320 2863 7075 293b 0.774786s (cpu); │ │ │ │ +00025220: 2030 2e35 3337 3138 3773 2028 7468 7265 0.537187s (thre │ │ │ │ 00025230: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00025240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025250: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00025260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -10305,16 +10305,16 @@ │ │ │ │ 00028400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028410: 2d2d 2b0a 7c69 3220 3a20 7469 6d65 2052 --+.|i2 : time R │ │ │ │ 00028420: 2720 3d20 696e 7465 6772 616c 436c 6f73 ' = integralClos │ │ │ │ 00028430: 7572 6528 522c 2056 6572 626f 7369 7479 ure(R, Verbosity │ │ │ │ 00028440: 203d 3e20 3229 2020 2020 2020 2020 2020 => 2) │ │ │ │ 00028450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028460: 2020 7c0a 7c20 5b6a 6163 6f62 6961 6e20 |.| [jacobian │ │ │ │ -00028470: 7469 6d65 202e 3030 3035 3734 3131 3620 time .000574116 │ │ │ │ -00028480: 7365 6320 236d 696e 6f72 7320 335d 2020 sec #minors 3] │ │ │ │ +00028470: 7469 6d65 202e 3030 3036 3336 3332 2073 time .00063632 s │ │ │ │ +00028480: 6563 2023 6d69 6e6f 7273 2033 5d20 2020 ec #minors 3] │ │ │ │ 00028490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000284a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000284b0: 2020 7c0a 7c69 6e74 6567 7261 6c20 636c |.|integral cl │ │ │ │ 000284c0: 6f73 7572 6520 6e76 6172 7320 3320 6e75 osure nvars 3 nu │ │ │ │ 000284d0: 6d67 656e 7320 3120 6973 2053 3220 636f mgens 1 is S2 co │ │ │ │ 000284e0: 6469 6d20 3120 636f 6469 6d4a 2032 2020 dim 1 codimJ 2 │ │ │ │ 000284f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -10326,180 +10326,180 @@ │ │ │ │ 00028550: 2020 7c0a 7c20 5b73 7465 7020 303a 2020 |.| [step 0: │ │ │ │ 00028560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000285a0: 2020 7c0a 7c20 2020 2020 2072 6164 6963 |.| radic │ │ │ │ 000285b0: 616c 2028 7573 6520 6d69 6e70 7269 6d65 al (use minprime │ │ │ │ -000285c0: 7329 202e 3030 3235 3431 3335 2073 6563 s) .00254135 sec │ │ │ │ +000285c0: 7329 202e 3030 3330 3533 3235 2073 6563 s) .00305325 sec │ │ │ │ 000285d0: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 000285e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000285f0: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -00028600: 6572 313a 2020 2e30 3037 3939 3031 2073 er1: .0079901 s │ │ │ │ +00028600: 6572 313a 2020 2e30 3130 3439 3439 2073 er1: .0104949 s │ │ │ │ 00028610: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 00028620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028640: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -00028650: 6572 323a 2020 2e30 3039 3032 3236 3420 er2: .00902264 │ │ │ │ -00028660: 7365 636f 6e64 7320 2020 2020 2020 2020 seconds │ │ │ │ +00028650: 6572 323a 2020 2e30 3131 3830 3537 2073 er2: .0118057 s │ │ │ │ +00028660: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 00028670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028690: 2020 7c0a 7c20 2020 2020 206d 696e 7072 |.| minpr │ │ │ │ -000286a0: 6573 3a20 2020 2e30 3038 3331 3839 3320 es: .00831893 │ │ │ │ -000286b0: 7365 636f 6e64 7320 2020 2020 2020 2020 seconds │ │ │ │ +000286a0: 6573 3a20 2020 2e30 3131 3532 3336 2073 es: .0115236 s │ │ │ │ +000286b0: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 000286c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000286d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000286e0: 2020 7c0a 7c20 2074 696d 6520 2e30 3339 |.| time .039 │ │ │ │ -000286f0: 3432 3237 2073 6563 2020 2366 7261 6374 4227 sec #fract │ │ │ │ -00028700: 696f 6e73 2034 5d20 2020 2020 2020 2020 ions 4] │ │ │ │ +000286e0: 2020 7c0a 7c20 2074 696d 6520 2e30 3531 |.| time .051 │ │ │ │ +000286f0: 3432 3120 7365 6320 2023 6672 6163 7469 421 sec #fracti │ │ │ │ +00028700: 6f6e 7320 345d 2020 2020 2020 2020 2020 ons 4] │ │ │ │ 00028710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028730: 2020 7c0a 7c20 5b73 7465 7020 313a 2020 |.| [step 1: │ │ │ │ 00028740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028780: 2020 7c0a 7c20 2020 2020 2072 6164 6963 |.| radic │ │ │ │ 00028790: 616c 2028 7573 6520 6d69 6e70 7269 6d65 al (use minprime │ │ │ │ -000287a0: 7329 202e 3030 3235 3932 3838 2073 6563 s) .00259288 sec │ │ │ │ +000287a0: 7329 202e 3030 3330 3133 3734 2073 6563 s) .00301374 sec │ │ │ │ 000287b0: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 000287c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000287d0: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -000287e0: 6572 313a 2020 2e30 3132 3331 3733 2073 er1: .0123173 s │ │ │ │ +000287e0: 6572 313a 2020 2e30 3134 3633 3834 2073 er1: .0146384 s │ │ │ │ 000287f0: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 00028800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028820: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -00028830: 6572 323a 2020 2e30 3132 3632 3132 2073 er2: .0126212 s │ │ │ │ +00028830: 6572 323a 2020 2e30 3132 3737 3136 2073 er2: .0127716 s │ │ │ │ 00028840: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 00028850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028870: 2020 7c0a 7c20 2020 2020 206d 696e 7072 |.| minpr │ │ │ │ -00028880: 6573 3a20 2020 2e30 3131 3931 3137 2073 es: .0119117 s │ │ │ │ +00028880: 6573 3a20 2020 2e30 3133 3933 3636 2073 es: .0139366 s │ │ │ │ 00028890: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 000288a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000288b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000288c0: 2020 7c0a 7c20 2074 696d 6520 2e30 3530 |.| time .050 │ │ │ │ -000288d0: 3738 3336 2073 6563 2020 2366 7261 6374 7836 sec #fract │ │ │ │ +000288c0: 2020 7c0a 7c20 2074 696d 6520 2e30 3537 |.| time .057 │ │ │ │ +000288d0: 3931 3534 2073 6563 2020 2366 7261 6374 9154 sec #fract │ │ │ │ 000288e0: 696f 6e73 2034 5d20 2020 2020 2020 2020 ions 4] │ │ │ │ 000288f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028910: 2020 7c0a 7c20 5b73 7465 7020 323a 2020 |.| [step 2: │ │ │ │ 00028920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028960: 2020 7c0a 7c20 2020 2020 2072 6164 6963 |.| radic │ │ │ │ 00028970: 616c 2028 7573 6520 6d69 6e70 7269 6d65 al (use minprime │ │ │ │ -00028980: 7329 202e 3030 3232 3735 3436 2073 6563 s) .00227546 sec │ │ │ │ +00028980: 7329 202e 3030 3238 3635 3336 2073 6563 s) .00286536 sec │ │ │ │ 00028990: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 000289a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000289b0: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -000289c0: 6572 313a 2020 2e30 3131 3338 3838 2073 er1: .0113888 s │ │ │ │ +000289c0: 6572 313a 2020 2e30 3134 3832 3534 2073 er1: .0148254 s │ │ │ │ 000289d0: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 000289e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000289f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a00: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -00028a10: 6572 323a 2020 2e30 3039 3536 3739 2073 er2: .0095679 s │ │ │ │ -00028a20: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ +00028a10: 6572 323a 2020 2e30 3132 3235 3720 7365 er2: .012257 se │ │ │ │ +00028a20: 636f 6e64 7320 2020 2020 2020 2020 2020 conds │ │ │ │ 00028a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a50: 2020 7c0a 7c20 2020 2020 206d 696e 7072 |.| minpr │ │ │ │ -00028a60: 6573 3a20 2020 2e30 3039 3132 3837 3220 es: .00912872 │ │ │ │ -00028a70: 7365 636f 6e64 7320 2020 2020 2020 2020 seconds │ │ │ │ +00028a60: 6573 3a20 2020 2e30 3131 3335 3339 2073 es: .0113539 s │ │ │ │ +00028a70: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 00028a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028aa0: 2020 7c0a 7c20 2074 696d 6520 2e30 3433 |.| time .043 │ │ │ │ -00028ab0: 3237 3435 2073 6563 2020 2366 7261 6374 2745 sec #fract │ │ │ │ -00028ac0: 696f 6e73 2035 5d20 2020 2020 2020 2020 ions 5] │ │ │ │ +00028aa0: 2020 7c0a 7c20 2074 696d 6520 2e30 3534 |.| time .054 │ │ │ │ +00028ab0: 3533 2073 6563 2020 2366 7261 6374 696f 53 sec #fractio │ │ │ │ +00028ac0: 6e73 2035 5d20 2020 2020 2020 2020 2020 ns 5] │ │ │ │ 00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028af0: 2020 7c0a 7c20 5b73 7465 7020 333a 2020 |.| [step 3: │ │ │ │ 00028b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b40: 2020 7c0a 7c20 2020 2020 2072 6164 6963 |.| radic │ │ │ │ 00028b50: 616c 2028 7573 6520 6d69 6e70 7269 6d65 al (use minprime │ │ │ │ -00028b60: 7329 202e 3030 3232 3938 3838 2073 6563 s) .00229888 sec │ │ │ │ +00028b60: 7329 202e 3030 3239 3034 3337 2073 6563 s) .00290437 sec │ │ │ │ 00028b70: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b90: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -00028ba0: 6572 313a 2020 2e31 3036 3830 3520 7365 er1: .106805 se │ │ │ │ +00028ba0: 6572 313a 2020 2e31 3733 3336 3520 7365 er1: .173365 se │ │ │ │ 00028bb0: 636f 6e64 7320 2020 2020 2020 2020 2020 conds │ │ │ │ 00028bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028be0: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -00028bf0: 6572 323a 2020 2e30 3132 3936 3832 2073 er2: .0129682 s │ │ │ │ +00028bf0: 6572 323a 2020 2e30 3135 3732 3235 2073 er2: .0157225 s │ │ │ │ 00028c00: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 00028c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c30: 2020 7c0a 7c20 2020 2020 206d 696e 7072 |.| minpr │ │ │ │ -00028c40: 6573 3a20 2020 2e30 3135 3839 3837 2073 es: .0158987 s │ │ │ │ +00028c40: 6573 3a20 2020 2e30 3138 3438 3132 2073 es: .0184812 s │ │ │ │ 00028c50: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 00028c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c80: 2020 7c0a 7c20 2074 696d 6520 2e31 3530 |.| time .150 │ │ │ │ -00028c90: 3231 3820 7365 6320 2023 6672 6163 7469 218 sec #fracti │ │ │ │ +00028c80: 2020 7c0a 7c20 2074 696d 6520 2e32 3235 |.| time .225 │ │ │ │ +00028c90: 3132 3620 7365 6320 2023 6672 6163 7469 126 sec #fracti │ │ │ │ 00028ca0: 6f6e 7320 355d 2020 2020 2020 2020 2020 ons 5] │ │ │ │ 00028cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028cd0: 2020 7c0a 7c20 5b73 7465 7020 343a 2020 |.| [step 4: │ │ │ │ 00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d20: 2020 7c0a 7c20 2020 2020 2072 6164 6963 |.| radic │ │ │ │ 00028d30: 616c 2028 7573 6520 6d69 6e70 7269 6d65 al (use minprime │ │ │ │ -00028d40: 7329 202e 3030 3231 3936 3134 2073 6563 s) .00219614 sec │ │ │ │ +00028d40: 7329 202e 3030 3239 3639 3733 2073 6563 s) .00296973 sec │ │ │ │ 00028d50: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d70: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -00028d80: 6572 313a 2020 2e30 3038 3433 3539 3320 er1: .00843593 │ │ │ │ -00028d90: 7365 636f 6e64 7320 2020 2020 2020 2020 seconds │ │ │ │ +00028d80: 6572 313a 2020 2e30 3131 3237 3533 2073 er1: .0112753 s │ │ │ │ +00028d90: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 00028da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028dc0: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -00028dd0: 6572 323a 2020 2e30 3135 3738 3331 2073 er2: .0157831 s │ │ │ │ +00028dd0: 6572 323a 2020 2e30 3139 3333 3338 2073 er2: .0193338 s │ │ │ │ 00028de0: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 00028df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e10: 2020 7c0a 7c20 2020 2020 206d 696e 7072 |.| minpr │ │ │ │ -00028e20: 6573 3a20 2020 2e30 3132 3334 3238 2073 es: .0123428 s │ │ │ │ -00028e30: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ +00028e20: 6573 3a20 2020 2e30 3134 3330 3720 7365 es: .014307 se │ │ │ │ +00028e30: 636f 6e64 7320 2020 2020 2020 2020 2020 conds │ │ │ │ 00028e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e60: 2020 7c0a 7c20 2074 696d 6520 2e30 3530 |.| time .050 │ │ │ │ -00028e70: 3937 3832 2073 6563 2020 2366 7261 6374 9782 sec #fract │ │ │ │ +00028e60: 2020 7c0a 7c20 2074 696d 6520 2e30 3632 |.| time .062 │ │ │ │ +00028e70: 3637 3032 2073 6563 2020 2366 7261 6374 6702 sec #fract │ │ │ │ 00028e80: 696f 6e73 2035 5d20 2020 2020 2020 2020 ions 5] │ │ │ │ 00028e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028eb0: 2020 7c0a 7c20 5b73 7465 7020 353a 2020 |.| [step 5: │ │ │ │ 00028ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f00: 2020 7c0a 7c20 2020 2020 2072 6164 6963 |.| radic │ │ │ │ 00028f10: 616c 2028 7573 6520 6d69 6e70 7269 6d65 al (use minprime │ │ │ │ -00028f20: 7329 202e 3030 3235 3233 3533 2073 6563 s) .00252353 sec │ │ │ │ +00028f20: 7329 202e 3030 3331 3734 3338 2073 6563 s) .00317438 sec │ │ │ │ 00028f30: 6f6e 6473 2020 2020 2020 2020 2020 2020 onds │ │ │ │ 00028f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f50: 2020 7c0a 7c20 2020 2020 2069 646c 697a |.| idliz │ │ │ │ -00028f60: 6572 313a 2020 2e30 3038 3935 3433 3720 er1: .00895437 │ │ │ │ -00028f70: 7365 636f 6e64 7320 2020 2020 2020 2020 seconds │ │ │ │ +00028f60: 6572 313a 2020 2e30 3130 3430 3737 2073 er1: .0104077 s │ │ │ │ +00028f70: 6563 6f6e 6473 2020 2020 2020 2020 2020 econds │ │ │ │ 00028f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028fa0: 2020 7c0a 7c20 2074 696d 6520 2e30 3138 |.| time .018 │ │ │ │ -00028fb0: 3533 3638 2073 6563 2020 2366 7261 6374 5368 sec #fract │ │ │ │ +00028fa0: 2020 7c0a 7c20 2074 696d 6520 2e30 3231 |.| time .021 │ │ │ │ +00028fb0: 3932 3231 2073 6563 2020 2366 7261 6374 9221 sec #fract │ │ │ │ 00028fc0: 696f 6e73 2035 5d20 2020 2020 2020 2020 ions 5] │ │ │ │ 00028fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ff0: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00029000: 3335 3733 3573 2028 6370 7529 3b20 302e 35735s (cpu); 0. │ │ │ │ -00029010: 3331 3337 3637 7320 2874 6872 6561 6429 313767s (thread) │ │ │ │ +00029000: 3437 3834 3539 7320 2863 7075 293b 2030 478459s (cpu); 0 │ │ │ │ +00029010: 2e33 3737 3633 7320 2874 6872 6561 6429 .37763s (thread) │ │ │ │ 00029020: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00029030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029040: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00029050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -10998,18 +10998,18 @@ │ │ │ │ 0002af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0002af70: 0a7c 6934 203a 2074 696d 6520 696e 7465 .|i4 : time inte │ │ │ │ 0002af80: 6772 616c 436c 6f73 7572 6520 4a20 2020 gralClosure J │ │ │ │ 0002af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002afb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002afc0: 0a7c 202d 2d20 7573 6564 2030 2e39 3033 .| -- used 0.903 │ │ │ │ -0002afd0: 3735 3273 2028 6370 7529 3b20 302e 3734 752s (cpu); 0.74 │ │ │ │ -0002afe0: 3836 3432 7320 2874 6872 6561 6429 3b20 8642s (thread); │ │ │ │ -0002aff0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +0002afc0: 0a7c 202d 2d20 7573 6564 2031 2e36 3238 .| -- used 1.628 │ │ │ │ +0002afd0: 3932 7320 2863 7075 293b 2030 2e39 3131 92s (cpu); 0.911 │ │ │ │ +0002afe0: 3032 3473 2028 7468 7265 6164 293b 2030 024s (thread); 0 │ │ │ │ +0002aff0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0002b000: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002b010: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002b020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b050: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002b060: 0a7c 2020 2020 2020 2020 2020 2020 2032 .| 2 │ │ │ │ @@ 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2028 7468 7265 6164 293b 2030 028s (thread); 0 │ │ │ │ +0002b360: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0002b370: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002b380: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002b390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b3c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002b3d0: 0a7c 2020 2020 2020 2020 2020 2020 2032 .| 2 │ │ ├── ./usr/share/info/InvariantRing.info.gz │ │ │ ├── InvariantRing.info │ │ │ │ @@ -3301,16 +3301,16 @@ │ │ │ │ 0000ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ce50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ce60: 2d2d 2b0a 7c69 3520 3a20 656c 6170 7365 --+.|i5 : elapse │ │ │ │ 0000ce70: 6454 696d 6520 6571 7569 7661 7269 616e dTime equivarian │ │ │ │ 0000ce80: 7448 696c 6265 7274 5365 7269 6573 2854 tHilbertSeries(T │ │ │ │ 0000ce90: 2c20 4f72 6465 7220 3d3e 2035 2920 2020 , Order => 5) │ │ │ │ 0000cea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ceb0: 2020 7c0a 7c20 2d2d 202e 3030 3236 3234 |.| -- .002624 │ │ │ │ -0000cec0: 3739 7320 656c 6170 7365 6420 2020 2020 79s elapsed │ │ │ │ +0000ceb0: 2020 7c0a 7c20 2d2d 202e 3030 3332 3032 |.| -- .003202 │ │ │ │ +0000cec0: 3632 7320 656c 6170 7365 6420 2020 2020 62s elapsed │ │ │ │ 0000ced0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cf00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000cf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000cf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3431,16 +3431,16 @@ │ │ │ │ 0000d660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d680: 2d2d 2b0a 7c69 3720 3a20 656c 6170 7365 --+.|i7 : elapse │ │ │ │ 0000d690: 6454 696d 6520 6571 7569 7661 7269 616e dTime equivarian │ │ │ │ 0000d6a0: 7448 696c 6265 7274 5365 7269 6573 2854 tHilbertSeries(T │ │ │ │ 0000d6b0: 2c20 4f72 6465 7220 3d3e 2035 293b 2020 , Order => 5); │ │ │ │ 0000d6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d6d0: 2020 7c0a 7c20 2d2d 202e 3030 3034 3633 |.| -- .000463 │ │ │ │ -0000d6e0: 3939 3973 2065 6c61 7073 6564 2020 2020 999s elapsed │ │ │ │ +0000d6d0: 2020 7c0a 7c20 2d2d 202e 3030 3035 3439 |.| -- .000549 │ │ │ │ +0000d6e0: 3135 3173 2065 6c61 7073 6564 2020 2020 151s elapsed │ │ │ │ 0000d6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d720: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0000d730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000d750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -7199,31 +7199,31 @@ │ │ │ │ 0001c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0001c200: 0a7c 6934 203a 2074 696d 6520 5031 3d70 .|i4 : time P1=p │ │ │ │ 0001c210: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ 0001c220: 2043 3478 4332 2020 2020 2020 2020 2020 C4xC2 │ │ │ │ 0001c230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c240: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001c250: 0a7c 202d 2d20 7573 6564 2030 2e38 3637 .| -- used 0.867 │ │ │ │ -0001c260: 3430 3473 2028 6370 7529 3b20 302e 3538 404s (cpu); 0.58 │ │ │ │ -0001c270: 3735 3935 7320 2874 6872 6561 6429 3b20 7595s (thread); │ │ │ │ -0001c280: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +0001c250: 0a7c 202d 2d20 7573 6564 2031 2e30 3735 .| -- used 1.075 │ │ │ │ +0001c260: 3232 7320 2863 7075 293b 2030 2e36 3638 22s (cpu); 0.668 │ │ │ │ +0001c270: 3239 3273 2028 7468 7265 6164 293b 2030 292s (thread); 0 │ │ │ │ +0001c280: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 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 2020 2020 2020 207c | │ │ │ │ 0001c2f0: 0a7c 2020 2020 2020 2032 2020 2032 2020 .| 2 2 │ │ │ │ -0001c300: 2020 3220 2020 3220 3220 2020 2020 2020 2 2 2 │ │ │ │ +0001c300: 2020 3220 2020 3320 2020 2020 2020 3320 2 3 3 │ │ │ │ 0001c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c330: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0001c340: 0a7c 6f34 203d 207b 7a20 2c20 7820 202b .|o4 = {z , x + │ │ │ │ -0001c350: 2079 202c 2078 2079 207d 2020 2020 2020 y , x y } │ │ │ │ +0001c350: 2079 202c 2078 2079 202d 2078 2a79 207d y , x y - x*y } │ │ │ │ 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 207c | │ │ │ │ 0001c390: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7239,18 +7239,18 @@ │ │ │ │ 0001c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0001c480: 0a7c 6935 203a 2074 696d 6520 5032 3d70 .|i5 : time P2=p │ │ │ │ 0001c490: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ 0001c4a0: 2843 3478 4332 2c44 6164 653d 3e74 7275 (C4xC2,Dade=>tru │ │ │ │ 0001c4b0: 6529 2020 2020 2020 2020 2020 2020 2020 e) │ │ │ │ 0001c4c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001c4d0: 0a7c 202d 2d20 7573 6564 2030 2e36 3733 .| -- used 0.673 │ │ │ │ -0001c4e0: 3535 3773 2028 6370 7529 3b20 302e 3337 557s (cpu); 0.37 │ │ │ │ -0001c4f0: 3730 3873 2028 7468 7265 6164 293b 2030 708s (thread); 0 │ │ │ │ -0001c500: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ +0001c4d0: 0a7c 202d 2d20 7573 6564 2030 2e39 3736 .| -- used 0.976 │ │ │ │ +0001c4e0: 3232 3873 2028 6370 7529 3b20 302e 3439 228s (cpu); 0.49 │ │ │ │ +0001c4f0: 3736 3631 7320 2874 6872 6561 6429 3b20 7661s (thread); │ │ │ │ +0001c500: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 0001c510: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0001c520: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001c530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c560: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0001c570: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ @@ -7558,31 +7558,31 @@ │ │ │ │ 0001d850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0001d870: 0a7c 6936 203a 2074 696d 6520 7365 636f .|i6 : time seco │ │ │ │ 0001d880: 6e64 6172 7949 6e76 6172 6961 6e74 7328 ndaryInvariants( │ │ │ │ 0001d890: 5031 2c43 3478 4332 2920 2020 2020 2020 P1,C4xC2) │ │ │ │ 0001d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001d8c0: 0a7c 202d 2d20 7573 6564 2030 2e30 3331 .| -- used 0.031 │ │ │ │ -0001d8d0: 3233 3135 7320 2863 7075 293b 2030 2e30 2315s (cpu); 0.0 │ │ │ │ -0001d8e0: 3331 3233 3531 7320 2874 6872 6561 6429 312351s (thread) │ │ │ │ +0001d8c0: 0a7c 202d 2d20 7573 6564 2030 2e30 3239 .| -- used 0.029 │ │ │ │ +0001d8d0: 3830 3137 7320 2863 7075 293b 2030 2e30 8017s (cpu); 0.0 │ │ │ │ +0001d8e0: 3239 3830 3635 7320 2874 6872 6561 6429 298065s (thread) │ │ │ │ 0001d8f0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0001d900: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0001d910: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001d920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d950: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001d960: 0a7c 2020 2020 2020 2020 2020 3320 2020 .| 3 │ │ │ │ -0001d970: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ +0001d960: 0a7c 2020 2020 2020 2020 2020 3420 2020 .| 4 │ │ │ │ +0001d970: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 0001d980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d9a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001d9b0: 0a7c 6f36 203d 207b 312c 2078 2079 202d .|o6 = {1, x y - │ │ │ │ -0001d9c0: 2078 2a79 207d 2020 2020 2020 2020 2020 x*y } │ │ │ │ +0001d9b0: 0a7c 6f36 203d 207b 312c 2078 2020 2b20 .|o6 = {1, x + │ │ │ │ +0001d9c0: 7920 7d20 2020 2020 2020 2020 2020 2020 y } │ │ │ │ 0001d9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d9f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0001da00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001da10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001da30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7598,17 +7598,17 @@ │ │ │ │ 0001dad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001dae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0001daf0: 0a7c 6937 203a 2074 696d 6520 7365 636f .|i7 : time seco │ │ │ │ 0001db00: 6e64 6172 7949 6e76 6172 6961 6e74 7328 ndaryInvariants( │ │ │ │ 0001db10: 5032 2c43 3478 4332 2920 2020 2020 2020 P2,C4xC2) │ │ │ │ 0001db20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001db30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001db40: 0a7c 202d 2d20 7573 6564 2031 2e39 3734 .| -- used 1.974 │ │ │ │ -0001db50: 3473 2028 6370 7529 3b20 312e 3331 3730 4s (cpu); 1.3170 │ │ │ │ -0001db60: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ +0001db40: 0a7c 202d 2d20 7573 6564 2032 2e37 3637 .| -- used 2.767 │ │ │ │ +0001db50: 3939 7320 2863 7075 293b 2031 2e36 3832 99s (cpu); 1.682 │ │ │ │ +0001db60: 3173 2028 7468 7265 6164 293b 2030 7320 1s (thread); 0s │ │ │ │ 0001db70: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0001db80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0001db90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001dbd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -8867,16 +8867,16 @@ │ │ │ │ 00022a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022a40: 2d2d 2d2b 0a7c 6934 203a 2065 6c61 7073 ---+.|i4 : elaps │ │ │ │ 00022a50: 6564 5469 6d65 2069 6e76 6172 6961 6e74 edTime invariant │ │ │ │ 00022a60: 7320 5334 2020 2020 2020 2020 2020 2020 s S4 │ │ │ │ 00022a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a90: 2020 207c 0a7c 202d 2d20 2e37 3230 3539 |.| -- .72059 │ │ │ │ -00022aa0: 3273 2065 6c61 7073 6564 2020 2020 2020 2s elapsed │ │ │ │ +00022a90: 2020 207c 0a7c 202d 2d20 2e36 3731 3537 |.| -- .67157 │ │ │ │ +00022aa0: 3473 2065 6c61 7073 6564 2020 2020 2020 4s elapsed │ │ │ │ 00022ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ae0: 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8957,16 +8957,16 @@ │ │ │ │ 00022fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022fe0: 2020 207c 0a7c 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 207c 0a7c 202d 2d20 2e35 3737 3530 |.| -- .57750 │ │ │ │ -00023040: 3773 2065 6c61 7073 6564 2020 2020 2020 7s elapsed │ │ │ │ +00023030: 2020 207c 0a7c 202d 2d20 2e35 3538 3130 |.| -- .55810 │ │ │ │ +00023040: 3973 2065 6c61 7073 6564 2020 2020 2020 9s elapsed │ │ │ │ 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 207c 0a7c 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 2020 2020 │ │ │ │ @@ -9894,16 +9894,16 @@ │ │ │ │ 00026a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00026a70: 0a7c 6934 203a 2065 6c61 7073 6564 5469 .|i4 : elapsedTi │ │ │ │ 00026a80: 6d65 2069 6e76 6172 6961 6e74 7320 5334 me invariants S4 │ │ │ │ 00026a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ab0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00026ac0: 0a7c 202d 2d20 2e36 3130 3338 3373 2065 .| -- .610383s e │ │ │ │ -00026ad0: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ +00026ac0: 0a7c 202d 2d20 2e36 3033 3432 7320 656c .| -- .60342s el │ │ │ │ +00026ad0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00026b10: 0a7c 2020 2020 2020 2020 2020 2020 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 │ │ │ │ @@ -9959,15 +9959,15 @@ │ │ │ │ 00026e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00026e80: 0a7c 6935 203a 2065 6c61 7073 6564 5469 .|i5 : elapsedTi │ │ │ │ 00026e90: 6d65 2069 6e76 6172 6961 6e74 7328 5334 me invariants(S4 │ │ │ │ 00026ea0: 2c55 7365 4c69 6e65 6172 416c 6765 6272 ,UseLinearAlgebr │ │ │ │ 00026eb0: 613d 3e74 7275 6529 2020 2020 2020 2020 a=>true) │ │ │ │ 00026ec0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00026ed0: 0a7c 202d 2d20 2e30 3737 3938 3139 7320 .| -- .0779819s │ │ │ │ +00026ed0: 0a7c 202d 2d20 2e30 3834 3034 3231 7320 .| -- .0840421s │ │ │ │ 00026ee0: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00026ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026f10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00026f20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00026f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -15074,4894 +15074,4896 @@ │ │ │ │ 0003ae10: 676f 7269 7468 6d73 3a0a 6873 6f70 2061 gorithms:.hsop a │ │ │ │ 0003ae20: 6c67 6f72 6974 686d 732c 2066 6f72 2061 lgorithms, for a │ │ │ │ 0003ae30: 2064 6973 6375 7373 696f 6e20 636f 6d70 discussion comp │ │ │ │ 0003ae40: 6172 696e 6720 7468 6520 7477 6f20 616c aring the two al │ │ │ │ 0003ae50: 676f 7269 7468 6d73 2e0a 0a2b 2d2d 2d2d gorithms...+---- │ │ │ │ 0003ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ae90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003aea0: 2b0a 7c69 3120 3a20 413d 6d61 7472 6978 +.|i1 : A=matrix │ │ │ │ -0003aeb0: 7b7b 302c 312c 307d 2c7b 302c 302c 317d {{0,1,0},{0,0,1} │ │ │ │ -0003aec0: 2c7b 312c 302c 307d 7d3b 2020 2020 2020 ,{1,0,0}}; │ │ │ │ +0003ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0003ae90: 203a 2041 3d6d 6174 7269 787b 7b30 2c31 : A=matrix{{0,1 │ │ │ │ +0003aea0: 2c30 7d2c 7b30 2c30 2c31 7d2c 7b31 2c30 ,0},{0,0,1},{1,0 │ │ │ │ +0003aeb0: 2c30 7d7d 3b20 2020 2020 2020 207c 0a7c ,0}}; |.| │ │ │ │ +0003aec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003aed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003aee0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0003aef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003af00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003aee0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003aef0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003af00: 3320 2020 2020 2020 3320 2020 2020 2020 3 3 │ │ │ │ 0003af10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003af20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0003af30: 7c20 2020 2020 2020 2020 2020 2020 2033 | 3 │ │ │ │ -0003af40: 2020 2020 2020 2033 2020 2020 2020 2020 3 │ │ │ │ -0003af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003af70: 2020 2020 207c 0a7c 6f31 203a 204d 6174 |.|o1 : Mat │ │ │ │ -0003af80: 7269 7820 5a5a 2020 3c2d 2d20 5a5a 2020 rix ZZ <-- ZZ │ │ │ │ -0003af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003afb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0003afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b000: 2d2d 2d2b 0a7c 6932 203a 2042 3d6d 6174 ---+.|i2 : B=mat │ │ │ │ -0003b010: 7269 787b 7b30 2c31 2c30 7d2c 7b31 2c30 rix{{0,1,0},{1,0 │ │ │ │ -0003b020: 2c30 7d2c 7b30 2c30 2c31 7d7d 3b20 2020 ,0},{0,0,1}}; │ │ │ │ -0003b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b040: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0003b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b090: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003b0a0: 2020 3320 2020 2020 2020 3320 2020 2020 3 3 │ │ │ │ -0003b0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003af20: 207c 0a7c 6f31 203a 204d 6174 7269 7820 |.|o1 : Matrix │ │ │ │ +0003af30: 5a5a 2020 3c2d 2d20 5a5a 2020 2020 2020 ZZ <-- ZZ │ │ │ │ +0003af40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003af50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0003af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003af80: 2d2d 2d2d 2d2b 0a7c 6932 203a 2042 3d6d -----+.|i2 : B=m │ │ │ │ +0003af90: 6174 7269 787b 7b30 2c31 2c30 7d2c 7b31 atrix{{0,1,0},{1 │ │ │ │ +0003afa0: 2c30 2c30 7d2c 7b30 2c30 2c31 7d7d 3b20 ,0,0},{0,0,1}}; │ │ │ │ +0003afb0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0003afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003afd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003afe0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003aff0: 2020 2020 2020 2020 2020 3320 2020 2020 3 │ │ │ │ +0003b000: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0003b010: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +0003b020: 203a 204d 6174 7269 7820 5a5a 2020 3c2d : Matrix ZZ <- │ │ │ │ +0003b030: 2d20 5a5a 2020 2020 2020 2020 2020 2020 - ZZ │ │ │ │ +0003b040: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0003b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003b080: 0a7c 6933 203a 2053 333d 6669 6e69 7465 .|i3 : S3=finite │ │ │ │ +0003b090: 4163 7469 6f6e 287b 412c 427d 2c51 515b Action({A,B},QQ[ │ │ │ │ +0003b0a0: 782c 792c 7a5d 2920 2020 2020 2020 2020 x,y,z]) │ │ │ │ +0003b0b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b0d0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ -0003b0e0: 4d61 7472 6978 205a 5a20 203c 2d2d 205a Matrix ZZ <-- Z │ │ │ │ -0003b0f0: 5a20 2020 2020 2020 2020 2020 2020 2020 Z │ │ │ │ -0003b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b110: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003b120: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0003b130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b160: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 5333 ------+.|i3 : S3 │ │ │ │ -0003b170: 3d66 696e 6974 6541 6374 696f 6e28 7b41 =finiteAction({A │ │ │ │ -0003b180: 2c42 7d2c 5151 5b78 2c79 2c7a 5d29 2020 ,B},QQ[x,y,z]) │ │ │ │ +0003b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b0e0: 2020 207c 0a7c 6f33 203d 2051 515b 782e |.|o3 = QQ[x. │ │ │ │ +0003b0f0: 2e7a 5d20 3c2d 207b 7c20 3020 3120 3020 .z] <- {| 0 1 0 │ │ │ │ +0003b100: 7c2c 207c 2030 2031 2030 207c 7d20 2020 |, | 0 1 0 |} │ │ │ │ +0003b110: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003b120: 2020 2020 2020 2020 2020 7c20 3020 3020 | 0 0 │ │ │ │ +0003b130: 3120 7c20 207c 2031 2030 2030 207c 2020 1 | | 1 0 0 | │ │ │ │ +0003b140: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0003b150: 2020 2020 2020 2020 2020 2020 7c20 3120 | 1 │ │ │ │ +0003b160: 3020 3020 7c20 207c 2030 2030 2031 207c 0 0 | | 0 0 1 | │ │ │ │ +0003b170: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b1a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003b1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b1f0: 2020 2020 7c0a 7c6f 3320 3d20 5151 5b78 |.|o3 = QQ[x │ │ │ │ -0003b200: 2e2e 7a5d 203c 2d20 7b7c 2030 2031 2030 ..z] <- {| 0 1 0 │ │ │ │ -0003b210: 207c 2c20 7c20 3020 3120 3020 7c7d 2020 |, | 0 1 0 |} │ │ │ │ -0003b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b230: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0003b240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b250: 7c20 3020 3020 3120 7c20 207c 2031 2030 | 0 0 1 | | 1 0 │ │ │ │ -0003b260: 2030 207c 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ -0003b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b280: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003b290: 2020 2020 2020 207c 2031 2030 2030 207c | 1 0 0 | │ │ │ │ -0003b2a0: 2020 7c20 3020 3020 3120 7c20 2020 2020 | 0 0 1 | │ │ │ │ -0003b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b2c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0003b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b1a0: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0003b1b0: 203a 2046 696e 6974 6547 726f 7570 4163 : FiniteGroupAc │ │ │ │ +0003b1c0: 7469 6f6e 2020 2020 2020 2020 2020 2020 tion │ │ │ │ +0003b1d0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0003b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003b210: 0a7c 6934 203a 2070 7269 6d61 7279 496e .|i4 : primaryIn │ │ │ │ +0003b220: 7661 7269 616e 7473 2053 3320 2020 2020 variants S3 │ │ │ │ +0003b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b290: 2020 2020 2020 2020 2033 2020 2020 3320 3 3 │ │ │ │ +0003b2a0: 2020 2033 207c 0a7c 6f34 203d 207b 7820 3 |.|o4 = {x │ │ │ │ +0003b2b0: 2b20 7920 2b20 7a2c 2078 2a79 202b 2078 + y + z, x*y + x │ │ │ │ +0003b2c0: 2a7a 202b 2079 2a7a 2c20 7820 202b 2079 *z + y*z, x + y │ │ │ │ +0003b2d0: 2020 2b20 7a20 7d7c 0a7c 2020 2020 2020 + z }|.| │ │ │ │ 0003b2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b310: 7c0a 7c6f 3320 3a20 4669 6e69 7465 4772 |.|o3 : FiniteGr │ │ │ │ -0003b320: 6f75 7041 6374 696f 6e20 2020 2020 2020 oupAction │ │ │ │ -0003b330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b350: 2020 2020 2020 207c 0a2b 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 2d2d 2d2d ---------------- │ │ │ │ -0003b390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0003b3a0: 7c69 3420 3a20 7072 696d 6172 7949 6e76 |i4 : primaryInv │ │ │ │ -0003b3b0: 6172 6961 6e74 7320 5333 2020 2020 2020 ariants S3 │ │ │ │ -0003b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b3e0: 2020 2020 207c 0a7c 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 7c20 |.| │ │ │ │ -0003b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b440: 2032 2020 2020 3220 2020 2032 2020 2032 2 2 2 2 │ │ │ │ -0003b450: 2020 2020 2020 2032 2020 2020 3220 2020 2 2 │ │ │ │ -0003b460: 2020 3220 2020 2020 2020 3220 2020 2020 2 2 │ │ │ │ -0003b470: 2032 207c 0a7c 6f34 203d 207b 7820 2b20 2 |.|o4 = {x + │ │ │ │ -0003b480: 7920 2b20 7a2c 2078 2020 2b20 7920 202b y + z, x + y + │ │ │ │ -0003b490: 207a 202c 2078 2079 202b 2078 2a79 2020 z , x y + x*y │ │ │ │ -0003b4a0: 2b20 7820 7a20 2b20 7920 7a20 2b20 782a + x z + y z + x* │ │ │ │ -0003b4b0: 7a20 202b 2079 2a7a 207d 7c0a 7c20 2020 z + y*z }|.| │ │ │ │ +0003b300: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ +0003b310: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ +0003b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b330: 2020 2020 2020 2020 2020 207c 0a2b 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 2d2b 0a0a -------------+.. │ │ │ │ +0003b370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b390: 2020 2053 330a 4265 6c6f 772c 2074 6865 S3.Below, the │ │ │ │ +0003b3a0: 2069 6e76 6172 6961 6e74 2072 696e 6720 invariant ring │ │ │ │ +0003b3b0: 5151 5b78 2c79 2c7a 5d20 2020 6973 2063 QQ[x,y,z] is c │ │ │ │ +0003b3c0: 616c 6375 6c61 7465 6420 7769 7468 204b alculated with K │ │ │ │ +0003b3d0: 2062 6569 6e67 2074 6865 2066 6965 6c64 being the field │ │ │ │ +0003b3e0: 2077 6974 680a 3130 3120 656c 656d 656e with.101 elemen │ │ │ │ +0003b3f0: 7473 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ts...+---------- │ │ │ │ +0003b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b440: 2d2d 2d2b 0a7c 6935 203a 204b 3d47 4628 ---+.|i5 : K=GF( │ │ │ │ +0003b450: 3130 3129 2020 2020 2020 2020 2020 2020 101) │ │ │ │ +0003b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b490: 2020 207c 0a7c 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 2020 2020 │ │ │ │ 0003b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b4e0: 2020 207c 0a7c 6f35 203d 204b 2020 2020 |.|o5 = K │ │ │ │ 0003b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b500: 207c 0a7c 6f34 203a 204c 6973 7420 2020 |.|o4 : List │ │ │ │ +0003b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b540: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0003b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0003b590: 0a0a 2020 2020 2020 2020 2020 2020 2020 .. │ │ │ │ +0003b530: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b580: 2020 207c 0a7c 6f35 203a 2047 616c 6f69 |.|o5 : Galoi │ │ │ │ +0003b590: 7346 6965 6c64 2020 2020 2020 2020 2020 sField │ │ │ │ 0003b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b5b0: 2020 2020 2053 330a 4265 6c6f 772c 2074 S3.Below, t │ │ │ │ -0003b5c0: 6865 2069 6e76 6172 6961 6e74 2072 696e he invariant rin │ │ │ │ -0003b5d0: 6720 5151 5b78 2c79 2c7a 5d20 2020 6973 g QQ[x,y,z] is │ │ │ │ -0003b5e0: 2063 616c 6375 6c61 7465 6420 7769 7468 calculated with │ │ │ │ -0003b5f0: 204b 2062 6569 6e67 2074 6865 2066 6965 K being the fie │ │ │ │ -0003b600: 6c64 2077 6974 680a 3130 3120 656c 656d ld with.101 elem │ │ │ │ -0003b610: 656e 7473 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d ents...+-------- │ │ │ │ -0003b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b660: 2d2d 2d2d 2d2b 0a7c 6935 203a 204b 3d47 -----+.|i5 : K=G │ │ │ │ -0003b670: 4628 3130 3129 2020 2020 2020 2020 2020 F(101) │ │ │ │ +0003b5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b5d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0003b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b620: 2d2d 2d2b 0a7c 6936 203a 2053 333d 6669 ---+.|i6 : S3=fi │ │ │ │ +0003b630: 6e69 7465 4163 7469 6f6e 287b 412c 427d niteAction({A,B} │ │ │ │ +0003b640: 2c4b 5b78 2c79 2c7a 5d29 2020 2020 2020 ,K[x,y,z]) │ │ │ │ +0003b650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b670: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 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 2020 2020 │ │ │ │ -0003b6b0: 2020 2020 207c 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2020 │ │ │ │ -0003b8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b800: 2020 207c 0a7c 6f36 203a 2046 696e 6974 |.|o6 : Finit │ │ │ │ +0003b810: 6547 726f 7570 4163 7469 6f6e 2020 2020 eGroupAction │ │ │ │ +0003b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b850: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0003b860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003b8a0: 2d2d 2d2b 0a7c 6937 203a 2070 7269 6d61 ---+.|i7 : prima │ │ │ │ +0003b8b0: 7279 496e 7661 7269 616e 7473 2853 332c ryInvariants(S3, │ │ │ │ +0003b8c0: 4461 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|.| │ │ │ │ -0003b990: 2020 2020 2020 2020 207c 2031 2030 2030 | 1 0 0 │ │ │ │ -0003b9a0: 207c 2020 7c20 3020 3020 3120 7c20 2020 | | 0 0 1 | │ │ │ │ -0003b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b9d0: 2020 2020 207c 0a7c 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 2020 2020 │ │ │ │ -0003ba20: 2020 2020 207c 0a7c 6f36 203a 2046 696e |.|o6 : Fin │ │ │ │ -0003ba30: 6974 6547 726f 7570 4163 7469 6f6e 2020 iteGroupAction │ │ │ │ -0003ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ba50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ba60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ba70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0003ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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2020 207c 0a7c 2020 2020 202d 2d2d + |.| --- │ │ │ │ -0003c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c1f0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0003c200: 3220 3420 2020 2020 2020 2035 2020 2020 2 4 5 │ │ │ │ -0003c210: 2020 2020 3520 2020 2020 3620 2020 2020 5 6 │ │ │ │ -0003c220: 3620 2020 2020 2035 2020 2020 2020 2034 6 5 4 │ │ │ │ -0003c230: 2032 2020 2020 2020 3220 3420 2020 2020 2 2 4 │ │ │ │ -0003c240: 2020 2035 207c 0a7c 2020 2020 2031 3379 5 |.| 13y │ │ │ │ -0003c250: 207a 2020 2b20 3332 782a 7a20 202b 2033 z + 32x*z + 3 │ │ │ │ -0003c260: 3279 2a7a 2020 2b20 397a 202c 2031 3378 2y*z + 9z , 13x │ │ │ │ -0003c270: 2020 2d20 3433 7820 7920 2b20 3435 7820 - 43x y + 45x │ │ │ │ -0003c280: 7920 202b 2034 3578 2079 2020 2d20 3433 y + 45x y - 43 │ │ │ │ -0003c290: 782a 7920 207c 0a7c 2020 2020 202d 2d2d x*y |.| --- │ │ │ │ -0003c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c2e0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0003c2f0: 2020 3620 2020 2020 2035 2020 2020 2020 6 5 │ │ │ │ -0003c300: 2034 2020 2020 2020 2020 2033 2032 2020 4 3 2 │ │ │ │ -0003c310: 2020 2020 2032 2033 2020 2020 2020 2020 2 3 │ │ │ │ -0003c320: 2034 2020 2020 2020 2035 2020 2020 2020 4 5 │ │ │ │ -0003c330: 2034 2032 207c 0a7c 2020 2020 202b 2031 4 2 |.| + 1 │ │ │ │ -0003c340: 3379 2020 2d20 3433 7820 7a20 2d20 3233 3y - 43x z - 23 │ │ │ │ -0003c350: 7820 792a 7a20 2d20 3339 7820 7920 7a20 x y*z - 39x y z │ │ │ │ -0003c360: 2d20 3339 7820 7920 7a20 2d20 3233 782a - 39x y z - 23x* │ │ │ │ -0003c370: 7920 7a20 2d20 3433 7920 7a20 2b20 3435 y z - 43y z + 45 │ │ │ │ -0003c380: 7820 7a20 207c 0a7c 2020 2020 202d 2d2d x z |.| --- │ │ │ │ -0003c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c3d0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0003c3e0: 2020 3320 2020 3220 2020 2020 2032 2032 3 2 2 2 │ │ │ │ -0003c3f0: 2032 2020 2020 2020 2020 3320 3220 2020 2 3 2 │ │ │ │ -0003c400: 2020 2034 2032 2020 2020 2020 3220 2020 4 2 2 │ │ │ │ -0003c410: 3320 2020 2020 2020 2032 2033 2020 2020 3 2 3 │ │ │ │ -0003c420: 2020 3220 347c 0a7c 2020 2020 202d 2033 2 4|.| - 3 │ │ │ │ -0003c430: 3978 2079 2a7a 2020 2b20 3239 7820 7920 9x y*z + 29x y │ │ │ │ -0003c440: 7a20 202d 2033 3978 2a79 207a 2020 2b20 z - 39x*y z + │ │ │ │ -0003c450: 3435 7920 7a20 202d 2033 3978 2079 2a7a 45y z - 39x y*z │ │ │ │ -0003c460: 2020 2d20 3339 782a 7920 7a20 202b 2034 - 39x*y z + 4 │ │ │ │ -0003c470: 3578 207a 207c 0a7c 2020 2020 202d 2d2d 5x z |.| --- │ │ │ │ -0003c480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003c4c0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0003c4d0: 2020 2020 2020 3420 2020 2020 2032 2034 4 2 4 │ │ │ │ -0003c4e0: 2020 2020 2020 2020 3520 2020 2020 2020 5 │ │ │ │ -0003c4f0: 2035 2020 2020 2020 3620 2020 2020 2020 5 6 │ │ │ │ -0003c500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003c510: 2020 2020 207c 0a7c 2020 2020 202d 2032 |.| - 2 │ │ │ │ -0003c520: 3378 2a79 2a7a 2020 2b20 3435 7920 7a20 3x*y*z + 45y z │ │ │ │ -0003c530: 202d 2034 3378 2a7a 2020 2d20 3433 792a - 43x*z - 43y* │ │ │ │ -0003c540: 7a20 202b 2031 337a 207d 2020 2020 2020 z + 13z } │ │ │ │ +0003b930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003b940: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003b950: 2036 2020 2020 2020 3520 2020 2020 2020 6 5 │ │ │ │ +0003b960: 3420 3220 2020 2020 2033 2033 2020 2020 4 2 3 3 │ │ │ │ +0003b970: 2020 3220 3420 2020 2020 2020 2035 2020 2 4 5 │ │ │ │ +0003b980: 2020 2020 3620 2020 2020 2035 2020 2020 6 5 │ │ │ │ +0003b990: 2020 207c 0a7c 6f37 203d 207b 2d20 3435 |.|o7 = {- 45 │ │ │ │ +0003b9a0: 7820 202d 2032 3878 2079 202d 2034 3578 x - 28x y - 45x │ │ │ │ +0003b9b0: 2079 2020 2d20 3432 7820 7920 202d 2034 y - 42x y - 4 │ │ │ │ +0003b9c0: 3578 2079 2020 2d20 3238 782a 7920 202d 5x y - 28x*y - │ │ │ │ +0003b9d0: 2034 3579 2020 2d20 3238 7820 7a20 2b20 45y - 28x z + │ │ │ │ +0003b9e0: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ +0003b9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ba00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ba10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ba20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ba30: 2d2d 2d7c 0a7c 2020 2020 2020 2020 3420 ---|.| 4 │ │ │ │ +0003ba40: 2020 2020 2020 2020 3320 3220 2020 2020 3 2 │ │ │ │ +0003ba50: 2020 3220 3320 2020 2020 2020 2020 3420 2 3 4 │ │ │ │ +0003ba60: 2020 2020 2020 3520 2020 2020 2020 3420 5 4 │ │ │ │ +0003ba70: 3220 2020 2020 2033 2020 2032 2020 2020 2 3 2 │ │ │ │ +0003ba80: 2020 207c 0a7c 2020 2020 2031 3178 2079 |.| 11x y │ │ │ │ +0003ba90: 2a7a 202b 2032 3378 2079 207a 202b 2032 *z + 23x y z + 2 │ │ │ │ +0003baa0: 3378 2079 207a 202b 2031 3178 2a79 207a 3x y z + 11x*y z │ │ │ │ +0003bab0: 202d 2032 3879 207a 202d 2034 3578 207a - 28y z - 45x z │ │ │ │ +0003bac0: 2020 2b20 3233 7820 792a 7a20 202b 2020 + 23x y*z + │ │ │ │ +0003bad0: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ +0003bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003baf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bb20: 2d2d 2d7c 0a7c 2020 2020 2020 2020 3220 ---|.| 2 │ │ │ │ +0003bb30: 3220 3220 2020 2020 2020 2033 2032 2020 2 2 3 2 │ │ │ │ +0003bb40: 2020 2020 3420 3220 2020 2020 2033 2033 4 2 3 3 │ │ │ │ +0003bb50: 2020 2020 2020 3220 2020 3320 2020 2020 2 3 │ │ │ │ +0003bb60: 2020 2032 2033 2020 2020 2020 3320 3320 2 3 3 3 │ │ │ │ +0003bb70: 2020 207c 0a7c 2020 2020 2031 3878 2079 |.| 18x y │ │ │ │ +0003bb80: 207a 2020 2b20 3233 782a 7920 7a20 202d z + 23x*y z - │ │ │ │ +0003bb90: 2034 3579 207a 2020 2d20 3432 7820 7a20 45y z - 42x z │ │ │ │ +0003bba0: 202b 2032 3378 2079 2a7a 2020 2b20 3233 + 23x y*z + 23 │ │ │ │ +0003bbb0: 782a 7920 7a20 202d 2034 3279 207a 2020 x*y z - 42y z │ │ │ │ +0003bbc0: 2d20 207c 0a7c 2020 2020 202d 2d2d 2d2d - |.| ----- │ │ │ │ +0003bbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bc10: 2d2d 2d7c 0a7c 2020 2020 2020 2020 3220 ---|.| 2 │ │ │ │ +0003bc20: 3420 2020 2020 2020 2020 2034 2020 2020 4 4 │ │ │ │ +0003bc30: 2020 3220 3420 2020 2020 2020 2035 2020 2 4 5 │ │ │ │ +0003bc40: 2020 2020 2020 3520 2020 2020 2036 2020 5 6 │ │ │ │ +0003bc50: 2020 3620 2020 2020 2035 2020 2020 2020 6 5 │ │ │ │ +0003bc60: 2020 207c 0a7c 2020 2020 2034 3578 207a |.| 45x z │ │ │ │ +0003bc70: 2020 2b20 3131 782a 792a 7a20 202d 2034 + 11x*y*z - 4 │ │ │ │ +0003bc80: 3579 207a 2020 2d20 3238 782a 7a20 202d 5y z - 28x*z - │ │ │ │ +0003bc90: 2032 3879 2a7a 2020 2d20 3435 7a20 2c20 28y*z - 45z , │ │ │ │ +0003bca0: 3978 2020 2b20 3332 7820 7920 2b20 2020 9x + 32x y + │ │ │ │ +0003bcb0: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ +0003bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003bd00: 2d2d 2d7c 0a7c 2020 2020 2020 2020 3420 ---|.| 4 │ │ │ │ +0003bd10: 3220 2020 2020 2033 2033 2020 2020 2020 2 3 3 │ │ │ │ +0003bd20: 3220 3420 2020 2020 2020 2035 2020 2020 2 4 5 │ │ │ │ +0003bd30: 2036 2020 2020 2020 3520 2020 2020 2020 6 5 │ │ │ │ +0003bd40: 3420 2020 2020 2020 2020 3320 3220 2020 4 3 2 │ │ │ │ +0003bd50: 2020 207c 0a7c 2020 2020 2031 3378 2079 |.| 13x y │ │ │ │ +0003bd60: 2020 2d20 3231 7820 7920 202b 2031 3378 - 21x y + 13x │ │ │ │ +0003bd70: 2079 2020 2b20 3332 782a 7920 202b 2039 y + 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+0003c020: 2035 207c 0a7c 2020 2020 2031 3379 207a 5 |.| 13y z │ │ │ │ +0003c030: 2020 2b20 3332 782a 7a20 202b 2033 3279 + 32x*z + 32y │ │ │ │ +0003c040: 2a7a 2020 2b20 397a 202c 2031 3378 2020 *z + 9z , 13x │ │ │ │ +0003c050: 2d20 3433 7820 7920 2b20 3435 7820 7920 - 43x y + 45x y │ │ │ │ +0003c060: 202b 2034 3578 2079 2020 2d20 3433 782a + 45x y - 43x* │ │ │ │ +0003c070: 7920 207c 0a7c 2020 2020 202d 2d2d 2d2d y |.| ----- │ │ │ │ +0003c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c0c0: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020 ---|.| │ │ │ │ +0003c0d0: 3620 2020 2020 2035 2020 2020 2020 2034 6 5 4 │ │ │ │ +0003c0e0: 2020 2020 2020 2020 2033 2032 2020 2020 3 2 │ │ │ │ +0003c0f0: 2020 2032 2033 2020 2020 2020 2020 2034 2 3 4 │ │ │ │ +0003c100: 2020 2020 2020 2035 2020 2020 2020 2034 5 4 │ │ │ │ +0003c110: 2032 207c 0a7c 2020 2020 202b 2031 3379 2 |.| + 13y │ │ │ │ +0003c120: 2020 2d20 3433 7820 7a20 2d20 3233 7820 - 43x z - 23x │ │ │ │ +0003c130: 792a 7a20 2d20 3339 7820 7920 7a20 2d20 y*z - 39x y z - │ │ │ │ +0003c140: 3339 7820 7920 7a20 2d20 3233 782a 7920 39x y z - 23x*y │ │ │ │ +0003c150: 7a20 2d20 3433 7920 7a20 2b20 3435 7820 z - 43y z + 45x │ │ │ │ +0003c160: 7a20 207c 0a7c 2020 2020 202d 2d2d 2d2d z |.| ----- │ │ │ │ +0003c170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c1b0: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020 ---|.| │ │ │ │ +0003c1c0: 3320 2020 3220 2020 2020 2032 2032 2032 3 2 2 2 2 │ │ │ │ +0003c1d0: 2020 2020 2020 2020 3320 3220 2020 2020 3 2 │ │ │ │ +0003c1e0: 2034 2032 2020 2020 2020 3220 2020 3320 4 2 2 3 │ │ │ │ +0003c1f0: 2020 2020 2020 2032 2033 2020 2020 2020 2 3 │ │ │ │ +0003c200: 3220 347c 0a7c 2020 2020 202d 2033 3978 2 4|.| - 39x │ │ │ │ +0003c210: 2079 2a7a 2020 2b20 3239 7820 7920 7a20 y*z + 29x y z │ │ │ │ +0003c220: 202d 2033 3978 2a79 207a 2020 2b20 3435 - 39x*y z + 45 │ │ │ │ +0003c230: 7920 7a20 202d 2033 3978 2079 2a7a 2020 y z - 39x y*z │ │ │ │ +0003c240: 2d20 3339 782a 7920 7a20 202b 2034 3578 - 39x*y z + 45x │ │ │ │ +0003c250: 207a 207c 0a7c 2020 2020 202d 2d2d 2d2d z |.| ----- │ │ │ │ +0003c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003c2a0: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020 ---|.| │ │ │ │ +0003c2b0: 2020 2020 3420 2020 2020 2032 2034 2020 4 2 4 │ │ │ │ +0003c2c0: 2020 2020 2020 3520 2020 2020 2020 2035 5 5 │ │ │ │ +0003c2d0: 2020 2020 2020 3620 2020 2020 2020 2020 6 │ │ │ │ +0003c2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003c2f0: 2020 207c 0a7c 2020 2020 202d 2032 3378 |.| - 23x │ │ │ │ +0003c300: 2a79 2a7a 2020 2b20 3435 7920 7a20 202d *y*z + 45y z - │ │ │ │ +0003c310: 2034 3378 2a7a 2020 2d20 3433 792a 7a20 43x*z - 43y*z │ │ │ │ +0003c320: 202b 2031 337a 207d 2020 2020 2020 2020 + 13z } │ │ │ │ +0003c330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003c340: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003c390: 2020 207c 0a7c 6f37 203a 204c 6973 7420 |.|o7 : List │ │ │ │ +0003c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003c3b0: 2020 2020 2020 2020 2020 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2920 666f 7220 7468 6520 696e 7661 7269 ) for the invari │ │ │ │ -0003d010: 616e 7420 7269 6e67 206f 6620 7468 6520 ant ring of the │ │ │ │ -0003d020: 6772 6f75 7020 6163 7469 6f6e 0a0a 4465 group action..De │ │ │ │ -0003d030: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -0003d040: 3d3d 3d3d 3d0a 0a2a 6e6f 7465 2044 6164 =====..*note Dad │ │ │ │ -0003d050: 653a 2070 7269 6d61 7279 496e 7661 7269 e: primaryInvari │ │ │ │ -0003d060: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ -0003d070: 5f63 6d44 6164 653d 3e5f 7064 5f70 645f _cmDade=>_pd_pd_ │ │ │ │ -0003d080: 7064 5f72 702c 2074 616b 6573 202a 6e6f pd_rp, takes *no │ │ │ │ -0003d090: 7465 0a42 6f6f 6c65 616e 3a20 284d 6163 te.Boolean: (Mac │ │ │ │ -0003d0a0: 6175 6c61 7932 446f 6329 426f 6f6c 6561 aulay2Doc)Boolea │ │ │ │ -0003d0b0: 6e2c 2076 616c 7565 7320 616e 6420 6973 n, values and is │ │ │ │ -0003d0c0: 2073 6574 2074 6f20 2a6e 6f74 6520 6661 set to *note fa │ │ │ │ -0003d0d0: 6c73 653a 0a28 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If *note D │ │ │ │ -0003d100: 6164 653a 0a70 7269 6d61 7279 496e 7661 ade:.primaryInva │ │ │ │ -0003d110: 7269 616e 7473 5f6c 705f 7064 5f70 645f riants_lp_pd_pd_ │ │ │ │ -0003d120: 7064 5f63 6d44 6164 653d 3e5f 7064 5f70 pd_cmDade=>_pd_p │ │ │ │ -0003d130: 645f 7064 5f72 702c 2069 7320 7365 7420 d_pd_rp, is set │ │ │ │ -0003d140: 746f 202a 6e6f 7465 2074 7275 653a 0a28 to *note true:.( │ │ │ │ -0003d150: 4d61 6361 756c 6179 3244 6f63 2974 7275 Macaulay2Doc)tru │ │ │ │ -0003d160: 652c 2c20 7468 656e 202a 6e6f 7465 2070 e,, then *note p │ │ │ │ -0003d170: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ -0003d180: 3a20 7072 696d 6172 7949 6e76 6172 6961 : primaryInvaria │ │ │ │ -0003d190: 6e74 732c 2077 696c 6c20 7573 650a 7468 nts, will use.th │ │ │ │ -0003d1a0: 6520 4461 6465 2061 6c67 6f72 6974 686d e Dade algorithm │ │ │ │ -0003d1b0: 2074 6f20 6361 6c63 756c 6174 6520 6120 to calculate a │ │ │ │ -0003d1c0: 686f 6d6f 6765 6e65 6f75 7320 7379 7374 homogeneous syst │ │ │ │ -0003d1d0: 656d 206f 6620 7061 7261 6d65 7465 7273 em of parameters │ │ │ │ -0003d1e0: 2028 6873 6f70 2920 666f 720a 7468 6520 (hsop) for.the │ │ │ │ -0003d1f0: 696e 7661 7269 616e 7420 7269 6e67 206f invariant ring o │ │ │ │ -0003d200: 6620 6120 6669 6e69 7465 2067 726f 7570 f a finite group │ │ │ │ -0003d210: 2e0a 0a54 6865 2065 7861 6d70 6c65 2062 ...The example b │ │ │ │ -0003d220: 656c 6f77 2063 6f6d 7075 7465 7320 7468 elow computes th │ │ │ │ -0003d230: 6520 696e 7661 7269 616e 7420 7269 6e67 e invariant ring │ │ │ │ -0003d240: 206f 6620 5333 2061 6374 696e 6720 6f6e of S3 acting on │ │ │ │ -0003d250: 2051 515b 782c 792c 7a5d 2062 790a 7065 QQ[x,y,z] by.pe │ │ │ │ -0003d260: 726d 7574 6174 696f 6e73 206f 6e20 7468 rmutations on th │ │ │ │ -0003d270: 6520 7661 7269 6162 6c65 732e 202a 6e6f e variables. *no │ │ │ │ -0003d280: 7465 2044 6164 653a 0a70 7269 6d61 7279 te Dade:.primary │ │ │ │ -0003d290: 496e 7661 7269 616e 7473 5f6c 705f 7064 Invariants_lp_pd │ │ │ │ -0003d2a0: 5f70 645f 7064 5f63 6d44 6164 653d 3e5f _pd_pd_cmDade=>_ │ │ │ │ -0003d2b0: 7064 5f70 645f 7064 5f72 702c 2069 7320 pd_pd_pd_rp, is │ │ │ │ -0003d2c0: 7365 7420 746f 202a 6e6f 7465 2074 7275 set to *note tru │ │ │ │ -0003d2d0: 653a 0a28 4d61 6361 756c 6179 3244 6f63 e:.(Macaulay2Doc │ │ │ │ -0003d2e0: 2974 7275 652c 2e0a 0a2b 2d2d 2d2d 2d2d )true,...+------ │ │ │ │ -0003d2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d330: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2041 -------+.|i1 : A │ │ │ │ -0003d340: 3d6d 6174 7269 787b 7b30 2c31 2c30 7d2c =matrix{{0,1,0}, │ │ │ │ -0003d350: 7b30 2c30 2c31 7d2c 7b31 2c30 2c30 7d7d {0,0,1},{1,0,0}} │ │ │ │ -0003d360: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +0003c780: 2020 6e2d 310a 6361 7264 696e 616c 6974 n-1.cardinalit │ │ │ │ +0003c790: 7920 6772 6561 7465 7220 7468 616e 207c y greater than | │ │ │ │ +0003c7a0: 477c 2020 202c 2077 6865 7265 206e 2069 G| , where n i │ │ │ │ +0003c7b0: 7320 7468 6520 6e75 6d62 6572 206f 6620 s the number of │ │ │ │ +0003c7c0: 7661 7269 6162 6c65 7320 696e 2074 6865 variables in the │ │ │ │ +0003c7d0: 0a70 6f6c 796e 6f6d 6961 6c20 7269 6e67 .polynomial ring │ │ │ │ +0003c7e0: 202e 2055 7369 6e67 2061 2067 726f 756e . Using a groun │ │ │ │ +0003c7f0: 6420 6669 656c 6420 736d 616c 6c65 7220 d field smaller │ │ │ │ +0003c800: 7468 616e 2074 6869 7320 7275 6e73 2074 than this runs t │ │ │ │ +0003c810: 6865 2072 6973 6b20 6f66 2074 6865 0a61 he risk of the.a │ │ │ │ +0003c820: 6c67 6f72 6974 686d 2067 6574 7469 6e67 lgorithm getting │ │ │ │ +0003c830: 2073 7475 636b 2069 6e20 616e 2069 6e66 stuck in an inf │ │ │ │ +0003c840: 696e 6974 6520 6c6f 6f70 3b20 2a6e 6f74 inite loop; *not │ │ │ │ +0003c850: 6520 7072 696d 6172 7949 6e76 6172 6961 e primaryInvaria │ │ │ │ +0003c860: 6e74 733a 0a70 7269 6d61 7279 496e 7661 nts:.primaryInva │ │ │ │ +0003c870: 7269 616e 7473 2c20 6469 7370 6c61 7973 riants, displays │ │ │ │ +0003c880: 2061 2077 6172 6e69 6e67 206d 6573 7361 a warning messa │ │ │ │ +0003c890: 6765 2061 736b 696e 6720 7468 6520 7573 ge asking the us │ │ │ │ +0003c8a0: 6572 2077 6865 7468 6572 2074 6865 7920 er whether they │ │ │ │ +0003c8b0: 7769 7368 0a74 6f20 636f 6e74 696e 7565 wish.to continue │ │ │ │ +0003c8c0: 2077 6974 6820 7468 6520 636f 6d70 7574 with the comput │ │ │ │ +0003c8d0: 6174 696f 6e20 696e 2074 6869 7320 6361 ation in this ca │ │ │ │ +0003c8e0: 7365 2e20 5365 6520 2a6e 6f74 6520 6873 se. See *note hs │ │ │ │ +0003c8f0: 6f70 2061 6c67 6f72 6974 686d 733a 2068 op algorithms: h │ │ │ │ +0003c900: 736f 700a 616c 676f 7269 7468 6d73 2c20 sop.algorithms, │ │ │ │ +0003c910: 666f 7220 6120 6469 7363 7573 7369 6f6e for a discussion │ │ │ │ +0003c920: 206f 6e20 7468 6520 4461 6465 2061 6c67 on the Dade alg │ │ │ │ +0003c930: 6f72 6974 686d 2e0a 0a57 6179 7320 746f orithm...Ways to │ │ │ │ +0003c940: 2075 7365 2070 7269 6d61 7279 496e 7661 use primaryInva │ │ │ │ +0003c950: 7269 616e 7473 3a0a 3d3d 3d3d 3d3d 3d3d riants:.======== │ │ │ │ +0003c960: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0003c970: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2270 7269 ======.. * "pri │ │ │ │ +0003c980: 6d61 7279 496e 7661 7269 616e 7473 2846 maryInvariants(F │ │ │ │ +0003c990: 696e 6974 6547 726f 7570 4163 7469 6f6e initeGroupAction │ │ │ │ +0003c9a0: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +0003c9b0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +0003c9c0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +0003c9d0: 626a 6563 7420 2a6e 6f74 6520 7072 696d bject *note prim │ │ │ │ +0003c9e0: 6172 7949 6e76 6172 6961 6e74 733a 2070 aryInvariants: p │ │ │ │ +0003c9f0: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ +0003ca00: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +0003ca10: 686f 640a 6675 6e63 7469 6f6e 2077 6974 hod.function wit │ │ │ │ +0003ca20: 6820 6f70 7469 6f6e 733a 2028 4d61 6361 h options: (Maca │ │ │ │ +0003ca30: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ +0003ca40: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ +0003ca50: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ +0003ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003caa0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0003cab0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0003cac0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0003cad0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0003cae0: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ +0003caf0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+ds/M2/Macaul │ │ │ │ +0003cb00: 6179 322f 7061 636b 6167 6573 2f0a 496e ay2/packages/.In │ │ │ │ +0003cb10: 7661 7269 616e 7452 696e 672f 4861 7765 variantRing/Hawe │ │ │ │ +0003cb20: 7344 6f63 2e6d 323a 3137 353a 302e 0a1f sDoc.m2:175:0... │ │ │ │ +0003cb30: 0a46 696c 653a 2049 6e76 6172 6961 6e74 .File: Invariant │ │ │ │ +0003cb40: 5269 6e67 2e69 6e66 6f2c 204e 6f64 653a Ring.info, Node: │ │ │ │ +0003cb50: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ +0003cb60: 7473 5f6c 705f 7064 5f70 645f 7064 5f63 ts_lp_pd_pd_pd_c │ │ │ │ +0003cb70: 6d44 6164 653d 3e5f 7064 5f70 645f 7064 mDade=>_pd_pd_pd │ │ │ │ +0003cb80: 5f72 702c 204e 6578 743a 2070 7269 6d61 _rp, Next: prima │ │ │ │ +0003cb90: 7279 496e 7661 7269 616e 7473 5f6c 705f ryInvariants_lp_ │ │ │ │ +0003cba0: 7064 5f70 645f 7064 5f63 6d44 6567 7265 pd_pd_pd_cmDegre │ │ │ │ +0003cbb0: 6556 6563 746f 723d 3e5f 7064 5f70 645f eVector=>_pd_pd_ │ │ │ │ +0003cbc0: 7064 5f72 702c 2050 7265 763a 2070 7269 pd_rp, Prev: pri │ │ │ │ +0003cbd0: 6d61 7279 496e 7661 7269 616e 7473 2c20 maryInvariants, │ │ │ │ +0003cbe0: 5570 3a20 546f 700a 0a70 7269 6d61 7279 Up: Top..primary │ │ │ │ +0003cbf0: 496e 7661 7269 616e 7473 282e 2e2e 2c44 Invariants(...,D │ │ │ │ +0003cc00: 6164 653d 3e2e 2e2e 2920 2d2d 2061 6e20 ade=>...) -- an │ │ │ │ +0003cc10: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ +0003cc20: 7420 666f 7220 7072 696d 6172 7949 6e76 t for primaryInv │ │ │ │ +0003cc30: 6172 6961 6e74 7320 6465 7465 726d 696e ariants determin │ │ │ │ +0003cc40: 696e 6720 7768 6574 6865 7220 746f 2075 ing whether to u │ │ │ │ +0003cc50: 7365 2074 6865 2044 6164 6520 616c 676f se the Dade algo │ │ │ │ +0003cc60: 7269 7468 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a rithm.********** │ │ │ │ +0003cc70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003cc80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003cc90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003cca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003ccb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003ccc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003ccd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0003cce0: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ +0003ccf0: 2020 2020 2020 2020 7072 696d 6172 7949 primaryI │ │ │ │ +0003cd00: 6e76 6172 6961 6e74 7320 470a 2020 2a20 nvariants G. * │ │ │ │ +0003cd10: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ +0003cd20: 472c 2061 6e20 696e 7374 616e 6365 206f G, an instance o │ │ │ │ +0003cd30: 6620 7468 6520 7479 7065 202a 6e6f 7465 f the type *note │ │ │ │ +0003cd40: 2046 696e 6974 6547 726f 7570 4163 7469 FiniteGroupActi │ │ │ │ +0003cd50: 6f6e 3a20 4669 6e69 7465 4772 6f75 7041 on: FiniteGroupA │ │ │ │ +0003cd60: 6374 696f 6e2c 0a20 202a 204f 7574 7075 ction,. * Outpu │ │ │ │ +0003cd70: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e ts:. * a *n │ │ │ │ +0003cd80: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ +0003cd90: 6c61 7932 446f 6329 4c69 7374 2c2c 2020 lay2Doc)List,, │ │ │ │ +0003cda0: 636f 6e73 6973 7469 6e67 206f 6620 6120 consisting of a │ │ │ │ +0003cdb0: 686f 6d6f 6765 6e65 6f75 7320 7379 7374 homogeneous syst │ │ │ │ +0003cdc0: 656d 0a20 2020 2020 2020 206f 6620 7061 em. of pa │ │ │ │ +0003cdd0: 7261 6d65 7465 7273 2028 6873 6f70 2920 rameters (hsop) │ │ │ │ +0003cde0: 666f 7220 7468 6520 696e 7661 7269 616e for the invarian │ │ │ │ +0003cdf0: 7420 7269 6e67 206f 6620 7468 6520 6772 t ring of the gr │ │ │ │ +0003ce00: 6f75 7020 6163 7469 6f6e 0a0a 4465 7363 oup action..Desc │ │ │ │ +0003ce10: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +0003ce20: 3d3d 3d0a 0a2a 6e6f 7465 2044 6164 653a ===..*note Dade: │ │ │ │ +0003ce30: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ +0003ce40: 7473 5f6c 705f 7064 5f70 645f 7064 5f63 ts_lp_pd_pd_pd_c │ │ │ │ +0003ce50: 6d44 6164 653d 3e5f 7064 5f70 645f 7064 mDade=>_pd_pd_pd │ │ │ │ +0003ce60: 5f72 702c 2074 616b 6573 202a 6e6f 7465 _rp, takes *note │ │ │ │ +0003ce70: 0a42 6f6f 6c65 616e 3a20 284d 6163 6175 .Boolean: (Macau │ │ │ │ +0003ce80: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c lay2Doc)Boolean, │ │ │ │ +0003ce90: 2076 616c 7565 7320 616e 6420 6973 2073 values and is s │ │ │ │ +0003cea0: 6574 2074 6f20 2a6e 6f74 6520 6661 6c73 et to *note fals │ │ │ │ +0003ceb0: 653a 0a28 4d61 6361 756c 6179 3244 6f63 e:.(Macaulay2Doc │ │ │ │ +0003cec0: 2966 616c 7365 2c20 6279 2064 6566 6175 )false, by defau │ │ │ │ +0003ced0: 6c74 2e20 4966 202a 6e6f 7465 2044 6164 lt. If *note Dad │ │ │ │ +0003cee0: 653a 0a70 7269 6d61 7279 496e 7661 7269 e:.primaryInvari │ │ │ │ +0003cef0: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ +0003cf00: 5f63 6d44 6164 653d 3e5f 7064 5f70 645f _cmDade=>_pd_pd_ │ │ │ │ +0003cf10: 7064 5f72 702c 2069 7320 7365 7420 746f pd_rp, is set to │ │ │ │ +0003cf20: 202a 6e6f 7465 2074 7275 653a 0a28 4d61 *note true:.(Ma │ │ │ │ +0003cf30: 6361 756c 6179 3244 6f63 2974 7275 652c caulay2Doc)true, │ │ │ │ +0003cf40: 2c20 7468 656e 202a 6e6f 7465 2070 7269 , then *note pri │ │ │ │ +0003cf50: 6d61 7279 496e 7661 7269 616e 7473 3a20 maryInvariants: │ │ │ │ +0003cf60: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ +0003cf70: 732c 2077 696c 6c20 7573 650a 7468 6520 s, will use.the │ │ │ │ +0003cf80: 4461 6465 2061 6c67 6f72 6974 686d 2074 Dade algorithm t │ │ │ │ +0003cf90: 6f20 6361 6c63 756c 6174 6520 6120 686f o calculate a ho │ │ │ │ +0003cfa0: 6d6f 6765 6e65 6f75 7320 7379 7374 656d mogeneous system │ │ │ │ +0003cfb0: 206f 6620 7061 7261 6d65 7465 7273 2028 of parameters ( │ │ │ │ +0003cfc0: 6873 6f70 2920 666f 720a 7468 6520 696e hsop) for.the in │ │ │ │ +0003cfd0: 7661 7269 616e 7420 7269 6e67 206f 6620 variant ring of │ │ │ │ +0003cfe0: 6120 6669 6e69 7465 2067 726f 7570 2e0a a finite group.. │ │ │ │ +0003cff0: 0a54 6865 2065 7861 6d70 6c65 2062 656c .The example bel │ │ │ │ +0003d000: 6f77 2063 6f6d 7075 7465 7320 7468 6520 ow computes the │ │ │ │ +0003d010: 696e 7661 7269 616e 7420 7269 6e67 206f invariant ring o │ │ │ │ +0003d020: 6620 5333 2061 6374 696e 6720 6f6e 2051 f S3 acting on Q │ │ │ │ +0003d030: 515b 782c 792c 7a5d 2062 790a 7065 726d Q[x,y,z] by.perm │ │ │ │ +0003d040: 7574 6174 696f 6e73 206f 6e20 7468 6520 utations on the │ │ │ │ +0003d050: 7661 7269 6162 6c65 732e 202a 6e6f 7465 variables. *note │ │ │ │ +0003d060: 2044 6164 653a 0a70 7269 6d61 7279 496e Dade:.primaryIn │ │ │ │ +0003d070: 7661 7269 616e 7473 5f6c 705f 7064 5f70 variants_lp_pd_p │ │ │ │ +0003d080: 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7a20 202b 2036 611320x*y z + 6 │ │ │ │ +0003e6c0: 3932 3334 3330 3079 207a 2020 2b20 3132 9234300y z + 12 │ │ │ │ +0003e6d0: 3637 3239 3030 3078 207a 2020 2b20 3539 6729000x z + 59 │ │ │ │ +0003e6e0: 3636 3131 3332 3078 2079 2a7a 2020 2b20 6611320x y*z + │ │ │ │ +0003e6f0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d |.| --- │ │ │ │ +0003e700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e740: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +0003e750: 2020 2020 2020 2020 2032 2033 2020 2020 2 3 │ │ │ │ +0003e760: 2020 2020 2020 2020 2033 2033 2020 2020 3 3 │ │ │ │ +0003e770: 2020 2020 2020 2020 3220 3420 2020 2020 2 4 │ │ │ │ +0003e780: 2020 2020 2020 2020 2020 2020 3420 2020 4 │ │ │ │ +0003e790: 2020 2020 207c 0a7c 2020 2020 2035 3936 |.| 596 │ │ │ │ +0003e7a0: 3631 3133 3230 782a 7920 7a20 202b 2031 611320x*y z + 1 │ │ │ │ +0003e7b0: 3236 3732 3930 3030 7920 7a20 202b 2036 26729000y z + 6 │ │ │ │ +0003e7c0: 3932 3334 3330 3078 207a 2020 2b20 3138 9234300x z + 18 │ │ │ │ +0003e7d0: 3034 3137 3630 3078 2a79 2a7a 2020 2b20 0417600x*y*z + │ │ │ │ +0003e7e0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d |.| --- │ │ │ │ +0003e7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e830: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +0003e840: 2020 2020 2020 3220 3420 2020 2020 2020 2 4 │ │ │ │ +0003e850: 2020 2020 2020 2035 2020 2020 2020 2020 5 │ │ │ │ +0003e860: 2020 2020 2020 3520 2020 2020 2020 2020 5 │ │ │ │ +0003e870: 2036 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +0003e880: 2020 2020 207c 0a7c 2020 2020 2036 3932 |.| 692 │ │ │ │ +0003e890: 3334 3330 3079 207a 2020 2b20 3133 3832 34300y z + 1382 │ │ │ │ +0003e8a0: 3430 3030 782a 7a20 202b 2031 3338 3234 4000x*z + 13824 │ │ │ │ +0003e8b0: 3030 3079 2a7a 2020 2b20 3831 3030 3030 000y*z + 810000 │ │ │ │ +0003e8c0: 7a20 7d20 2020 2020 2020 2020 2020 2020 z } │ │ │ │ +0003e8d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003e8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e920: 2020 2020 207c 0a7c 6f34 203a 204c 6973 |.|o4 : Lis │ │ │ │ +0003e930: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +0003e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e970: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0003e980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e9c0: 2d2d 2d2d 2d2b 0a0a 436f 6d70 6172 6520 -----+..Compare │ │ │ │ +0003e9d0: 7468 6973 2072 6573 756c 7420 746f 2074 this result to t │ │ │ │ +0003e9e0: 6865 2068 736f 7020 6f75 7470 7574 2077 he hsop output w │ │ │ │ +0003e9f0: 6865 6e20 4461 6465 2069 7320 6c65 6674 hen Dade is left │ │ │ │ +0003ea00: 2074 6f20 6974 7320 6465 6661 756c 7420 to its default │ │ │ │ +0003ea10: 7661 6c75 650a 2a6e 6f74 6520 6661 6c73 value.*note fals │ │ │ │ +0003ea20: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ +0003ea30: 2966 616c 7365 2c2e 0a0a 2b2d 2d2d 2d2d )false,...+----- │ │ │ │ 0003ea40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ea50: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 -------|.| │ │ │ │ -0003ea60: 2020 2020 2020 2020 3220 3420 2020 2020 2 4 │ │ │ │ -0003ea70: 2020 2020 2020 2020 2035 2020 2020 2020 5 │ │ │ │ -0003ea80: 2020 2020 2020 2020 3520 2020 2020 2020 5 │ │ │ │ -0003ea90: 2020 2036 2020 2020 2020 2020 2020 2020 6 │ │ │ │ -0003eaa0: 2020 2020 2020 207c 0a7c 2020 2020 2036 |.| 6 │ │ │ │ -0003eab0: 3932 3334 3330 3079 207a 2020 2b20 3133 9234300y z + 13 │ │ │ │ -0003eac0: 3832 3430 3030 782a 7a20 202b 2031 3338 824000x*z + 138 │ │ │ │ -0003ead0: 3234 3030 3079 2a7a 2020 2b20 3831 3030 24000y*z + 8100 │ │ │ │ -0003eae0: 3030 7a20 7d20 2020 2020 2020 2020 2020 00z } │ │ │ │ -0003eaf0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0003eb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ea60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ea70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003ea80: 0a7c 6935 203a 2070 7269 6d61 7279 496e .|i5 : primaryIn │ │ │ │ +0003ea90: 7661 7269 616e 7473 2853 3329 2020 2020 variants(S3) │ │ │ │ +0003eaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eac0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0003ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eb00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0003eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eb40: 2020 2020 2020 207c 0a7c 6f34 203a 204c |.|o4 : L │ │ │ │ -0003eb50: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ -0003eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eb90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0003eba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ebb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ebc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ebd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ebe0: 2d2d 2d2d 2d2d 2d2b 0a0a 436f 6d70 6172 -------+..Compar │ │ │ │ -0003ebf0: 6520 7468 6973 2072 6573 756c 7420 746f e this result to │ │ │ │ -0003ec00: 2074 6865 2068 736f 7020 6f75 7470 7574 the hsop output │ │ │ │ -0003ec10: 2077 6865 6e20 4461 6465 2069 7320 6c65 when Dade is le │ │ │ │ -0003ec20: 6674 2074 6f20 6974 7320 6465 6661 756c ft to its defaul │ │ │ │ -0003ec30: 7420 7661 6c75 650a 2a6e 6f74 6520 6661 t value.*note fa │ │ │ │ -0003ec40: 6c73 653a 2028 4d61 6361 756c 6179 3244 lse: (Macaulay2D │ │ │ │ -0003ec50: 6f63 2966 616c 7365 2c2e 0a0a 2b2d 2d2d oc)false,...+--- │ │ │ │ +0003eb20: 2020 3220 2020 2032 2020 2020 3220 2020 2 2 2 │ │ │ │ +0003eb30: 3220 2020 2020 2020 3220 2020 2032 2020 2 2 2 │ │ │ │ +0003eb40: 2020 2032 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ +0003eb50: 2020 3220 7c0a 7c6f 3520 3d20 7b78 202b 2 |.|o5 = {x + │ │ │ │ +0003eb60: 2079 202b 207a 2c20 7820 202b 2079 2020 y + z, x + y │ │ │ │ +0003eb70: 2b20 7a20 2c20 7820 7920 2b20 782a 7920 + z , x y + x*y │ │ │ │ +0003eb80: 202b 2078 207a 202b 2079 207a 202b 2078 + x z + y z + x │ │ │ │ +0003eb90: 2a7a 2020 2b20 792a 7a20 7d7c 0a7c 2020 *z + y*z }|.| │ │ │ │ +0003eba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ebd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ebe0: 2020 7c0a 7c6f 3520 3a20 4c69 7374 2020 |.|o5 : List │ │ │ │ +0003ebf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ec20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003ec30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003eca0: 2d2b 0a7c 6935 203a 2070 7269 6d61 7279 -+.|i5 : primary │ │ │ │ -0003ecb0: 496e 7661 7269 616e 7473 2853 3329 2020 Invariants(S3) │ │ │ │ -0003ecc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ecd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ece0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0003ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ed00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ed10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ed20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003ed30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003ed40: 2020 2020 3220 2020 2032 2020 2020 3220 2 2 2 │ │ │ │ -0003ed50: 2020 3220 2020 2020 2020 3220 2020 2032 2 2 2 │ │ │ │ -0003ed60: 2020 2020 2032 2020 2020 2020 2032 2020 2 2 │ │ │ │ -0003ed70: 2020 2020 3220 7c0a 7c6f 3520 3d20 7b78 2 |.|o5 = {x │ │ │ │ -0003ed80: 202b 2079 202b 207a 2c20 7820 202b 2079 + y + z, x + y │ │ │ │ -0003ed90: 2020 2b20 7a20 2c20 7820 7920 2b20 782a + z , x y + x* │ │ │ │ -0003eda0: 7920 202b 2078 207a 202b 2079 207a 202b y + x z + y z + │ │ │ │ -0003edb0: 2078 2a7a 2020 2b20 792a 7a20 7d7c 0a7c x*z + y*z }|.| │ │ │ │ +0003ec70: 2b0a 0a20 2020 2020 2020 2020 2020 2020 +.. │ │ │ │ +0003ec80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ec90: 2020 2020 2020 5333 0a42 656c 6f77 2c20 S3.Below, │ │ │ │ +0003eca0: 7468 6520 696e 7661 7269 616e 7420 7269 the invariant ri │ │ │ │ +0003ecb0: 6e67 2051 515b 782c 792c 7a5d 2020 2069 ng QQ[x,y,z] i │ │ │ │ +0003ecc0: 7320 6361 6c63 756c 6174 6564 2077 6974 s calculated wit │ │ │ │ +0003ecd0: 6820 4b20 6265 696e 6720 7468 6520 6669 h K being the fi │ │ │ │ +0003ece0: 656c 6420 7769 7468 0a31 3031 2065 6c65 eld with.101 ele │ │ │ │ +0003ecf0: 6d65 6e74 732e 0a0a 2b2d 2d2d 2d2d 2d2d ments...+------- │ │ │ │ +0003ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ed10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ed20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ed30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ed40: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4b3d ------+.|i6 : K= │ │ │ │ +0003ed50: 4746 2831 3031 2920 2020 2020 2020 2020 GF(101) │ │ │ │ +0003ed60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ed70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ed80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ed90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0003eda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003edb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003edc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003edd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ede0: 2020 2020 2020 7c0a 7c6f 3620 3d20 4b20 |.|o6 = K │ │ │ │ 0003edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ee00: 2020 2020 7c0a 7c6f 3520 3a20 4c69 7374 |.|o5 : List │ │ │ │ +0003ee00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ee10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ee40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0003ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ee60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ee90: 2d2d 2b0a 0a20 2020 2020 2020 2020 2020 --+.. │ │ │ │ +0003ee30: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0003ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ee60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ee80: 2020 2020 2020 7c0a 7c6f 3620 3a20 4761 |.|o6 : Ga │ │ │ │ +0003ee90: 6c6f 6973 4669 656c 6420 2020 2020 2020 loisField │ │ │ │ 0003eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eeb0: 2020 2020 2020 2020 5333 0a42 656c 6f77 S3.Below │ │ │ │ -0003eec0: 2c20 7468 6520 696e 7661 7269 616e 7420 , the invariant │ │ │ │ -0003eed0: 7269 6e67 2051 515b 782c 792c 7a5d 2020 ring QQ[x,y,z] │ │ │ │ -0003eee0: 2069 7320 6361 6c63 756c 6174 6564 2077 is calculated w │ │ │ │ -0003eef0: 6974 6820 4b20 6265 696e 6720 7468 6520 ith K being the │ │ │ │ -0003ef00: 6669 656c 6420 7769 7468 0a31 3031 2065 field with.101 e │ │ │ │ -0003ef10: 6c65 6d65 6e74 732e 0a0a 2b2d 2d2d 2d2d lements...+----- │ │ │ │ -0003ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ef40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ef50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ef60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ -0003ef70: 4b3d 4746 2831 3031 2920 2020 2020 2020 K=GF(101) │ │ │ │ +0003eeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eed0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0003eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ef10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ef20: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 5333 ------+.|i7 : S3 │ │ │ │ +0003ef30: 3d66 696e 6974 6541 6374 696f 6e28 7b41 =finiteAction({A │ │ │ │ +0003ef40: 2c42 7d2c 4b5b 782c 792c 7a5d 2920 2020 ,B},K[x,y,z]) │ │ │ │ +0003ef50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ef60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ef70: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0003ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003efb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0003efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003efd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003efb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003efc0: 2020 2020 2020 7c0a 7c6f 3720 3d20 4b5b |.|o7 = K[ │ │ │ │ +0003efd0: 782e 2e7a 5d20 3c2d 207b 7c20 3020 3120 x..z] <- {| 0 1 │ │ │ │ +0003efe0: 3020 7c2c 207c 2030 2031 2030 207c 7d20 0 |, | 0 1 0 |} │ │ │ │ 0003eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f000: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -0003f010: 4b20 2020 2020 2020 2020 2020 2020 2020 K │ │ │ │ -0003f020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f010: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0003f020: 2020 2020 2020 2020 2020 7c20 3020 3020 | 0 0 │ │ │ │ +0003f030: 3120 7c20 207c 2031 2030 2030 207c 2020 1 | | 1 0 0 | │ │ │ │ 0003f040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f050: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0003f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f060: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0003f070: 2020 2020 2020 2020 2020 7c20 3120 3020 | 1 0 │ │ │ │ +0003f080: 3020 7c20 207c 2030 2030 2031 207c 2020 0 | | 0 0 1 | │ │ │ │ 0003f090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f0a0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ -0003f0b0: 4761 6c6f 6973 4669 656c 6420 2020 2020 GaloisField │ │ │ │ +0003f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f0b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0003f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003f0d0: 2020 2020 2020 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+0003fc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fc90: 2020 2020 2020 7c0a 7c6f 3820 3a20 4c69 |.|o8 : Li │ │ │ │ +0003fca0: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ +0003fcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fce0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0003fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003fd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003fd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003fd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003fd30: 2d2d 2d2d 2d2d 2b0a 0a46 6f72 206d 6f72 ------+..For mor │ │ │ │ +0003fd40: 6520 696e 666f 726d 6174 696f 6e20 6162 e information ab │ │ │ │ +0003fd50: 6f75 7420 7468 6520 616c 676f 7269 7468 out the algorith │ │ │ │ +0003fd60: 6d73 2075 7365 6420 746f 2063 616c 6375 ms used to calcu │ │ │ │ +0003fd70: 6c61 7465 2061 2068 736f 7020 696e 0a70 late a hsop in.p │ │ │ │ +0003fd80: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ +0003fd90: 2c20 7365 6520 2a6e 6f74 6520 6873 6f70 , see *note hsop │ │ │ │ +0003fda0: 2061 6c67 6f72 6974 686d 733a 2068 736f algorithms: hso │ │ │ │ +0003fdb0: 7020 616c 676f 7269 7468 6d73 2c2e 0a0a p algorithms,... │ │ │ │ +0003fdc0: 4361 7665 6174 0a3d 3d3d 3d3d 3d0a 0a20 Caveat.======.. │ │ │ │ +0003fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fe00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fe10: 2020 2020 2020 2020 2020 0a43 7572 7265 .Curre │ │ │ │ +0003fe20: 6e74 6c79 2075 7365 7273 2063 616e 206f ntly users can o │ │ │ │ +0003fe30: 6e6c 7920 7573 6520 2a6e 6f74 6520 7072 nly use *note pr │ │ │ │ +0003fe40: 696d 6172 7949 6e76 6172 6961 6e74 733a imaryInvariants: │ │ │ │ +0003fe50: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ +0003fe60: 7473 2c20 746f 0a20 2020 2020 2020 2020 ts, to. │ │ │ │ 0003fe70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fe80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fe90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003fea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003feb0: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20 |.|o8 : │ │ │ │ -0003fec0: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ -0003fed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003fee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003fef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ff00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0003ff10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ff20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ff30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ff50: 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 6f72 206d --------+..For m │ │ │ │ -0003ff60: 6f72 6520 696e 666f 726d 6174 696f 6e20 ore information │ │ │ │ -0003ff70: 6162 6f75 7420 7468 6520 616c 676f 7269 about the algori │ │ │ │ -0003ff80: 7468 6d73 2075 7365 6420 746f 2063 616c thms used to cal │ │ │ │ -0003ff90: 6375 6c61 7465 2061 2068 736f 7020 696e culate a hsop in │ │ │ │ -0003ffa0: 0a70 7269 6d61 7279 496e 7661 7269 616e .primaryInvarian │ │ │ │ -0003ffb0: 7473 2c20 7365 6520 2a6e 6f74 6520 6873 ts, see *note hs │ │ │ │ -0003ffc0: 6f70 2061 6c67 6f72 6974 686d 733a 2068 op algorithms: h │ │ │ │ -0003ffd0: 736f 7020 616c 676f 7269 7468 6d73 2c2e sop algorithms,. │ │ │ │ -0003ffe0: 0a0a 4361 7665 6174 0a3d 3d3d 3d3d 3d0a ..Caveat.======. │ │ │ │ -0003fff0: 0a20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -00040000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003feb0: 2020 2020 0a63 616c 6375 6c61 7465 2061 .calculate a │ │ │ │ +0003fec0: 2068 736f 7020 666f 7220 7468 6520 696e hsop for the in │ │ │ │ +0003fed0: 7661 7269 616e 7420 7269 6e67 206f 7665 variant ring ove │ │ │ │ +0003fee0: 7220 6120 6669 6e69 7465 2066 6965 6c64 r a finite field │ │ │ │ +0003fef0: 2062 7920 7573 696e 6720 7468 6520 4461 by using the Da │ │ │ │ +0003ff00: 6465 0a20 2020 2020 2020 2020 2020 2020 de. │ │ │ │ +0003ff10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ff20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ff30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ff40: 2020 2020 2020 2020 200a 616c 676f 7269 .algori │ │ │ │ +0003ff50: 7468 6d2e 2055 7365 7273 2073 686f 756c thm. Users shoul │ │ │ │ +0003ff60: 6420 656e 7465 7220 7468 6520 6669 6e69 d enter the fini │ │ │ │ +0003ff70: 7465 2066 6965 6c64 2061 7320 6120 2a6e te field as a *n │ │ │ │ +0003ff80: 6f74 6520 4761 6c6f 6973 4669 656c 643a ote GaloisField: │ │ │ │ +0003ff90: 0a20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ +0003ffa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ffb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ffc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ffd0: 2020 2020 200a 284d 6163 6175 6c61 7932 .(Macaulay2 │ │ │ │ +0003ffe0: 446f 6329 4761 6c6f 6973 4669 656c 642c Doc)GaloisField, │ │ │ │ +0003fff0: 206f 7220 6120 7175 6f74 6965 6e74 2066 or a quotient f │ │ │ │ +00040000: 6965 6c64 206f 6620 7468 6520 666f 726d ield of the form │ │ │ │ +00040010: 202a 6e6f 7465 205a 5a3a 0a20 2020 2020 *note ZZ:. │ │ │ │ 00040020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040030: 2020 2020 2020 2020 2020 2020 0a43 7572 .Cur │ │ │ │ -00040040: 7265 6e74 6c79 2075 7365 7273 2063 616e rently users can │ │ │ │ -00040050: 206f 6e6c 7920 7573 6520 2a6e 6f74 6520 only use *note │ │ │ │ -00040060: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ -00040070: 733a 2070 7269 6d61 7279 496e 7661 7269 s: primaryInvari │ │ │ │ -00040080: 616e 7473 2c20 746f 0a20 2020 2020 2020 ants, to. │ │ │ │ -00040090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000400a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040060: 2020 0a28 4d61 6361 756c 6179 3244 6f63 .(Macaulay2Doc │ │ │ │ +00040070: 295a 5a2c 2f70 2061 6e64 2061 7265 2061 )ZZ,/p and are a │ │ │ │ +00040080: 6476 6973 6564 2074 6f20 656e 7375 7265 dvised to ensure │ │ │ │ +00040090: 2074 6861 7420 7468 6520 6772 6f75 6e64 that the ground │ │ │ │ +000400a0: 2066 6965 6c64 2068 6173 0a20 2020 2020 field has. │ │ │ │ 000400b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000400c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000400d0: 2020 2020 2020 0a63 616c 6375 6c61 7465 .calculate │ │ │ │ -000400e0: 2061 2068 736f 7020 666f 7220 7468 6520 a hsop for the │ │ │ │ -000400f0: 696e 7661 7269 616e 7420 7269 6e67 206f invariant ring o │ │ │ │ -00040100: 7665 7220 6120 6669 6e69 7465 2066 6965 ver a finite fie │ │ │ │ -00040110: 6c64 2062 7920 7573 696e 6720 7468 6520 ld by using the │ │ │ │ -00040120: 4461 6465 0a20 2020 2020 2020 2020 2020 Dade. │ │ │ │ -00040130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040160: 2020 2020 2020 2020 2020 200a 616c 676f .algo │ │ │ │ -00040170: 7269 7468 6d2e 2055 7365 7273 2073 686f rithm. Users sho │ │ │ │ -00040180: 756c 6420 656e 7465 7220 7468 6520 6669 uld enter the fi │ │ │ │ -00040190: 6e69 7465 2066 6965 6c64 2061 7320 6120 nite field as a │ │ │ │ -000401a0: 2a6e 6f74 6520 4761 6c6f 6973 4669 656c *note GaloisFiel │ │ │ │ -000401b0: 643a 0a20 2020 2020 2020 2020 2020 2020 d:. │ │ │ │ -000401c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000401d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000401e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000401f0: 2020 2020 2020 200a 284d 6163 6175 6c61 .(Macaula │ │ │ │ -00040200: 7932 446f 6329 4761 6c6f 6973 4669 656c y2Doc)GaloisFiel │ │ │ │ -00040210: 642c 206f 7220 6120 7175 6f74 6965 6e74 d, or a quotient │ │ │ │ -00040220: 2066 6965 6c64 206f 6620 7468 6520 666f field of the fo │ │ │ │ -00040230: 726d 202a 6e6f 7465 205a 5a3a 0a20 2020 rm *note ZZ:. │ │ │ │ -00040240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040280: 2020 2020 0a28 4d61 6361 756c 6179 3244 .(Macaulay2D │ │ │ │ -00040290: 6f63 295a 5a2c 2f70 2061 6e64 2061 7265 oc)ZZ,/p and are │ │ │ │ -000402a0: 2061 6476 6973 6564 2074 6f20 656e 7375 advised to ensu │ │ │ │ -000402b0: 7265 2074 6861 7420 7468 6520 6772 6f75 re that the grou │ │ │ │ -000402c0: 6e64 2066 6965 6c64 2068 6173 0a20 2020 nd field has. │ │ │ │ -000402d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000402e0: 2020 2020 2020 2020 206e 2d31 0a63 6172 n-1.car │ │ │ │ -000402f0: 6469 6e61 6c69 7479 2067 7265 6174 6572 dinality greater │ │ │ │ -00040300: 2074 6861 6e20 7c47 7c20 2020 2c20 7768 than |G| , wh │ │ │ │ -00040310: 6572 6520 6e20 6973 2074 6865 206e 756d ere n is the num │ │ │ │ -00040320: 6265 7220 6f66 2076 6172 6961 626c 6573 ber of variables │ │ │ │ -00040330: 2069 6e20 7468 650a 706f 6c79 6e6f 6d69 in the.polynomi │ │ │ │ -00040340: 616c 2072 696e 6720 522e 2055 7369 6e67 al ring R. Using │ │ │ │ -00040350: 2061 2067 726f 756e 6420 6669 656c 6420 a ground field │ │ │ │ -00040360: 736d 616c 6c65 7220 7468 616e 2074 6869 smaller than thi │ │ │ │ -00040370: 7320 7275 6e73 2074 6865 2072 6973 6b20 s runs the risk │ │ │ │ -00040380: 6f66 2074 6865 0a61 6c67 6f72 6974 686d of the.algorithm │ │ │ │ -00040390: 2067 6574 7469 6e67 2073 7475 636b 2069 getting stuck i │ │ │ │ -000403a0: 6e20 616e 2069 6e66 696e 6974 6520 6c6f n an infinite lo │ │ │ │ -000403b0: 6f70 3b20 2a6e 6f74 6520 7072 696d 6172 op; *note primar │ │ │ │ -000403c0: 7949 6e76 6172 6961 6e74 733a 0a70 7269 yInvariants:.pri │ │ │ │ -000403d0: 6d61 7279 496e 7661 7269 616e 7473 2c20 maryInvariants, │ │ │ │ -000403e0: 6469 7370 6c61 7973 2061 2077 6172 6e69 displays a warni │ │ │ │ -000403f0: 6e67 206d 6573 7361 6765 2061 736b 696e ng message askin │ │ │ │ -00040400: 6720 7468 6520 7573 6572 2077 6865 7468 g the user wheth │ │ │ │ -00040410: 6572 2074 6865 7920 7769 7368 0a74 6f20 er they wish.to │ │ │ │ -00040420: 636f 6e74 696e 7565 2077 6974 6820 7468 continue with th │ │ │ │ -00040430: 6520 636f 6d70 7574 6174 696f 6e20 696e e computation in │ │ │ │ -00040440: 2074 6869 7320 6361 7365 2e20 5365 6520 this case. See │ │ │ │ -00040450: 2a6e 6f74 6520 6873 6f70 2061 6c67 6f72 *note hsop algor │ │ │ │ -00040460: 6974 686d 733a 2068 736f 700a 616c 676f ithms: hsop.algo │ │ │ │ -00040470: 7269 7468 6d73 2c20 666f 7220 6120 6469 rithms, for a di │ │ │ │ -00040480: 7363 7573 7369 6f6e 206f 6e20 7468 6520 scussion on the │ │ │ │ -00040490: 4461 6465 2061 6c67 6f72 6974 686d 2e0a Dade algorithm.. │ │ │ │ -000404a0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -000404b0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6873 ==.. * *note hs │ │ │ │ -000404c0: 6f70 2061 6c67 6f72 6974 686d 733a 2068 op algorithms: h │ │ │ │ -000404d0: 736f 7020 616c 676f 7269 7468 6d73 2c20 sop algorithms, │ │ │ │ -000404e0: 2d2d 2061 6e20 6f76 6572 7669 6577 206f -- an overview o │ │ │ │ -000404f0: 6620 7468 6520 616c 676f 7269 7468 6d73 f the algorithms │ │ │ │ -00040500: 0a20 2020 2075 7365 6420 696e 2070 7269 . used in pri │ │ │ │ -00040510: 6d61 7279 496e 7661 7269 616e 7473 0a20 maryInvariants. │ │ │ │ -00040520: 202a 202a 6e6f 7465 2070 7269 6d61 7279 * *note primary │ │ │ │ -00040530: 496e 7661 7269 616e 7473 3a20 7072 696d Invariants: prim │ │ │ │ -00040540: 6172 7949 6e76 6172 6961 6e74 732c 202d aryInvariants, - │ │ │ │ -00040550: 2d20 636f 6d70 7574 6573 2061 206c 6973 - computes a lis │ │ │ │ -00040560: 7420 6f66 2070 7269 6d61 7279 0a20 2020 t of primary. │ │ │ │ -00040570: 2069 6e76 6172 6961 6e74 7320 666f 7220 invariants for │ │ │ │ -00040580: 7468 6520 696e 7661 7269 616e 7420 7269 the invariant ri │ │ │ │ -00040590: 6e67 206f 6620 6120 6669 6e69 7465 2067 ng of a finite g │ │ │ │ -000405a0: 726f 7570 0a0a 4675 6e63 7469 6f6e 7320 roup..Functions │ │ │ │ -000405b0: 7769 7468 206f 7074 696f 6e61 6c20 6172 with optional ar │ │ │ │ -000405c0: 6775 6d65 6e74 206e 616d 6564 2044 6164 gument named Dad │ │ │ │ -000405d0: 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d e:.============= │ │ │ │ -000405e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000405f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00040600: 0a20 202a 202a 6e6f 7465 2070 7269 6d61 . * *note prima │ │ │ │ -00040610: 7279 496e 7661 7269 616e 7473 282e 2e2e ryInvariants(... │ │ │ │ -00040620: 2c44 6164 653d 3e2e 2e2e 293a 0a20 2020 ,Dade=>...):. │ │ │ │ -00040630: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ -00040640: 7473 5f6c 705f 7064 5f70 645f 7064 5f63 ts_lp_pd_pd_pd_c │ │ │ │ -00040650: 6d44 6164 653d 3e5f 7064 5f70 645f 7064 mDade=>_pd_pd_pd │ │ │ │ -00040660: 5f72 702c 202d 2d20 616e 206f 7074 696f _rp, -- an optio │ │ │ │ -00040670: 6e61 6c20 6172 6775 6d65 6e74 0a20 2020 nal argument. │ │ │ │ -00040680: 2066 6f72 2070 7269 6d61 7279 496e 7661 for primaryInva │ │ │ │ -00040690: 7269 616e 7473 2064 6574 6572 6d69 6e69 riants determini │ │ │ │ -000406a0: 6e67 2077 6865 7468 6572 2074 6f20 7573 ng whether to us │ │ │ │ -000406b0: 6520 7468 6520 4461 6465 2061 6c67 6f72 e the Dade algor │ │ │ │ -000406c0: 6974 686d 0a0a 4675 7274 6865 7220 696e ithm..Further in │ │ │ │ -000406d0: 666f 726d 6174 696f 6e0a 3d3d 3d3d 3d3d formation.====== │ │ │ │ -000406e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -000406f0: 202a 2044 6566 6175 6c74 2076 616c 7565 * Default value │ │ │ │ -00040700: 3a20 2a6e 6f74 6520 6661 6c73 653a 2028 : *note false: ( │ │ │ │ -00040710: 4d61 6361 756c 6179 3244 6f63 2966 616c Macaulay2Doc)fal │ │ │ │ -00040720: 7365 2c0a 2020 2a20 4675 6e63 7469 6f6e se,. * Function │ │ │ │ -00040730: 3a20 2a6e 6f74 6520 7072 696d 6172 7949 : *note primaryI │ │ │ │ -00040740: 6e76 6172 6961 6e74 733a 2070 7269 6d61 nvariants: prima │ │ │ │ -00040750: 7279 496e 7661 7269 616e 7473 2c20 2d2d ryInvariants, -- │ │ │ │ -00040760: 2063 6f6d 7075 7465 7320 6120 6c69 7374 computes a list │ │ │ │ -00040770: 206f 660a 2020 2020 7072 696d 6172 7920 of. primary │ │ │ │ -00040780: 696e 7661 7269 616e 7473 2066 6f72 2074 invariants for t │ │ │ │ -00040790: 6865 2069 6e76 6172 6961 6e74 2072 696e he invariant rin │ │ │ │ -000407a0: 6720 6f66 2061 2066 696e 6974 6520 6772 g of a finite gr │ │ │ │ -000407b0: 6f75 700a 2020 2a20 4f70 7469 6f6e 206b oup. * Option k │ │ │ │ -000407c0: 6579 3a20 2a6e 6f74 6520 4461 6465 3a20 ey: *note Dade: │ │ │ │ -000407d0: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ -000407e0: 735f 6c70 5f70 645f 7064 5f70 645f 636d s_lp_pd_pd_pd_cm │ │ │ │ -000407f0: 4461 6465 3d3e 5f70 645f 7064 5f70 645f Dade=>_pd_pd_pd_ │ │ │ │ -00040800: 7270 2c0a 2020 2020 2d2d 2061 6e20 6f70 rp,. -- an op │ │ │ │ -00040810: 7469 6f6e 616c 2061 7267 756d 656e 7420 tional argument │ │ │ │ -00040820: 666f 7220 7072 696d 6172 7949 6e76 6172 for primaryInvar │ │ │ │ -00040830: 6961 6e74 7320 6465 7465 726d 696e 696e iants determinin │ │ │ │ -00040840: 6720 7768 6574 6865 7220 746f 2075 7365 g whether to use │ │ │ │ -00040850: 0a20 2020 2074 6865 2044 6164 6520 616c . the Dade al │ │ │ │ -00040860: 676f 7269 7468 6d0a 2d2d 2d2d 2d2d 2d2d gorithm.-------- │ │ │ │ -00040870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000408a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000408b0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -000408c0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -000408d0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -000408e0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -000408f0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00040900: 3235 2e31 312b 6473 2f4d 322f 4d61 6361 25.11+ds/M2/Maca │ │ │ │ -00040910: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -00040920: 496e 7661 7269 616e 7452 696e 672f 4861 InvariantRing/Ha │ │ │ │ -00040930: 7765 7344 6f63 2e6d 323a 3234 343a 302e wesDoc.m2:244:0. │ │ │ │ -00040940: 0a1f 0a46 696c 653a 2049 6e76 6172 6961 ...File: Invaria │ │ │ │ -00040950: 6e74 5269 6e67 2e69 6e66 6f2c 204e 6f64 ntRing.info, Nod │ │ │ │ -00040960: 653a 2070 7269 6d61 7279 496e 7661 7269 e: primaryInvari │ │ │ │ -00040970: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ -00040980: 5f63 6d44 6567 7265 6556 6563 746f 723d _cmDegreeVector= │ │ │ │ -00040990: 3e5f 7064 5f70 645f 7064 5f72 702c 204e >_pd_pd_pd_rp, N │ │ │ │ -000409a0: 6578 743a 2072 616e 6b5f 6c70 4469 6167 ext: rank_lpDiag │ │ │ │ -000409b0: 6f6e 616c 4163 7469 6f6e 5f72 702c 2050 onalAction_rp, P │ │ │ │ -000409c0: 7265 763a 2070 7269 6d61 7279 496e 7661 rev: primaryInva │ │ │ │ -000409d0: 7269 616e 7473 5f6c 705f 7064 5f70 645f riants_lp_pd_pd_ │ │ │ │ -000409e0: 7064 5f63 6d44 6164 653d 3e5f 7064 5f70 pd_cmDade=>_pd_p │ │ │ │ -000409f0: 645f 7064 5f72 702c 2055 703a 2054 6f70 d_pd_rp, Up: Top │ │ │ │ -00040a00: 0a0a 7072 696d 6172 7949 6e76 6172 6961 ..primaryInvaria │ │ │ │ -00040a10: 6e74 7328 2e2e 2e2c 4465 6772 6565 5665 nts(...,DegreeVe │ │ │ │ -00040a20: 6374 6f72 3d3e 2e2e 2e29 202d 2d20 616e ctor=>...) -- an │ │ │ │ -00040a30: 206f 7074 696f 6e61 6c20 6172 6775 6d65 optional argume │ │ │ │ -00040a40: 6e74 2066 6f72 2070 7269 6d61 7279 496e nt for primaryIn │ │ │ │ -00040a50: 7661 7269 616e 7473 2074 6861 7420 6669 variants that fi │ │ │ │ -00040a60: 6e64 7320 696e 7661 7269 616e 7473 206f nds invariants o │ │ │ │ -00040a70: 6620 6365 7274 6169 6e20 6465 6772 6565 f certain degree │ │ │ │ -00040a80: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a s.************** │ │ │ │ -00040a90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00040aa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00040ab0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00040ac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00040ad0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00040ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00040af0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00040b00: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ -00040b10: 2020 2020 2020 2070 7269 6d61 7279 496e primaryIn │ │ │ │ -00040b20: 7661 7269 616e 7473 2047 0a20 202a 2049 variants G. * I │ │ │ │ -00040b30: 6e70 7574 733a 0a20 2020 2020 202a 2047 nputs:. * G │ │ │ │ -00040b40: 2c20 616e 2069 6e73 7461 6e63 6520 6f66 , an instance of │ │ │ │ -00040b50: 2074 6865 2074 7970 6520 2a6e 6f74 6520 the type *note │ │ │ │ -00040b60: 4669 6e69 7465 4772 6f75 7041 6374 696f FiniteGroupActio │ │ │ │ -00040b70: 6e3a 2046 696e 6974 6547 726f 7570 4163 n: FiniteGroupAc │ │ │ │ -00040b80: 7469 6f6e 2c0a 2020 2a20 4f75 7470 7574 tion,. * Output │ │ │ │ -00040b90: 733a 0a20 2020 2020 202a 2061 202a 6e6f s:. * a *no │ │ │ │ -00040ba0: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ -00040bb0: 6179 3244 6f63 294c 6973 742c 2c20 2063 ay2Doc)List,, c │ │ │ │ -00040bc0: 6f6e 7369 7374 696e 6720 6f66 2061 2068 onsisting of a h │ │ │ │ -00040bd0: 6f6d 6f67 656e 656f 7573 2073 7973 7465 omogeneous syste │ │ │ │ -00040be0: 6d0a 2020 2020 2020 2020 6f66 2070 6172 m. of par │ │ │ │ -00040bf0: 616d 6574 6572 7320 2868 736f 7029 2066 ameters (hsop) f │ │ │ │ -00040c00: 6f72 2074 6865 2069 6e76 6172 6961 6e74 or the invariant │ │ │ │ -00040c10: 2072 696e 6720 6f66 2074 6865 2067 726f ring of the gro │ │ │ │ -00040c20: 7570 2061 6374 696f 6e0a 0a44 6573 6372 up action..Descr │ │ │ │ -00040c30: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00040c40: 3d3d 0a0a 2020 2020 2020 2020 2020 2020 ==.. │ │ │ │ -00040c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040c80: 2020 2020 2020 2020 2020 2020 2020 0a42 .B │ │ │ │ -00040c90: 7920 6465 6661 756c 742c 202a 6e6f 7465 y default, *note │ │ │ │ -00040ca0: 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6365 206f 6620 6120 existence of a │ │ │ │ -00040da0: 686f 6d6f 6765 6e65 6f75 7320 7379 7374 homogeneous syst │ │ │ │ -00040db0: 656d 206f 6620 7061 7261 6d65 7465 7273 em of parameters │ │ │ │ -00040dc0: 0a20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -00040dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040e00: 2020 2020 2020 2020 2020 2020 2020 0a20 . │ │ │ │ -00040e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040e50: 2020 2020 2020 2020 2020 2020 200a 2868 .(h │ │ │ │ -00040e60: 736f 7029 2028 6620 2c2e 2e2e 2c66 2029 sop) (f ,...,f ) │ │ │ │ -00040e70: 2077 6974 6820 706f 7369 7469 7665 2064 with positive d │ │ │ │ -00040e80: 6567 7265 6573 2063 6f72 7265 7370 6f6e egrees correspon │ │ │ │ -00040e90: 6469 6e67 2074 6f20 2864 202c 2e2e 2e2c ding to (d ,..., │ │ │ │ -00040ea0: 6420 2920 696e 202a 6e6f 7465 0a20 2020 d ) in *note. │ │ │ │ -00040eb0: 2020 2020 2020 3120 2020 2020 206e 2020 1 n │ │ │ │ -00040ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040ee0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ -00040ef0: 6e0a 2020 2020 2020 2020 2020 2020 2020 n. │ │ │ │ -00040f00: 2020 2020 2020 206e 0a5a 5a3a 2028 4d61 n.ZZ: (Ma │ │ │ │ -00040f10: 6361 756c 6179 3244 6f63 295a 5a2c 202e caulay2Doc)ZZ, . │ │ │ │ -00040f20: 2049 6620 6974 2069 7320 6b6e 6f77 6e20 If it is known │ │ │ │ -00040f30: 7468 6174 2061 2068 736f 7020 6578 6973 that a hsop exis │ │ │ │ -00040f40: 7473 2066 6f72 2061 2063 6572 7461 696e ts for a certain │ │ │ │ -00040f50: 0a63 6f6c 6c65 6374 696f 6e20 6f66 2064 .collection of d │ │ │ │ -00040f60: 6567 7265 6573 2c20 7468 6973 2063 616e egrees, this can │ │ │ │ -00040f70: 2062 6520 6173 7369 676e 6564 2c20 6173 be assigned, as │ │ │ │ -00040f80: 2061 202a 6e6f 7465 204c 6973 743a 0a28 a *note List:.( │ │ │ │ -00040f90: 4d61 6361 756c 6179 3244 6f63 294c 6973 Macaulay2Doc)Lis │ │ │ │ -00040fa0: 742c 2c20 746f 2074 6865 206f 7074 696f t,, to the optio │ │ │ │ -00040fb0: 6e61 6c20 6172 6775 6d65 6e74 202a 6e6f nal argument *no │ │ │ │ -00040fc0: 7465 2044 6567 7265 6556 6563 746f 723a te DegreeVector: │ │ │ │ -00040fd0: 0a70 7269 6d61 7279 496e 7661 7269 616e .primaryInvarian │ │ │ │ -00040fe0: 7473 5f6c 705f 7064 5f70 645f 7064 5f63 ts_lp_pd_pd_pd_c │ │ │ │ -00040ff0: 6d44 6567 7265 6556 6563 746f 723d 3e5f mDegreeVector=>_ │ │ │ │ -00041000: 7064 5f70 645f 7064 5f72 702c 2e20 2a6e pd_pd_pd_rp,. *n │ │ │ │ -00041010: 6f74 650a 7072 696d 6172 7949 6e76 6172 ote.primaryInvar │ │ │ │ -00041020: 6961 6e74 733a 2070 7269 6d61 7279 496e iants: primaryIn │ │ │ │ -00041030: 7661 7269 616e 7473 2c20 7769 6c6c 2074 variants, will t │ │ │ │ -00041040: 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If howe │ │ │ │ -00041080: 7665 7220 6e6f 2073 7563 6820 6873 6f70 ver no such hsop │ │ │ │ -00041090: 2065 7869 7374 732c 202a 6e6f 7465 2070 exists, *note p │ │ │ │ -000410a0: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ -000410b0: 3a0a 7072 696d 6172 7949 6e76 6172 6961 :.primaryInvaria │ │ │ │ -000410c0: 6e74 732c 206f 7574 7075 7473 2061 6e20 nts, outputs an │ │ │ │ -000410d0: 6572 726f 7220 6d65 7373 6167 652e 0a0a error message... │ │ │ │ -000410e0: 4e6f 7465 2074 6861 7420 7468 6520 2a6e Note that the *n │ │ │ │ -000410f0: 6f74 6520 4c69 7374 3a20 284d 6163 6175 ote List: (Macau │ │ │ │ -00041100: 6c61 7932 446f 6329 4c69 7374 2c20 6173 lay2Doc)List, as │ │ │ │ -00041110: 7369 676e 6564 2074 6f20 2a6e 6f74 6520 signed to *note │ │ │ │ -00041120: 4465 6772 6565 5665 6374 6f72 3a0a 7072 DegreeVector:.pr │ │ │ │ -00041130: 696d 6172 7949 6e76 6172 6961 6e74 735f imaryInvariants_ │ │ │ │ -00041140: 6c70 5f70 645f 7064 5f70 645f 636d 4465 lp_pd_pd_pd_cmDe │ │ │ │ -00041150: 6772 6565 5665 6374 6f72 3d3e 5f70 645f greeVector=>_pd_ │ │ │ │ -00041160: 7064 5f70 645f 7270 2c20 6973 2069 676e pd_pd_rp, is ign │ │ │ │ -00041170: 6f72 6564 2069 6620 2a6e 6f74 650a 4461 ored if *note.Da │ │ │ │ -00041180: 6465 3a20 7072 696d 6172 7949 6e76 6172 de: primaryInvar │ │ │ │ -00041190: 6961 6e74 735f 6c70 5f70 645f 7064 5f70 iants_lp_pd_pd_p │ │ │ │ -000411a0: 645f 636d 4461 6465 3d3e 5f70 645f 7064 d_cmDade=>_pd_pd │ │ │ │ -000411b0: 5f70 645f 7270 2c20 6973 2073 6574 2074 _pd_rp, is set t │ │ │ │ -000411c0: 6f20 2a6e 6f74 6520 7472 7565 3a0a 284d o *note true:.(M │ │ │ │ -000411d0: 6163 6175 6c61 7932 446f 6329 7472 7565 acaulay2Doc)true │ │ │ │ -000411e0: 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...+----------- │ │ │ │ -000411f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041220: 2d2d 2d2d 2d2b 0a7c 6931 203a 2041 3d6d -----+.|i1 : A=m │ │ │ │ -00041230: 6174 7269 787b 7b30 2c31 2c30 7d2c 7b30 atrix{{0,1,0},{0 │ │ │ │ -00041240: 2c30 2c31 7d2c 7b31 2c30 2c30 7d7d 3b20 ,0,1},{1,0,0}}; │ │ │ │ -00041250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041260: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000400c0: 2020 2020 2020 206e 2d31 0a63 6172 6469 n-1.cardi │ │ │ │ +000400d0: 6e61 6c69 7479 2067 7265 6174 6572 2074 nality greater t │ │ │ │ +000400e0: 6861 6e20 7c47 7c20 2020 2c20 7768 6572 han |G| , wher │ │ │ │ +000400f0: 6520 6e20 6973 2074 6865 206e 756d 6265 e n is the numbe │ │ │ │ +00040100: 7220 6f66 2076 6172 6961 626c 6573 2069 r of variables i │ │ │ │ +00040110: 6e20 7468 650a 706f 6c79 6e6f 6d69 616c n the.polynomial │ │ │ │ +00040120: 2072 696e 6720 522e 2055 7369 6e67 2061 ring R. 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See *n │ │ │ │ +00040230: 6f74 6520 6873 6f70 2061 6c67 6f72 6974 ote hsop algorit │ │ │ │ +00040240: 686d 733a 2068 736f 700a 616c 676f 7269 hms: hsop.algori │ │ │ │ +00040250: 7468 6d73 2c20 666f 7220 6120 6469 7363 thms, for a disc │ │ │ │ +00040260: 7573 7369 6f6e 206f 6e20 7468 6520 4461 ussion on the Da │ │ │ │ +00040270: 6465 2061 6c67 6f72 6974 686d 2e0a 0a53 de algorithm...S │ │ │ │ +00040280: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00040290: 0a0a 2020 2a20 2a6e 6f74 6520 6873 6f70 .. * *note hsop │ │ │ │ +000402a0: 2061 6c67 6f72 6974 686d 733a 2068 736f algorithms: hso │ │ │ │ +000402b0: 7020 616c 676f 7269 7468 6d73 2c20 2d2d p algorithms, -- │ │ │ │ +000402c0: 2061 6e20 6f76 6572 7669 6577 206f 6620 an overview of │ │ │ │ +000402d0: 7468 6520 616c 676f 7269 7468 6d73 0a20 the algorithms. │ │ │ │ +000402e0: 2020 2075 7365 6420 696e 2070 7269 6d61 used in prima │ │ │ │ +000402f0: 7279 496e 7661 7269 616e 7473 0a20 202a ryInvariants. * │ │ │ │ +00040300: 202a 6e6f 7465 2070 7269 6d61 7279 496e *note primaryIn │ │ │ │ +00040310: 7661 7269 616e 7473 3a20 7072 696d 6172 variants: primar │ │ │ │ +00040320: 7949 6e76 6172 6961 6e74 732c 202d 2d20 yInvariants, -- │ │ │ │ +00040330: 636f 6d70 7574 6573 2061 206c 6973 7420 computes a list │ │ │ │ +00040340: 6f66 2070 7269 6d61 7279 0a20 2020 2069 of primary. i │ │ │ │ +00040350: 6e76 6172 6961 6e74 7320 666f 7220 7468 nvariants for th │ │ │ │ +00040360: 6520 696e 7661 7269 616e 7420 7269 6e67 e invariant ring │ │ │ │ +00040370: 206f 6620 6120 6669 6e69 7465 2067 726f of a finite gro │ │ │ │ +00040380: 7570 0a0a 4675 6e63 7469 6f6e 7320 7769 up..Functions wi │ │ │ │ +00040390: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ +000403a0: 6d65 6e74 206e 616d 6564 2044 6164 653a ment named Dade: │ │ │ │ +000403b0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +000403c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000403d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +000403e0: 202a 202a 6e6f 7465 2070 7269 6d61 7279 * *note primary │ │ │ │ +000403f0: 496e 7661 7269 616e 7473 282e 2e2e 2c44 Invariants(...,D │ │ │ │ +00040400: 6164 653d 3e2e 2e2e 293a 0a20 2020 2070 ade=>...):. p │ │ │ │ +00040410: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ +00040420: 5f6c 705f 7064 5f70 645f 7064 5f63 6d44 _lp_pd_pd_pd_cmD │ │ │ │ +00040430: 6164 653d 3e5f 7064 5f70 645f 7064 5f72 ade=>_pd_pd_pd_r │ │ │ │ +00040440: 702c 202d 2d20 616e 206f 7074 696f 6e61 p, -- an optiona │ │ │ │ +00040450: 6c20 6172 6775 6d65 6e74 0a20 2020 2066 l argument. f │ │ │ │ +00040460: 6f72 2070 7269 6d61 7279 496e 7661 7269 or primaryInvari │ │ │ │ +00040470: 616e 7473 2064 6574 6572 6d69 6e69 6e67 ants determining │ │ │ │ +00040480: 2077 6865 7468 6572 2074 6f20 7573 6520 whether to use │ │ │ │ +00040490: 7468 6520 4461 6465 2061 6c67 6f72 6974 the Dade algorit │ │ │ │ +000404a0: 686d 0a0a 4675 7274 6865 7220 696e 666f hm..Further info │ │ │ │ +000404b0: 726d 6174 696f 6e0a 3d3d 3d3d 3d3d 3d3d rmation.======== │ │ │ │ +000404c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +000404d0: 2044 6566 6175 6c74 2076 616c 7565 3a20 Default value: │ │ │ │ +000404e0: 2a6e 6f74 6520 6661 6c73 653a 2028 4d61 *note false: (Ma │ │ │ │ +000404f0: 6361 756c 6179 3244 6f63 2966 616c 7365 caulay2Doc)false │ │ │ │ +00040500: 2c0a 2020 2a20 4675 6e63 7469 6f6e 3a20 ,. * Function: │ │ │ │ +00040510: 2a6e 6f74 6520 7072 696d 6172 7949 6e76 *note primaryInv │ │ │ │ +00040520: 6172 6961 6e74 733a 2070 7269 6d61 7279 ariants: primary │ │ │ │ +00040530: 496e 7661 7269 616e 7473 2c20 2d2d 2063 Invariants, -- c │ │ │ │ +00040540: 6f6d 7075 7465 7320 6120 6c69 7374 206f omputes a list o │ │ │ │ +00040550: 660a 2020 2020 7072 696d 6172 7920 696e f. primary in │ │ │ │ +00040560: 7661 7269 616e 7473 2066 6f72 2074 6865 variants for the │ │ │ │ +00040570: 2069 6e76 6172 6961 6e74 2072 696e 6720 invariant ring │ │ │ │ +00040580: 6f66 2061 2066 696e 6974 6520 6772 6f75 of a finite grou │ │ │ │ +00040590: 700a 2020 2a20 4f70 7469 6f6e 206b 6579 p. * Option key │ │ │ │ +000405a0: 3a20 2a6e 6f74 6520 4461 6465 3a20 7072 : *note Dade: pr │ │ │ │ +000405b0: 696d 6172 7949 6e76 6172 6961 6e74 735f imaryInvariants_ │ │ │ │ +000405c0: 6c70 5f70 645f 7064 5f70 645f 636d 4461 lp_pd_pd_pd_cmDa │ │ │ │ +000405d0: 6465 3d3e 5f70 645f 7064 5f70 645f 7270 de=>_pd_pd_pd_rp │ │ │ │ +000405e0: 2c0a 2020 2020 2d2d 2061 6e20 6f70 7469 ,. -- an opti │ │ │ │ +000405f0: 6f6e 616c 2061 7267 756d 656e 7420 666f onal argument fo │ │ │ │ +00040600: 7220 7072 696d 6172 7949 6e76 6172 6961 r primaryInvaria │ │ │ │ +00040610: 6e74 7320 6465 7465 726d 696e 696e 6720 nts determining │ │ │ │ +00040620: 7768 6574 6865 7220 746f 2075 7365 0a20 whether to use. │ │ │ │ +00040630: 2020 2074 6865 2044 6164 6520 616c 676f the Dade algo │ │ │ │ +00040640: 7269 7468 6d0a 2d2d 2d2d 2d2d 2d2d 2d2d rithm.---------- │ │ │ │ +00040650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00040660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00040670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00040680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00040690: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +000406a0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +000406b0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +000406c0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +000406d0: 682f 6d61 6361 756c 6179 322d 312e 3235 h/macaulay2-1.25 │ │ │ │ +000406e0: 2e31 312b 6473 2f4d 322f 4d61 6361 756c .11+ds/M2/Macaul │ │ │ │ +000406f0: 6179 322f 7061 636b 6167 6573 2f0a 496e ay2/packages/.In │ │ │ │ +00040700: 7661 7269 616e 7452 696e 672f 4861 7765 variantRing/Hawe │ │ │ │ +00040710: 7344 6f63 2e6d 323a 3234 343a 302e 0a1f sDoc.m2:244:0... │ │ │ │ +00040720: 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7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ +00040a90: 3a20 7072 696d 6172 7949 6e76 6172 6961 : primaryInvaria │ │ │ │ +00040aa0: 6e74 732c 2075 7365 7320 616e 206f 7074 nts, uses an opt │ │ │ │ +00040ab0: 696d 6973 696e 670a 2020 2020 2020 2020 imising. │ │ │ │ +00040ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040b00: 2020 0a20 2020 2020 2020 2020 2020 2020 . │ │ │ │ +00040b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040b50: 0a61 6c67 6f72 6974 686d 2077 6869 6368 .algorithm which │ │ │ │ +00040b60: 2074 6573 7473 2066 6f72 2074 6865 2065 tests for the e │ │ │ │ +00040b70: 7869 7374 656e 6365 206f 6620 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2020 2020 2020 |.| │ │ │ │ +00041480: 2020 2020 207c 2031 2030 2030 207c 2020 | 1 0 0 | │ │ │ │ +00041490: 7c20 3020 3020 3120 7c20 2020 2020 2020 | 0 0 1 | │ │ │ │ +000414a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000414b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000414c0: 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041500: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00041510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041510: 7c0a 7c6f 3320 3a20 4669 6e69 7465 4772 |.|o3 : FiniteGr │ │ │ │ +00041520: 6f75 7041 6374 696f 6e20 2020 2020 2020 oupAction │ │ │ │ 00041530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041540: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ -00041550: 2051 515b 782e 2e7a 5d20 3c2d 207b 7c20 QQ[x..z] <- {| │ │ │ │ -00041560: 3020 3120 3020 7c2c 207c 2030 2031 2030 0 1 0 |, | 0 1 0 │ │ │ │ -00041570: 207c 7d20 2020 2020 2020 2020 2020 2020 |} │ │ │ │ -00041580: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00041590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000415a0: 207c 2030 2030 2031 207c 2020 7c20 3120 | 0 0 1 | | 1 │ │ │ │ -000415b0: 3020 3020 7c20 2020 2020 2020 2020 2020 0 0 | │ │ │ │ -000415c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000415d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000415e0: 2020 2020 7c20 3120 3020 3020 7c20 207c | 1 0 0 | | │ │ │ │ -000415f0: 2030 2030 2031 207c 2020 2020 2020 2020 0 0 1 | │ │ │ │ -00041600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041610: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041560: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00041570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000415a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000415b0: 2b0a 7c69 3420 3a20 7072 696d 6172 7949 +.|i4 : primaryI │ │ │ │ +000415c0: 6e76 6172 6961 6e74 7328 5333 2c44 6567 nvariants(S3,Deg │ │ │ │ +000415d0: 7265 6556 6563 746f 723d 3e7b 332c 332c reeVector=>{3,3, │ │ │ │ +000415e0: 347d 2920 2020 2020 2020 2020 2020 2020 4}) │ │ │ │ +000415f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041600: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041650: 2020 2020 207c 0a7c 6f33 203a 2046 696e |.|o3 : Fin │ │ │ │ -00041660: 6974 6547 726f 7570 4163 7469 6f6e 2020 iteGroupAction │ │ │ │ -00041670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041690: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -000416a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000416b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000416c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000416d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -000416e0: 203a 2070 7269 6d61 7279 496e 7661 7269 : primaryInvari │ │ │ │ -000416f0: 616e 7473 2853 332c 4465 6772 6565 5665 ants(S3,DegreeVe │ │ │ │ -00041700: 6374 6f72 3d3e 7b33 2c33 2c34 7d29 2020 ctor=>{3,3,4}) │ │ │ │ -00041710: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00041720: 7c20 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 2020 2020 │ │ │ │ -00041760: 207c 0a7c 2020 2020 2020 2032 2020 2020 |.| 2 │ │ │ │ -00041770: 2020 2032 2020 2020 3220 2020 2020 3220 2 2 2 │ │ │ │ -00041780: 2020 2020 2020 3220 2020 2020 2032 2020 2 2 │ │ │ │ -00041790: 2020 2020 2020 2020 3420 2020 2034 2020 4 4 │ │ │ │ -000417a0: 2020 3420 7c0a 7c6f 3420 3d20 7b78 2079 4 |.|o4 = {x y │ │ │ │ -000417b0: 202b 2078 2a79 2020 2b20 7820 7a20 2b20 + x*y + x z + │ │ │ │ -000417c0: 7920 7a20 2b20 782a 7a20 202b 2079 2a7a y z + x*z + y*z │ │ │ │ -000417d0: 202c 2078 2a79 2a7a 2c20 7820 202b 2079 , x*y*z, x + y │ │ │ │ -000417e0: 2020 2b20 7a20 7d7c 0a7c 2020 2020 2020 + z }|.| │ │ │ │ +00041650: 7c0a 7c20 2020 2020 2020 3320 2020 2033 |.| 3 3 │ │ │ │ +00041660: 2020 2020 3320 2020 3220 2020 2020 2020 3 2 │ │ │ │ +00041670: 3220 2020 2032 2020 2020 2032 2020 2020 2 2 2 │ 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│ │ │ │ +00041760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041790: 7c0a 7c20 2020 2020 7920 7a20 2b20 782a |.| y z + x* │ │ │ │ +000417a0: 7a20 202b 2079 2a7a 207d 2020 2020 2020 z + y*z } │ │ │ │ +000417b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000417c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000417d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000417e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000417f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041820: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -00041830: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +00041820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041830: 7c0a 7c6f 3420 3a20 4c69 7374 2020 2020 |.|o4 : List │ │ │ │ 00041840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041860: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00041870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041880: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00041890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000418a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000418b0: 2b0a 0a43 6176 6561 740a 3d3d 3d3d 3d3d +..Caveat.====== │ │ │ │ -000418c0: 0a0a 2020 2020 2020 2020 2020 2020 2020 .. │ │ │ │ -000418d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000418e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000418b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000418c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000418d0: 2b0a 0a43 6176 6561 740a 3d3d 3d3d 3d3d +..Caveat.====== │ │ │ │ +000418e0: 0a0a 2020 2020 2020 2020 2020 2020 2020 .. │ │ │ │ 000418f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041900: 2020 2020 2020 2020 2020 2020 200a 4375 .Cu │ │ │ │ -00041910: 7272 656e 746c 7920 7573 6572 7320 6361 rrently users ca │ │ │ │ -00041920: 6e20 6f6e 6c79 2075 7365 202a 6e6f 7465 n only use *note │ │ │ │ -00041930: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ -00041940: 7473 3a20 7072 696d 6172 7949 6e76 6172 ts: primaryInvar │ │ │ │ -00041950: 6961 6e74 732c 2074 6f0a 2020 2020 2020 iants, to. │ │ │ │ -00041960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041920: 2020 2020 2020 2020 2020 2020 200a 4375 .Cu │ │ │ │ +00041930: 7272 656e 746c 7920 7573 6572 7320 6361 rrently users ca │ │ │ │ +00041940: 6e20 6f6e 6c79 2075 7365 202a 6e6f 7465 n only use *note │ │ │ │ +00041950: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ +00041960: 7473 3a20 7072 696d 6172 7949 6e76 6172 ts: primaryInvar │ │ │ │ +00041970: 6961 6e74 732c 2074 6f0a 2020 2020 2020 iants, to. │ │ │ │ 00041980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000419a0: 2020 2020 2020 200a 6361 6c63 756c 6174 .calculat │ │ │ │ -000419b0: 6520 6120 6873 6f70 2066 6f72 2074 6865 e a hsop for the │ │ │ │ -000419c0: 2069 6e76 6172 6961 6e74 2072 696e 6720 invariant ring │ │ │ │ -000419d0: 6f76 6572 2061 2066 696e 6974 6520 6669 over a finite fi │ │ │ │ -000419e0: 656c 6420 6279 2075 7369 6e67 2074 6865 eld by using the │ │ │ │ -000419f0: 2044 6164 650a 2020 2020 2020 2020 2020 Dade. │ │ │ │ -00041a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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Users sh │ │ │ │ +00041a70: 6f75 6c64 2065 6e74 6572 2074 6865 2066 ould enter the f │ │ │ │ +00041a80: 696e 6974 6520 6669 656c 6420 6173 2061 inite field as a │ │ │ │ +00041a90: 202a 6e6f 7465 2047 616c 6f69 7346 6965 *note GaloisFie │ │ │ │ +00041aa0: 6c64 3a0a 2020 2020 2020 2020 2020 2020 ld:. │ │ │ │ 00041ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041ac0: 2020 2020 2020 2020 0a28 4d61 6361 756c .(Macaul │ │ │ │ -00041ad0: 6179 3244 6f63 2947 616c 6f69 7346 6965 ay2Doc)GaloisFie │ │ │ │ -00041ae0: 6c64 2c20 6f72 2061 2071 756f 7469 656e ld, or a quotien │ │ │ │ -00041af0: 7420 6669 656c 6420 6f66 2074 6865 2066 t field of the f │ │ │ │ -00041b00: 6f72 6d20 2a6e 6f74 6520 5a5a 3a0a 2020 orm *note ZZ:. │ │ │ │ -00041b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041ae0: 2020 2020 2020 2020 0a28 4d61 6361 756c .(Macaul │ │ │ │ +00041af0: 6179 3244 6f63 2947 616c 6f69 7346 6965 ay2Doc)GaloisFie │ │ │ │ +00041b00: 6c64 2c20 6f72 2061 2071 756f 7469 656e ld, or a quotien │ │ │ │ +00041b10: 7420 6669 656c 6420 6f66 2074 6865 2066 t field of the f │ │ │ │ +00041b20: 6f72 6d20 2a6e 6f74 6520 5a5a 3a0a 2020 orm *note ZZ:. │ │ │ │ 00041b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041b50: 2020 2020 200a 284d 6163 6175 6c61 7932 .(Macaulay2 │ │ │ │ -00041b60: 446f 6329 5a5a 2c2f 7020 616e 6420 6172 Doc)ZZ,/p and ar │ │ │ │ -00041b70: 6520 6164 7669 7365 6420 746f 2065 6e73 e advised to ens │ │ │ │ -00041b80: 7572 6520 7468 6174 2074 6865 2067 726f ure that the gro │ │ │ │ -00041b90: 756e 6420 6669 656c 6420 6861 730a 2020 und field has. │ │ │ │ -00041ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041bb0: 2020 2020 2020 2020 2020 6e2d 310a 6361 n-1.ca │ │ │ │ -00041bc0: 7264 696e 616c 6974 7920 6772 6561 7465 rdinality greate │ │ │ │ -00041bd0: 7220 7468 616e 207c 477c 2020 202c 2077 r than |G| , w │ │ │ │ -00041be0: 6865 7265 206e 2069 7320 7468 6520 6e75 here n is the nu │ │ │ │ -00041bf0: 6d62 6572 206f 6620 7661 7269 6162 6c65 mber of variable │ │ │ │ -00041c00: 7320 696e 2074 6865 0a70 6f6c 796e 6f6d s in the.polynom │ │ │ │ -00041c10: 6961 6c20 7269 6e67 2052 2e20 5573 696e ial ring R. Usin │ │ │ │ -00041c20: 6720 6120 6772 6f75 6e64 2066 6965 6c64 g a ground field │ │ │ │ -00041c30: 2073 6d61 6c6c 6572 2074 6861 6e20 7468 smaller than th │ │ │ │ -00041c40: 6973 2072 756e 7320 7468 6520 7269 736b is runs the risk │ │ │ │ -00041c50: 206f 6620 7468 650a 616c 676f 7269 7468 of the.algorith │ │ │ │ -00041c60: 6d20 6765 7474 696e 6720 7374 7563 6b20 m getting stuck │ │ │ │ -00041c70: 696e 2061 6e20 696e 6669 6e69 7465 206c in an infinite l │ │ │ │ -00041c80: 6f6f 703b 202a 6e6f 7465 2070 7269 6d61 oop; *note prima │ │ │ │ -00041c90: 7279 496e 7661 7269 616e 7473 3a0a 7072 ryInvariants:.pr │ │ │ │ -00041ca0: 696d 6172 7949 6e76 6172 6961 6e74 732c imaryInvariants, │ │ │ │ -00041cb0: 2064 6973 706c 6179 7320 6120 7761 726e displays a warn │ │ │ │ -00041cc0: 696e 6720 6d65 7373 6167 6520 6173 6b69 ing message aski │ │ │ │ -00041cd0: 6e67 2074 6865 2075 7365 7220 7768 6574 ng the user whet │ │ │ │ -00041ce0: 6865 7220 7468 6579 2077 6973 680a 746f her they wish.to │ │ │ │ -00041cf0: 2063 6f6e 7469 6e75 6520 7769 7468 2074 continue with t │ │ │ │ -00041d00: 6865 2063 6f6d 7075 7461 7469 6f6e 2069 he computation i │ │ │ │ -00041d10: 6e20 7468 6973 2063 6173 652e 2053 6565 n this case. See │ │ │ │ -00041d20: 202a 6e6f 7465 2068 736f 7020 616c 676f *note hsop algo │ │ │ │ -00041d30: 7269 7468 6d73 3a20 6873 6f70 0a61 6c67 rithms: hsop.alg │ │ │ │ -00041d40: 6f72 6974 686d 732c 2066 6f72 2061 2064 orithms, for a d │ │ │ │ -00041d50: 6973 6375 7373 696f 6e20 6f6e 2074 6865 iscussion on the │ │ │ │ -00041d60: 2044 6164 6520 616c 676f 7269 7468 6d2e Dade algorithm. │ │ │ │ -00041d70: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -00041d80: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2070 ===.. * *note p │ │ │ │ -00041d90: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ -00041da0: 3a20 7072 696d 6172 7949 6e76 6172 6961 : primaryInvaria │ │ │ │ -00041db0: 6e74 732c 202d 2d20 636f 6d70 7574 6573 nts, -- computes │ │ │ │ -00041dc0: 2061 206c 6973 7420 6f66 2070 7269 6d61 a list of prima │ │ │ │ -00041dd0: 7279 0a20 2020 2069 6e76 6172 6961 6e74 ry. invariant │ │ │ │ -00041de0: 7320 666f 7220 7468 6520 696e 7661 7269 s for the invari │ │ │ │ -00041df0: 616e 7420 7269 6e67 206f 6620 6120 6669 ant ring of a fi │ │ │ │ -00041e00: 6e69 7465 2067 726f 7570 0a0a 4675 6e63 nite group..Func │ │ │ │ -00041e10: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ -00041e20: 6e61 6c20 6172 6775 6d65 6e74 206e 616d nal argument nam │ │ │ │ -00041e30: 6564 2044 6567 7265 6556 6563 746f 723a ed DegreeVector: │ │ │ │ -00041e40: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00041e50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00041e60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00041e70: 3d3d 3d3d 3d0a 0a20 202a 2022 6869 726f =====.. * "hiro │ │ │ │ -00041e80: 6e61 6b61 4465 636f 6d70 6f73 6974 696f nakaDecompositio │ │ │ │ -00041e90: 6e28 2e2e 2e2c 4465 6772 6565 5665 6374 n(...,DegreeVect │ │ │ │ -00041ea0: 6f72 3d3e 2e2e 2e29 220a 2020 2a20 2a6e or=>...)". * *n │ │ │ │ -00041eb0: 6f74 6520 7072 696d 6172 7949 6e76 6172 ote primaryInvar │ │ │ │ -00041ec0: 6961 6e74 7328 2e2e 2e2c 4465 6772 6565 iants(...,Degree │ │ │ │ -00041ed0: 5665 6374 6f72 3d3e 2e2e 2e29 3a0a 2020 Vector=>...):. │ │ │ │ -00041ee0: 2020 7072 696d 6172 7949 6e76 6172 6961 primaryInvaria │ │ │ │ -00041ef0: 6e74 735f 6c70 5f70 645f 7064 5f70 645f nts_lp_pd_pd_pd_ │ │ │ │ -00041f00: 636d 4465 6772 6565 5665 6374 6f72 3d3e cmDegreeVector=> │ │ │ │ -00041f10: 5f70 645f 7064 5f70 645f 7270 2c20 2d2d _pd_pd_pd_rp, -- │ │ │ │ -00041f20: 2061 6e20 6f70 7469 6f6e 616c 0a20 2020 an optional. │ │ │ │ -00041f30: 2061 7267 756d 656e 7420 666f 7220 7072 argument for pr │ │ │ │ -00041f40: 696d 6172 7949 6e76 6172 6961 6e74 7320 imaryInvariants │ │ │ │ -00041f50: 7468 6174 2066 696e 6473 2069 6e76 6172 that finds invar │ │ │ │ -00041f60: 6961 6e74 7320 6f66 2063 6572 7461 696e iants of certain │ │ │ │ -00041f70: 2064 6567 7265 6573 0a0a 4675 7274 6865 degrees..Furthe │ │ │ │ -00041f80: 7220 696e 666f 726d 6174 696f 6e0a 3d3d r information.== │ │ │ │ -00041f90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00041fa0: 3d0a 0a20 202a 2044 6566 6175 6c74 2076 =.. * Default v │ │ │ │ -00041fb0: 616c 7565 3a20 300a 2020 2a20 4675 6e63 alue: 0. * Func │ │ │ │ -00041fc0: 7469 6f6e 3a20 2a6e 6f74 6520 7072 696d tion: *note prim │ │ │ │ -00041fd0: 6172 7949 6e76 6172 6961 6e74 733a 2070 aryInvariants: p │ │ │ │ -00041fe0: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ -00041ff0: 2c20 2d2d 2063 6f6d 7075 7465 7320 6120 , -- computes a │ │ │ │ -00042000: 6c69 7374 206f 660a 2020 2020 7072 696d list of. prim │ │ │ │ -00042010: 6172 7920 696e 7661 7269 616e 7473 2066 ary invariants f │ │ │ │ -00042020: 6f72 2074 6865 2069 6e76 6172 6961 6e74 or the invariant │ │ │ │ -00042030: 2072 696e 6720 6f66 2061 2066 696e 6974 ring of a finit │ │ │ │ -00042040: 6520 6772 6f75 700a 2020 2a20 4f70 7469 e group. * Opti │ │ │ │ -00042050: 6f6e 206b 6579 3a20 2a6e 6f74 6520 4465 on key: *note De │ │ │ │ -00042060: 6772 6565 5665 6374 6f72 3a0a 2020 2020 greeVector:. │ │ │ │ -00042070: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ -00042080: 735f 6c70 5f70 645f 7064 5f70 645f 636d s_lp_pd_pd_pd_cm │ │ │ │ -00042090: 4465 6772 6565 5665 6374 6f72 3d3e 5f70 DegreeVector=>_p │ │ │ │ -000420a0: 645f 7064 5f70 645f 7270 2c20 2d2d 2061 d_pd_pd_rp, -- a │ │ │ │ -000420b0: 6e20 6f70 7469 6f6e 616c 0a20 2020 2061 n optional. a │ │ │ │ -000420c0: 7267 756d 656e 7420 666f 7220 7072 696d rgument for prim │ │ │ │ -000420d0: 6172 7949 6e76 6172 6961 6e74 7320 7468 aryInvariants th │ │ │ │ -000420e0: 6174 2066 696e 6473 2069 6e76 6172 6961 at finds invaria │ │ │ │ -000420f0: 6e74 7320 6f66 2063 6572 7461 696e 2064 nts of certain d │ │ │ │ -00042100: 6567 7265 6573 0a2d 2d2d 2d2d 2d2d 2d2d egrees.--------- │ │ │ │ -00042110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041b70: 2020 2020 200a 284d 6163 6175 6c61 7932 .(Macaulay2 │ │ │ │ +00041b80: 446f 6329 5a5a 2c2f 7020 616e 6420 6172 Doc)ZZ,/p and ar │ │ │ │ +00041b90: 6520 6164 7669 7365 6420 746f 2065 6e73 e advised to ens │ │ │ │ +00041ba0: 7572 6520 7468 6174 2074 6865 2067 726f ure that the gro │ │ │ │ +00041bb0: 756e 6420 6669 656c 6420 6861 730a 2020 und field has. │ │ │ │ +00041bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041bd0: 2020 2020 2020 2020 2020 6e2d 310a 6361 n-1.ca │ │ │ │ +00041be0: 7264 696e 616c 6974 7920 6772 6561 7465 rdinality greate │ │ │ │ +00041bf0: 7220 7468 616e 207c 477c 2020 202c 2077 r than |G| , w │ │ │ │ +00041c00: 6865 7265 206e 2069 7320 7468 6520 6e75 here n is the nu │ │ │ │ +00041c10: 6d62 6572 206f 6620 7661 7269 6162 6c65 mber of variable │ │ │ │ +00041c20: 7320 696e 2074 6865 0a70 6f6c 796e 6f6d s in the.polynom │ │ │ │ +00041c30: 6961 6c20 7269 6e67 2052 2e20 5573 696e ial ring R. Usin │ │ │ │ +00041c40: 6720 6120 6772 6f75 6e64 2066 6965 6c64 g a ground field │ │ │ │ +00041c50: 2073 6d61 6c6c 6572 2074 6861 6e20 7468 smaller than th │ │ │ │ +00041c60: 6973 2072 756e 7320 7468 6520 7269 736b is runs the risk │ │ │ │ +00041c70: 206f 6620 7468 650a 616c 676f 7269 7468 of the.algorith │ │ │ │ +00041c80: 6d20 6765 7474 696e 6720 7374 7563 6b20 m getting stuck │ │ │ │ +00041c90: 696e 2061 6e20 696e 6669 6e69 7465 206c in an infinite l │ │ │ │ +00041ca0: 6f6f 703b 202a 6e6f 7465 2070 7269 6d61 oop; *note prima │ │ │ │ +00041cb0: 7279 496e 7661 7269 616e 7473 3a0a 7072 ryInvariants:.pr │ │ │ │ +00041cc0: 696d 6172 7949 6e76 6172 6961 6e74 732c imaryInvariants, │ │ │ │ +00041cd0: 2064 6973 706c 6179 7320 6120 7761 726e displays a warn │ │ │ │ +00041ce0: 696e 6720 6d65 7373 6167 6520 6173 6b69 ing message aski │ │ │ │ +00041cf0: 6e67 2074 6865 2075 7365 7220 7768 6574 ng the user whet │ │ │ │ +00041d00: 6865 7220 7468 6579 2077 6973 680a 746f her they wish.to │ │ │ │ +00041d10: 2063 6f6e 7469 6e75 6520 7769 7468 2074 continue with t │ │ │ │ +00041d20: 6865 2063 6f6d 7075 7461 7469 6f6e 2069 he computation i │ │ │ │ +00041d30: 6e20 7468 6973 2063 6173 652e 2053 6565 n this case. See │ │ │ │ +00041d40: 202a 6e6f 7465 2068 736f 7020 616c 676f *note hsop algo │ │ │ │ +00041d50: 7269 7468 6d73 3a20 6873 6f70 0a61 6c67 rithms: hsop.alg │ │ │ │ +00041d60: 6f72 6974 686d 732c 2066 6f72 2061 2064 orithms, for a d │ │ │ │ +00041d70: 6973 6375 7373 696f 6e20 6f6e 2074 6865 iscussion on the │ │ │ │ +00041d80: 2044 6164 6520 616c 676f 7269 7468 6d2e Dade algorithm. │ │ │ │ +00041d90: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +00041da0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2070 ===.. * *note p │ │ │ │ +00041db0: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ +00041dc0: 3a20 7072 696d 6172 7949 6e76 6172 6961 : primaryInvaria │ │ │ │ +00041dd0: 6e74 732c 202d 2d20 636f 6d70 7574 6573 nts, -- computes │ │ │ │ +00041de0: 2061 206c 6973 7420 6f66 2070 7269 6d61 a list of prima │ │ │ │ +00041df0: 7279 0a20 2020 2069 6e76 6172 6961 6e74 ry. invariant │ │ │ │ +00041e00: 7320 666f 7220 7468 6520 696e 7661 7269 s for the invari │ │ │ │ +00041e10: 616e 7420 7269 6e67 206f 6620 6120 6669 ant ring of a fi │ │ │ │ +00041e20: 6e69 7465 2067 726f 7570 0a0a 4675 6e63 nite group..Func │ │ │ │ +00041e30: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ +00041e40: 6e61 6c20 6172 6775 6d65 6e74 206e 616d nal argument nam │ │ │ │ +00041e50: 6564 2044 6567 7265 6556 6563 746f 723a ed DegreeVector: │ │ │ │ +00041e60: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00041e70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00041e80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00041e90: 3d3d 3d3d 3d0a 0a20 202a 2022 6869 726f =====.. * "hiro │ │ │ │ +00041ea0: 6e61 6b61 4465 636f 6d70 6f73 6974 696f nakaDecompositio │ │ │ │ +00041eb0: 6e28 2e2e 2e2c 4465 6772 6565 5665 6374 n(...,DegreeVect │ │ │ │ +00041ec0: 6f72 3d3e 2e2e 2e29 220a 2020 2a20 2a6e or=>...)". * *n │ │ │ │ +00041ed0: 6f74 6520 7072 696d 6172 7949 6e76 6172 ote primaryInvar │ │ │ │ +00041ee0: 6961 6e74 7328 2e2e 2e2c 4465 6772 6565 iants(...,Degree │ │ │ │ +00041ef0: 5665 6374 6f72 3d3e 2e2e 2e29 3a0a 2020 Vector=>...):. │ │ │ │ +00041f00: 2020 7072 696d 6172 7949 6e76 6172 6961 primaryInvaria │ │ │ │ +00041f10: 6e74 735f 6c70 5f70 645f 7064 5f70 645f nts_lp_pd_pd_pd_ │ │ │ │ +00041f20: 636d 4465 6772 6565 5665 6374 6f72 3d3e cmDegreeVector=> │ │ │ │ +00041f30: 5f70 645f 7064 5f70 645f 7270 2c20 2d2d _pd_pd_pd_rp, -- │ │ │ │ +00041f40: 2061 6e20 6f70 7469 6f6e 616c 0a20 2020 an optional. │ │ │ │ +00041f50: 2061 7267 756d 656e 7420 666f 7220 7072 argument for pr │ │ │ │ +00041f60: 696d 6172 7949 6e76 6172 6961 6e74 7320 imaryInvariants │ │ │ │ +00041f70: 7468 6174 2066 696e 6473 2069 6e76 6172 that finds invar │ │ │ │ +00041f80: 6961 6e74 7320 6f66 2063 6572 7461 696e iants of certain │ │ │ │ +00041f90: 2064 6567 7265 6573 0a0a 4675 7274 6865 degrees..Furthe │ │ │ │ +00041fa0: 7220 696e 666f 726d 6174 696f 6e0a 3d3d r information.== │ │ │ │ +00041fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00041fc0: 3d0a 0a20 202a 2044 6566 6175 6c74 2076 =.. * Default v │ │ │ │ +00041fd0: 616c 7565 3a20 300a 2020 2a20 4675 6e63 alue: 0. * Func │ │ │ │ +00041fe0: 7469 6f6e 3a20 2a6e 6f74 6520 7072 696d tion: *note prim │ │ │ │ +00041ff0: 6172 7949 6e76 6172 6961 6e74 733a 2070 aryInvariants: p │ │ │ │ +00042000: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ +00042010: 2c20 2d2d 2063 6f6d 7075 7465 7320 6120 , -- computes a │ │ │ │ +00042020: 6c69 7374 206f 660a 2020 2020 7072 696d list of. prim │ │ │ │ +00042030: 6172 7920 696e 7661 7269 616e 7473 2066 ary invariants f │ │ │ │ +00042040: 6f72 2074 6865 2069 6e76 6172 6961 6e74 or the invariant │ │ │ │ +00042050: 2072 696e 6720 6f66 2061 2066 696e 6974 ring of a finit │ │ │ │ +00042060: 6520 6772 6f75 700a 2020 2a20 4f70 7469 e group. * Opti │ │ │ │ +00042070: 6f6e 206b 6579 3a20 2a6e 6f74 6520 4465 on key: *note De │ │ │ │ +00042080: 6772 6565 5665 6374 6f72 3a0a 2020 2020 greeVector:. │ │ │ │ +00042090: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ +000420a0: 735f 6c70 5f70 645f 7064 5f70 645f 636d s_lp_pd_pd_pd_cm │ │ │ │ +000420b0: 4465 6772 6565 5665 6374 6f72 3d3e 5f70 DegreeVector=>_p │ │ │ │ +000420c0: 645f 7064 5f70 645f 7270 2c20 2d2d 2061 d_pd_pd_rp, -- a │ │ │ │ +000420d0: 6e20 6f70 7469 6f6e 616c 0a20 2020 2061 n optional. a │ │ │ │ +000420e0: 7267 756d 656e 7420 666f 7220 7072 696d rgument for prim │ │ │ │ +000420f0: 6172 7949 6e76 6172 6961 6e74 7320 7468 aryInvariants th │ │ │ │ +00042100: 6174 2066 696e 6473 2069 6e76 6172 6961 at finds invaria │ │ │ │ +00042110: 6e74 7320 6f66 2063 6572 7461 696e 2064 nts of certain d │ │ │ │ +00042120: 6567 7265 6573 0a2d 2d2d 2d2d 2d2d 2d2d egrees.--------- │ │ │ │ 00042130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00042140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042150: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00042160: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00042170: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00042180: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00042190: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -000421a0: 352e 3131 2b64 732f 4d32 2f4d 6163 6175 5.11+ds/M2/Macau │ │ │ │ -000421b0: 6c61 7932 2f70 6163 6b61 6765 732f 0a49 lay2/packages/.I │ │ │ │ -000421c0: 6e76 6172 6961 6e74 5269 6e67 2f48 6177 nvariantRing/Haw │ │ │ │ -000421d0: 6573 446f 632e 6d32 3a32 3938 3a30 2e0a esDoc.m2:298:0.. │ │ │ │ -000421e0: 1f0a 4669 6c65 3a20 496e 7661 7269 616e ..File: Invarian │ │ │ │ -000421f0: 7452 696e 672e 696e 666f 2c20 4e6f 6465 tRing.info, Node │ │ │ │ -00042200: 3a20 7261 6e6b 5f6c 7044 6961 676f 6e61 : rank_lpDiagona │ │ │ │ -00042210: 6c41 6374 696f 6e5f 7270 2c20 4e65 7874 lAction_rp, Next │ │ │ │ -00042220: 3a20 7265 6c61 7469 6f6e 735f 6c70 4669 : relations_lpFi │ │ │ │ -00042230: 6e69 7465 4772 6f75 7041 6374 696f 6e5f niteGroupAction_ │ │ │ │ -00042240: 7270 2c20 5072 6576 3a20 7072 696d 6172 rp, Prev: primar │ │ │ │ -00042250: 7949 6e76 6172 6961 6e74 735f 6c70 5f70 yInvariants_lp_p │ │ │ │ -00042260: 645f 7064 5f70 645f 636d 4465 6772 6565 d_pd_pd_cmDegree │ │ │ │ -00042270: 5665 6374 6f72 3d3e 5f70 645f 7064 5f70 Vector=>_pd_pd_p │ │ │ │ -00042280: 645f 7270 2c20 5570 3a20 546f 700a 0a72 d_rp, Up: Top..r │ │ │ │ -00042290: 616e 6b28 4469 6167 6f6e 616c 4163 7469 ank(DiagonalActi │ │ │ │ -000422a0: 6f6e 2920 2d2d 206f 6620 6120 6469 6167 on) -- of a diag │ │ │ │ -000422b0: 6f6e 616c 2061 6374 696f 6e0a 2a2a 2a2a onal action.**** │ │ │ │ -000422c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000422d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000422e0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 ********.. * Fu │ │ │ │ -000422f0: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 7261 nction: *note ra │ │ │ │ -00042300: 6e6b 3a20 284d 6163 6175 6c61 7932 446f nk: (Macaulay2Do │ │ │ │ -00042310: 6329 7261 6e6b 2c0a 2020 2a20 5573 6167 c)rank,. * Usag │ │ │ │ -00042320: 653a 200a 2020 2020 2020 2020 7261 6e6b e: . rank │ │ │ │ -00042330: 2044 0a20 202a 2049 6e70 7574 733a 0a20 D. * Inputs:. │ │ │ │ -00042340: 2020 2020 202a 2044 2c20 616e 2069 6e73 * D, an ins │ │ │ │ -00042350: 7461 6e63 6520 6f66 2074 6865 2074 7970 tance of the typ │ │ │ │ -00042360: 6520 2a6e 6f74 6520 4469 6167 6f6e 616c e *note Diagonal │ │ │ │ -00042370: 4163 7469 6f6e 3a20 4469 6167 6f6e 616c Action: Diagonal │ │ │ │ -00042380: 4163 7469 6f6e 2c0a 2020 2a20 4f75 7470 Action,. * Outp │ │ │ │ -00042390: 7574 733a 0a20 2020 2020 202a 2061 6e20 uts:. * an │ │ │ │ -000423a0: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ -000423b0: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ -000423c0: 2c20 7468 6520 7261 6e6b 206f 6620 7468 , the rank of th │ │ │ │ -000423d0: 6520 746f 7275 7320 6661 6374 6f72 206f e torus factor o │ │ │ │ -000423e0: 6620 610a 2020 2020 2020 2020 6469 6167 f a. diag │ │ │ │ -000423f0: 6f6e 616c 2061 6374 696f 6e0a 0a44 6573 onal action..Des │ │ │ │ -00042400: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00042410: 3d3d 3d3d 0a0a 5468 6973 2066 756e 6374 ====..This funct │ │ │ │ -00042420: 696f 6e20 6973 2070 726f 7669 6465 6420 ion is provided │ │ │ │ -00042430: 6279 2074 6865 2070 6163 6b61 6765 202a by the package * │ │ │ │ -00042440: 6e6f 7465 2049 6e76 6172 6961 6e74 5269 note InvariantRi │ │ │ │ -00042450: 6e67 3a20 546f 702c 2e20 0a0a 5573 6520 ng: Top,. ..Use │ │ │ │ -00042460: 7468 6973 2066 756e 6374 696f 6e20 746f this function to │ │ │ │ -00042470: 2072 6563 6f76 6572 2074 6865 2072 616e recover the ran │ │ │ │ -00042480: 6b20 6f66 2074 6865 2074 6f72 7573 2066 k of the torus f │ │ │ │ -00042490: 6163 746f 7220 6f66 2061 2064 6961 676f actor of a diago │ │ │ │ -000424a0: 6e61 6c20 6163 7469 6f6e 2e0a 0a54 6865 nal action...The │ │ │ │ -000424b0: 2066 6f6c 6c6f 7769 6e67 2065 7861 6d70 following examp │ │ │ │ -000424c0: 6c65 2064 6566 696e 6573 2061 6e20 6163 le defines an ac │ │ │ │ -000424d0: 7469 6f6e 206f 6620 6120 7477 6f2d 6469 tion of a two-di │ │ │ │ -000424e0: 6d65 6e73 696f 6e61 6c20 746f 7275 7320 mensional torus │ │ │ │ -000424f0: 6f6e 2061 0a70 6f6c 796e 6f6d 6961 6c20 on a.polynomial │ │ │ │ -00042500: 7269 6e67 2069 6e20 666f 7572 2076 6172 ring in four var │ │ │ │ -00042510: 6961 626c 6573 2e0a 0a2b 2d2d 2d2d 2d2d iables...+------ │ │ │ │ -00042520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042540: 2d2b 0a7c 6931 203a 2052 203d 2051 515b -+.|i1 : R = QQ[ │ │ │ │ -00042550: 785f 312e 2e78 5f34 5d20 2020 2020 2020 x_1..x_4] │ │ │ │ -00042560: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00042570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042590: 2020 2020 207c 0a7c 6f31 203d 2052 2020 |.|o1 = R │ │ │ │ +00042150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042170: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +00042180: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +00042190: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +000421a0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +000421b0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +000421c0: 352e 3131 2b64 732f 4d32 2f4d 6163 6175 5.11+ds/M2/Macau │ │ │ │ +000421d0: 6c61 7932 2f70 6163 6b61 6765 732f 0a49 lay2/packages/.I │ │ │ │ +000421e0: 6e76 6172 6961 6e74 5269 6e67 2f48 6177 nvariantRing/Haw │ │ │ │ +000421f0: 6573 446f 632e 6d32 3a32 3938 3a30 2e0a esDoc.m2:298:0.. │ │ │ │ +00042200: 1f0a 4669 6c65 3a20 496e 7661 7269 616e ..File: Invarian │ │ │ │ +00042210: 7452 696e 672e 696e 666f 2c20 4e6f 6465 tRing.info, Node │ │ │ │ +00042220: 3a20 7261 6e6b 5f6c 7044 6961 676f 6e61 : rank_lpDiagona │ │ │ │ +00042230: 6c41 6374 696f 6e5f 7270 2c20 4e65 7874 lAction_rp, Next │ │ │ │ +00042240: 3a20 7265 6c61 7469 6f6e 735f 6c70 4669 : relations_lpFi │ │ │ │ +00042250: 6e69 7465 4772 6f75 7041 6374 696f 6e5f niteGroupAction_ │ │ │ │ +00042260: 7270 2c20 5072 6576 3a20 7072 696d 6172 rp, Prev: primar │ │ │ │ +00042270: 7949 6e76 6172 6961 6e74 735f 6c70 5f70 yInvariants_lp_p │ │ │ │ +00042280: 645f 7064 5f70 645f 636d 4465 6772 6565 d_pd_pd_cmDegree │ │ │ │ +00042290: 5665 6374 6f72 3d3e 5f70 645f 7064 5f70 Vector=>_pd_pd_p │ │ │ │ +000422a0: 645f 7270 2c20 5570 3a20 546f 700a 0a72 d_rp, Up: Top..r │ │ │ │ +000422b0: 616e 6b28 4469 6167 6f6e 616c 4163 7469 ank(DiagonalActi │ │ │ │ +000422c0: 6f6e 2920 2d2d 206f 6620 6120 6469 6167 on) -- of a diag │ │ │ │ +000422d0: 6f6e 616c 2061 6374 696f 6e0a 2a2a 2a2a onal action.**** │ │ │ │ +000422e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000422f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00042300: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 ********.. * Fu │ │ │ │ +00042310: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 7261 nction: *note ra │ │ │ │ +00042320: 6e6b 3a20 284d 6163 6175 6c61 7932 446f nk: (Macaulay2Do │ │ │ │ +00042330: 6329 7261 6e6b 2c0a 2020 2a20 5573 6167 c)rank,. * Usag │ │ │ │ +00042340: 653a 200a 2020 2020 2020 2020 7261 6e6b e: . rank │ │ │ │ +00042350: 2044 0a20 202a 2049 6e70 7574 733a 0a20 D. * Inputs:. │ │ │ │ +00042360: 2020 2020 202a 2044 2c20 616e 2069 6e73 * D, an ins │ │ │ │ +00042370: 7461 6e63 6520 6f66 2074 6865 2074 7970 tance of the typ │ │ │ │ +00042380: 6520 2a6e 6f74 6520 4469 6167 6f6e 616c e *note Diagonal │ │ │ │ +00042390: 4163 7469 6f6e 3a20 4469 6167 6f6e 616c Action: Diagonal │ │ │ │ +000423a0: 4163 7469 6f6e 2c0a 2020 2a20 4f75 7470 Action,. * Outp │ │ │ │ +000423b0: 7574 733a 0a20 2020 2020 202a 2061 6e20 uts:. * an │ │ │ │ +000423c0: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ +000423d0: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ +000423e0: 2c20 7468 6520 7261 6e6b 206f 6620 7468 , the rank of th │ │ │ │ +000423f0: 6520 746f 7275 7320 6661 6374 6f72 206f e torus factor o │ │ │ │ +00042400: 6620 610a 2020 2020 2020 2020 6469 6167 f a. diag │ │ │ │ +00042410: 6f6e 616c 2061 6374 696f 6e0a 0a44 6573 onal action..Des │ │ │ │ +00042420: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +00042430: 3d3d 3d3d 0a0a 5468 6973 2066 756e 6374 ====..This funct │ │ │ │ +00042440: 696f 6e20 6973 2070 726f 7669 6465 6420 ion is provided │ │ │ │ +00042450: 6279 2074 6865 2070 6163 6b61 6765 202a by the package * │ │ │ │ +00042460: 6e6f 7465 2049 6e76 6172 6961 6e74 5269 note InvariantRi │ │ │ │ +00042470: 6e67 3a20 546f 702c 2e20 0a0a 5573 6520 ng: Top,. ..Use │ │ │ │ +00042480: 7468 6973 2066 756e 6374 696f 6e20 746f this function to │ │ │ │ +00042490: 2072 6563 6f76 6572 2074 6865 2072 616e recover the ran │ │ │ │ +000424a0: 6b20 6f66 2074 6865 2074 6f72 7573 2066 k of the torus f │ │ │ │ +000424b0: 6163 746f 7220 6f66 2061 2064 6961 676f actor of a diago │ │ │ │ +000424c0: 6e61 6c20 6163 7469 6f6e 2e0a 0a54 6865 nal action...The │ │ │ │ +000424d0: 2066 6f6c 6c6f 7769 6e67 2065 7861 6d70 following examp │ │ │ │ +000424e0: 6c65 2064 6566 696e 6573 2061 6e20 6163 le defines an ac │ │ │ │ +000424f0: 7469 6f6e 206f 6620 6120 7477 6f2d 6469 tion of a two-di │ │ │ │ +00042500: 6d65 6e73 696f 6e61 6c20 746f 7275 7320 mensional torus │ │ │ │ +00042510: 6f6e 2061 0a70 6f6c 796e 6f6d 6961 6c20 on a.polynomial │ │ │ │ +00042520: 7269 6e67 2069 6e20 666f 7572 2076 6172 ring in four var │ │ │ │ +00042530: 6961 626c 6573 2e0a 0a2b 2d2d 2d2d 2d2d iables...+------ │ │ │ │ +00042540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042560: 2d2b 0a7c 6931 203a 2052 203d 2051 515b -+.|i1 : R = QQ[ │ │ │ │ +00042570: 785f 312e 2e78 5f34 5d20 2020 2020 2020 x_1..x_4] │ │ │ │ +00042580: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00042590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000425a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000425b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000425c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000425d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000425e0: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ -000425f0: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ -00042600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042610: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00042620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00042640: 6932 203a 2057 203d 206d 6174 7269 787b i2 : W = matrix{ │ │ │ │ -00042650: 7b30 2c31 2c2d 312c 317d 2c7b 312c 302c {0,1,-1,1},{1,0, │ │ │ │ -00042660: 2d31 2c2d 317d 7d7c 0a7c 2020 2020 2020 -1,-1}}|.| │ │ │ │ -00042670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042690: 207c 0a7c 6f32 203d 207c 2030 2031 202d |.|o2 = | 0 1 - │ │ │ │ -000426a0: 3120 3120 207c 2020 2020 2020 2020 2020 1 1 | │ │ │ │ -000426b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000426c0: 2020 207c 2031 2030 202d 3120 2d31 207c | 1 0 -1 -1 | │ │ │ │ -000426d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000426e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000425b0: 2020 2020 207c 0a7c 6f31 203d 2052 2020 |.|o1 = R │ │ │ │ +000425c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000425d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000425e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000425f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042600: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ +00042610: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ +00042620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042630: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00042640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00042660: 6932 203a 2057 203d 206d 6174 7269 787b i2 : W = matrix{ │ │ │ │ +00042670: 7b30 2c31 2c2d 312c 317d 2c7b 312c 302c {0,1,-1,1},{1,0, │ │ │ │ +00042680: 2d31 2c2d 317d 7d7c 0a7c 2020 2020 2020 -1,-1}}|.| │ │ │ │ +00042690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000426a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000426b0: 207c 0a7c 6f32 203d 207c 2030 2031 202d |.|o2 = | 0 1 - │ │ │ │ +000426c0: 3120 3120 207c 2020 2020 2020 2020 2020 1 1 | │ │ │ │ +000426d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000426e0: 2020 207c 2031 2030 202d 3120 2d31 207c | 1 0 -1 -1 | │ │ │ │ 000426f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042700: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00042710: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00042720: 3220 2020 2020 2020 3420 2020 2020 2020 2 4 │ │ │ │ -00042730: 2020 2020 2020 2020 207c 0a7c 6f32 203a |.|o2 : │ │ │ │ -00042740: 204d 6174 7269 7820 5a5a 2020 3c2d 2d20 Matrix ZZ <-- │ │ │ │ -00042750: 5a5a 2020 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ -00042760: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00042770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00042790: 6933 203a 2054 203d 2064 6961 676f 6e61 i3 : T = diagona │ │ │ │ -000427a0: 6c41 6374 696f 6e28 572c 2052 2920 2020 lAction(W, R) │ │ │ │ -000427b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000427c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000427d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000427e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000427f0: 202a 2032 2020 2020 2020 2020 2020 2020 * 2 │ │ │ │ -00042800: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -00042810: 203d 2052 203c 2d20 2851 5120 2920 2076 = R <- (QQ ) v │ │ │ │ -00042820: 6961 2020 2020 2020 2020 2020 2020 2020 ia │ │ │ │ -00042830: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00042840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042850: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00042860: 0a7c 2020 2020 207c 2030 2031 202d 3120 .| | 0 1 -1 │ │ │ │ -00042870: 3120 207c 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ -00042880: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00042890: 207c 2031 2030 202d 3120 2d31 207c 2020 | 1 0 -1 -1 | │ │ │ │ -000428a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000428b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00042700: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00042710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042720: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00042730: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00042740: 3220 2020 2020 2020 3420 2020 2020 2020 2 4 │ │ │ │ +00042750: 2020 2020 2020 2020 207c 0a7c 6f32 203a |.|o2 : │ │ │ │ +00042760: 204d 6174 7269 7820 5a5a 2020 3c2d 2d20 Matrix ZZ <-- │ │ │ │ +00042770: 5a5a 2020 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00042780: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00042790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000427a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000427b0: 6933 203a 2054 203d 2064 6961 676f 6e61 i3 : T = diagona │ │ │ │ +000427c0: 6c41 6374 696f 6e28 572c 2052 2920 2020 lAction(W, R) │ │ │ │ +000427d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000427e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000427f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042800: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00042810: 202a 2032 2020 2020 2020 2020 2020 2020 * 2 │ │ │ │ +00042820: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +00042830: 203d 2052 203c 2d20 2851 5120 2920 2076 = R <- (QQ ) v │ │ │ │ +00042840: 6961 2020 2020 2020 2020 2020 2020 2020 ia │ │ │ │ +00042850: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00042860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042870: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00042880: 0a7c 2020 2020 207c 2030 2031 202d 3120 .| | 0 1 -1 │ │ │ │ +00042890: 3120 207c 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ +000428a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000428b0: 207c 2031 2030 202d 3120 2d31 207c 2020 | 1 0 -1 -1 | │ │ │ │ 000428c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000428d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000428e0: 6f33 203a 2044 6961 676f 6e61 6c41 6374 o3 : DiagonalAct │ │ │ │ -000428f0: 696f 6e20 2020 2020 2020 2020 2020 2020 ion │ │ │ │ -00042900: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00042910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042930: 2d2b 0a7c 6934 203a 2072 616e 6b20 5420 -+.|i4 : rank T │ │ │ │ -00042940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042950: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000428d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000428e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000428f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00042900: 6f33 203a 2044 6961 676f 6e61 6c41 6374 o3 : DiagonalAct │ │ │ │ +00042910: 696f 6e20 2020 2020 2020 2020 2020 2020 ion │ │ │ │ +00042920: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00042930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042950: 2d2b 0a7c 6934 203a 2072 616e 6b20 5420 -+.|i4 : rank T │ │ │ │ 00042960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042980: 2020 2020 207c 0a7c 6f34 203d 2032 2020 |.|o4 = 2 │ │ │ │ +00042970: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00042980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00042990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000429a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000429b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -000429c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000429d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ -000429e0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ -000429f0: 202a 202a 6e6f 7465 2044 6961 676f 6e61 * *note Diagona │ │ │ │ -00042a00: 6c41 6374 696f 6e3a 2044 6961 676f 6e61 lAction: Diagona │ │ │ │ -00042a10: 6c41 6374 696f 6e2c 202d 2d20 7468 6520 lAction, -- the │ │ │ │ -00042a20: 636c 6173 7320 6f66 2061 6c6c 2064 6961 class of all dia │ │ │ │ -00042a30: 676f 6e61 6c20 6163 7469 6f6e 730a 2020 gonal actions. │ │ │ │ -00042a40: 2a20 2a6e 6f74 6520 6469 6167 6f6e 616c * *note diagonal │ │ │ │ -00042a50: 4163 7469 6f6e 3a20 6469 6167 6f6e 616c Action: diagonal │ │ │ │ -00042a60: 4163 7469 6f6e 2c20 2d2d 2064 6961 676f Action, -- diago │ │ │ │ -00042a70: 6e61 6c20 6772 6f75 7020 6163 7469 6f6e nal group action │ │ │ │ -00042a80: 2076 6961 2077 6569 6768 7473 0a0a 5761 via weights..Wa │ │ │ │ -00042a90: 7973 2074 6f20 7573 6520 7468 6973 206d ys to use this m │ │ │ │ -00042aa0: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d ethod:.========= │ │ │ │ -00042ab0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00042ac0: 0a20 202a 202a 6e6f 7465 2072 616e 6b28 . * *note rank( │ │ │ │ -00042ad0: 4469 6167 6f6e 616c 4163 7469 6f6e 293a DiagonalAction): │ │ │ │ -00042ae0: 2072 616e 6b5f 6c70 4469 6167 6f6e 616c rank_lpDiagonal │ │ │ │ -00042af0: 4163 7469 6f6e 5f72 702c 202d 2d20 6f66 Action_rp, -- of │ │ │ │ -00042b00: 2061 2064 6961 676f 6e61 6c0a 2020 2020 a diagonal. │ │ │ │ -00042b10: 6163 7469 6f6e 0a2d 2d2d 2d2d 2d2d 2d2d action.--------- │ │ │ │ -00042b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000429a0: 2020 2020 207c 0a7c 6f34 203d 2032 2020 |.|o4 = 2 │ │ │ │ +000429b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000429c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000429d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000429e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000429f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ +00042a00: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ +00042a10: 202a 202a 6e6f 7465 2044 6961 676f 6e61 * *note Diagona │ │ │ │ +00042a20: 6c41 6374 696f 6e3a 2044 6961 676f 6e61 lAction: Diagona │ │ │ │ +00042a30: 6c41 6374 696f 6e2c 202d 2d20 7468 6520 lAction, -- the │ │ │ │ +00042a40: 636c 6173 7320 6f66 2061 6c6c 2064 6961 class of all dia │ │ │ │ +00042a50: 676f 6e61 6c20 6163 7469 6f6e 730a 2020 gonal actions. │ │ │ │ +00042a60: 2a20 2a6e 6f74 6520 6469 6167 6f6e 616c * *note diagonal │ │ │ │ +00042a70: 4163 7469 6f6e 3a20 6469 6167 6f6e 616c Action: diagonal │ │ │ │ +00042a80: 4163 7469 6f6e 2c20 2d2d 2064 6961 676f Action, -- diago │ │ │ │ +00042a90: 6e61 6c20 6772 6f75 7020 6163 7469 6f6e nal group action │ │ │ │ +00042aa0: 2076 6961 2077 6569 6768 7473 0a0a 5761 via weights..Wa │ │ │ │ +00042ab0: 7973 2074 6f20 7573 6520 7468 6973 206d ys to use this m │ │ │ │ +00042ac0: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d ethod:.========= │ │ │ │ +00042ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00042ae0: 0a20 202a 202a 6e6f 7465 2072 616e 6b28 . * *note rank( │ │ │ │ +00042af0: 4469 6167 6f6e 616c 4163 7469 6f6e 293a DiagonalAction): │ │ │ │ +00042b00: 2072 616e 6b5f 6c70 4469 6167 6f6e 616c rank_lpDiagonal │ │ │ │ +00042b10: 4163 7469 6f6e 5f72 702c 202d 2d20 6f66 Action_rp, -- of │ │ │ │ +00042b20: 2061 2064 6961 676f 6e61 6c0a 2020 2020 a diagonal. │ │ │ │ +00042b30: 6163 7469 6f6e 0a2d 2d2d 2d2d 2d2d 2d2d action.--------- │ │ │ │ 00042b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00042b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042b60: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00042b70: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00042b80: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00042b90: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00042ba0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00042bb0: 352e 3131 2b64 732f 4d32 2f4d 6163 6175 5.11+ds/M2/Macau │ │ │ │ -00042bc0: 6c61 7932 2f70 6163 6b61 6765 732f 0a49 lay2/packages/.I │ │ │ │ -00042bd0: 6e76 6172 6961 6e74 5269 6e67 2f41 6265 nvariantRing/Abe │ │ │ │ -00042be0: 6c69 616e 4772 6f75 7073 446f 632e 6d32 lianGroupsDoc.m2 │ │ │ │ -00042bf0: 3a33 3532 3a30 2e0a 1f0a 4669 6c65 3a20 :352:0....File: │ │ │ │ -00042c00: 496e 7661 7269 616e 7452 696e 672e 696e InvariantRing.in │ │ │ │ -00042c10: 666f 2c20 4e6f 6465 3a20 7265 6c61 7469 fo, Node: relati │ │ │ │ -00042c20: 6f6e 735f 6c70 4669 6e69 7465 4772 6f75 ons_lpFiniteGrou │ │ │ │ -00042c30: 7041 6374 696f 6e5f 7270 2c20 4e65 7874 pAction_rp, Next │ │ │ │ -00042c40: 3a20 7265 796e 6f6c 6473 4f70 6572 6174 : reynoldsOperat │ │ │ │ -00042c50: 6f72 2c20 5072 6576 3a20 7261 6e6b 5f6c or, Prev: rank_l │ │ │ │ -00042c60: 7044 6961 676f 6e61 6c41 6374 696f 6e5f pDiagonalAction_ │ │ │ │ -00042c70: 7270 2c20 5570 3a20 546f 700a 0a72 656c rp, Up: Top..rel │ │ │ │ -00042c80: 6174 696f 6e73 2846 696e 6974 6547 726f ations(FiniteGro │ │ │ │ -00042c90: 7570 4163 7469 6f6e 2920 2d2d 2072 656c upAction) -- rel │ │ │ │ -00042ca0: 6174 696f 6e73 206f 6620 6120 6669 6e69 ations of a fini │ │ │ │ -00042cb0: 7465 2067 726f 7570 0a2a 2a2a 2a2a 2a2a te group.******* │ │ │ │ -00042cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00042cd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00042b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042b80: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +00042b90: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +00042ba0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +00042bb0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +00042bc0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +00042bd0: 352e 3131 2b64 732f 4d32 2f4d 6163 6175 5.11+ds/M2/Macau │ │ │ │ +00042be0: 6c61 7932 2f70 6163 6b61 6765 732f 0a49 lay2/packages/.I │ │ │ │ +00042bf0: 6e76 6172 6961 6e74 5269 6e67 2f41 6265 nvariantRing/Abe │ │ │ │ +00042c00: 6c69 616e 4772 6f75 7073 446f 632e 6d32 lianGroupsDoc.m2 │ │ │ │ +00042c10: 3a33 3532 3a30 2e0a 1f0a 4669 6c65 3a20 :352:0....File: │ │ │ │ +00042c20: 496e 7661 7269 616e 7452 696e 672e 696e InvariantRing.in │ │ │ │ +00042c30: 666f 2c20 4e6f 6465 3a20 7265 6c61 7469 fo, Node: relati │ │ │ │ +00042c40: 6f6e 735f 6c70 4669 6e69 7465 4772 6f75 ons_lpFiniteGrou │ │ │ │ +00042c50: 7041 6374 696f 6e5f 7270 2c20 4e65 7874 pAction_rp, Next │ │ │ │ +00042c60: 3a20 7265 796e 6f6c 6473 4f70 6572 6174 : reynoldsOperat │ │ │ │ +00042c70: 6f72 2c20 5072 6576 3a20 7261 6e6b 5f6c or, Prev: rank_l │ │ │ │ +00042c80: 7044 6961 676f 6e61 6c41 6374 696f 6e5f pDiagonalAction_ │ │ │ │ +00042c90: 7270 2c20 5570 3a20 546f 700a 0a72 656c rp, Up: Top..rel │ │ │ │ +00042ca0: 6174 696f 6e73 2846 696e 6974 6547 726f ations(FiniteGro │ │ │ │ +00042cb0: 7570 4163 7469 6f6e 2920 2d2d 2072 656c upAction) -- rel │ │ │ │ +00042cc0: 6174 696f 6e73 206f 6620 6120 6669 6e69 ations of a fini │ │ │ │ +00042cd0: 7465 2067 726f 7570 0a2a 2a2a 2a2a 2a2a te group.******* │ │ │ │ 00042ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00042cf0: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469 ****.. * Functi │ │ │ │ -00042d00: 6f6e 3a20 2a6e 6f74 6520 7265 6c61 7469 on: *note relati │ │ │ │ -00042d10: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ -00042d20: 6f63 2972 656c 6174 696f 6e73 2c0a 2020 oc)relations,. │ │ │ │ -00042d30: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -00042d40: 2020 7265 6c61 7469 6f6e 7320 470a 2020 relations G. │ │ │ │ -00042d50: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ -00042d60: 2a20 472c 2061 6e20 696e 7374 616e 6365 * G, an instance │ │ │ │ -00042d70: 206f 6620 7468 6520 7479 7065 202a 6e6f of the type *no │ │ │ │ -00042d80: 7465 2046 696e 6974 6547 726f 7570 4163 te FiniteGroupAc │ │ │ │ -00042d90: 7469 6f6e 3a20 4669 6e69 7465 4772 6f75 tion: FiniteGrou │ │ │ │ -00042da0: 7041 6374 696f 6e2c 2c0a 2020 2020 2020 pAction,,. │ │ │ │ -00042db0: 2020 7468 6520 6163 7469 6f6e 206f 6620 the action of │ │ │ │ -00042dc0: 6120 6669 6e69 7465 2067 726f 7570 0a20 a finite group. │ │ │ │ -00042dd0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -00042de0: 2020 2a20 6120 2a6e 6f74 6520 6c69 7374 * a *note list │ │ │ │ -00042df0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00042e00: 4c69 7374 2c2c 2061 206c 6973 7420 6f66 List,, a list of │ │ │ │ -00042e10: 2072 656c 6174 696f 6e73 206f 6620 7468 relations of th │ │ │ │ -00042e20: 6520 6772 6f75 700a 0a44 6573 6372 6970 e group..Descrip │ │ │ │ -00042e30: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00042e40: 0a0a 5468 6973 2066 756e 6374 696f 6e20 ..This function │ │ │ │ -00042e50: 6973 2070 726f 7669 6465 6420 6279 2074 is provided by t │ │ │ │ -00042e60: 6865 2070 6163 6b61 6765 202a 6e6f 7465 he package *note │ │ │ │ -00042e70: 2049 6e76 6172 6961 6e74 5269 6e67 3a20 InvariantRing: │ │ │ │ -00042e80: 546f 702c 2e20 0a0a 5573 6520 7468 6973 Top,. ..Use this │ │ │ │ -00042e90: 2066 756e 6374 696f 6e20 746f 2067 6574 function to get │ │ │ │ -00042ea0: 2074 6865 2072 656c 6174 696f 6e73 2061 the relations a │ │ │ │ -00042eb0: 6d6f 6e67 2065 6c65 6d65 6e74 7320 6f66 mong elements of │ │ │ │ -00042ec0: 2061 2067 726f 7570 2e20 4561 6368 2065 a group. Each e │ │ │ │ -00042ed0: 6c65 6d65 6e74 0a69 7320 7265 7072 6573 lement.is repres │ │ │ │ -00042ee0: 656e 7465 6420 6279 2061 2077 6f72 6420 ented by a word │ │ │ │ -00042ef0: 6f66 206d 696e 696d 616c 206c 656e 6774 of minimal lengt │ │ │ │ -00042f00: 6820 696e 2074 6865 2043 6f78 6574 6572 h in the Coxeter │ │ │ │ -00042f10: 2067 656e 6572 6174 6f72 732e 2041 6e64 generators. And │ │ │ │ -00042f20: 2065 6163 680a 7265 6c61 7469 6f6e 2069 each.relation i │ │ │ │ -00042f30: 7320 7265 7072 6573 656e 7465 6420 6279 s represented by │ │ │ │ -00042f40: 2061 206c 6973 7420 6f66 2074 776f 2077 a list of two w │ │ │ │ -00042f50: 6f72 6473 2074 6861 7420 6571 7561 7465 ords that equate │ │ │ │ -00042f60: 7320 7468 6520 6772 6f75 7020 656c 656d s the group elem │ │ │ │ -00042f70: 656e 740a 7265 7072 6573 656e 7465 6420 ent.represented │ │ │ │ -00042f80: 6279 2074 686f 7365 2074 776f 2077 6f72 by those two wor │ │ │ │ -00042f90: 6473 2e0a 0a54 6865 2066 6f6c 6c6f 7769 ds...The followi │ │ │ │ -00042fa0: 6e67 2065 7861 6d70 6c65 2064 6566 696e ng example defin │ │ │ │ -00042fb0: 6573 2074 6865 2070 6572 6d75 7461 7469 es the permutati │ │ │ │ -00042fc0: 6f6e 2061 6374 696f 6e20 6f66 2061 2073 on action of a s │ │ │ │ -00042fd0: 796d 6d65 7472 6963 2067 726f 7570 206f ymmetric group o │ │ │ │ -00042fe0: 6e0a 7468 7265 6520 656c 656d 656e 7473 n.three elements │ │ │ │ -00042ff0: 2075 7369 6e67 2074 6872 6565 2074 7261 using three tra │ │ │ │ -00043000: 6e73 706f 7369 7469 6f6e 732e 0a0a 2b2d nspositions...+- │ │ │ │ -00043010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00042d00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00042d10: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469 ****.. * Functi │ │ │ │ +00042d20: 6f6e 3a20 2a6e 6f74 6520 7265 6c61 7469 on: *note relati │ │ │ │ +00042d30: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ +00042d40: 6f63 2972 656c 6174 696f 6e73 2c0a 2020 oc)relations,. │ │ │ │ +00042d50: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +00042d60: 2020 7265 6c61 7469 6f6e 7320 470a 2020 relations G. │ │ │ │ +00042d70: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +00042d80: 2a20 472c 2061 6e20 696e 7374 616e 6365 * G, an instance │ │ │ │ +00042d90: 206f 6620 7468 6520 7479 7065 202a 6e6f of the type *no │ │ │ │ +00042da0: 7465 2046 696e 6974 6547 726f 7570 4163 te FiniteGroupAc │ │ │ │ +00042db0: 7469 6f6e 3a20 4669 6e69 7465 4772 6f75 tion: FiniteGrou │ │ │ │ +00042dc0: 7041 6374 696f 6e2c 2c0a 2020 2020 2020 pAction,,. │ │ │ │ +00042dd0: 2020 7468 6520 6163 7469 6f6e 206f 6620 the action of │ │ │ │ +00042de0: 6120 6669 6e69 7465 2067 726f 7570 0a20 a finite group. │ │ │ │ +00042df0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +00042e00: 2020 2a20 6120 2a6e 6f74 6520 6c69 7374 * a *note list │ │ │ │ +00042e10: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00042e20: 4c69 7374 2c2c 2061 206c 6973 7420 6f66 List,, a list of │ │ │ │ +00042e30: 2072 656c 6174 696f 6e73 206f 6620 7468 relations of th │ │ │ │ +00042e40: 6520 6772 6f75 700a 0a44 6573 6372 6970 e group..Descrip │ │ │ │ +00042e50: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +00042e60: 0a0a 5468 6973 2066 756e 6374 696f 6e20 ..This function │ │ │ │ +00042e70: 6973 2070 726f 7669 6465 6420 6279 2074 is provided by t │ │ │ │ +00042e80: 6865 2070 6163 6b61 6765 202a 6e6f 7465 he package *note │ │ │ │ +00042e90: 2049 6e76 6172 6961 6e74 5269 6e67 3a20 InvariantRing: │ │ │ │ +00042ea0: 546f 702c 2e20 0a0a 5573 6520 7468 6973 Top,. ..Use this │ │ │ │ +00042eb0: 2066 756e 6374 696f 6e20 746f 2067 6574 function to get │ │ │ │ +00042ec0: 2074 6865 2072 656c 6174 696f 6e73 2061 the relations a │ │ │ │ +00042ed0: 6d6f 6e67 2065 6c65 6d65 6e74 7320 6f66 mong elements of │ │ │ │ +00042ee0: 2061 2067 726f 7570 2e20 4561 6368 2065 a group. Each e │ │ │ │ +00042ef0: 6c65 6d65 6e74 0a69 7320 7265 7072 6573 lement.is repres │ │ │ │ +00042f00: 656e 7465 6420 6279 2061 2077 6f72 6420 ented by a word │ │ │ │ +00042f10: 6f66 206d 696e 696d 616c 206c 656e 6774 of minimal lengt │ │ │ │ +00042f20: 6820 696e 2074 6865 2043 6f78 6574 6572 h in the Coxeter │ │ │ │ +00042f30: 2067 656e 6572 6174 6f72 732e 2041 6e64 generators. And │ │ │ │ +00042f40: 2065 6163 680a 7265 6c61 7469 6f6e 2069 each.relation i │ │ │ │ +00042f50: 7320 7265 7072 6573 656e 7465 6420 6279 s represented by │ │ │ │ +00042f60: 2061 206c 6973 7420 6f66 2074 776f 2077 a list of two w │ │ │ │ +00042f70: 6f72 6473 2074 6861 7420 6571 7561 7465 ords that equate │ │ │ │ +00042f80: 7320 7468 6520 6772 6f75 7020 656c 656d s the group elem │ │ │ │ +00042f90: 656e 740a 7265 7072 6573 656e 7465 6420 ent.represented │ │ │ │ +00042fa0: 6279 2074 686f 7365 2074 776f 2077 6f72 by those two wor │ │ │ │ +00042fb0: 6473 2e0a 0a54 6865 2066 6f6c 6c6f 7769 ds...The followi │ │ │ │ +00042fc0: 6e67 2065 7861 6d70 6c65 2064 6566 696e ng example defin │ │ │ │ +00042fd0: 6573 2074 6865 2070 6572 6d75 7461 7469 es the permutati │ │ │ │ +00042fe0: 6f6e 2061 6374 696f 6e20 6f66 2061 2073 on action of a s │ │ │ │ +00042ff0: 796d 6d65 7472 6963 2067 726f 7570 206f ymmetric group o │ │ │ │ +00043000: 6e0a 7468 7265 6520 656c 656d 656e 7473 n.three elements │ │ │ │ +00043010: 2075 7369 6e67 2074 6872 6565 2074 7261 using three tra │ │ │ │ +00043020: 6e73 706f 7369 7469 6f6e 732e 0a0a 2b2d nspositions...+- │ │ │ │ 00043030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00043040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00043060: 3120 3a20 5220 3d20 5151 5b78 5f31 2e2e 1 : R = QQ[x_1.. │ │ │ │ -00043070: 785f 335d 2020 2020 2020 2020 2020 2020 x_3] │ │ │ │ -00043080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000430a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00043050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00043080: 3120 3a20 5220 3d20 5151 5b78 5f31 2e2e 1 : R = QQ[x_1.. │ │ │ │ +00043090: 785f 335d 2020 2020 2020 2020 2020 2020 x_3] │ │ │ │ +000430a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000430b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000430c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000430c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000430d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000430e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000430f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00043100: 3120 3d20 5220 2020 2020 2020 2020 2020 1 = R │ │ │ │ -00043110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000430f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043110: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00043120: 3120 3d20 5220 2020 2020 2020 2020 2020 1 = R │ │ │ │ 00043130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043140: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00043140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043160: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00043170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043190: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000431a0: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269 1 : PolynomialRi │ │ │ │ -000431b0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -000431c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000431d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000431e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -000431f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000431a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000431b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000431c0: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269 1 : PolynomialRi │ │ │ │ +000431d0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ +000431e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000431f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043200: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00043210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00043220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00043240: 3220 3a20 4c20 3d20 7b6d 6174 7269 7820 2 : L = {matrix │ │ │ │ -00043250: 7b7b 302c 312c 307d 2c7b 312c 302c 307d {{0,1,0},{1,0,0} │ │ │ │ -00043260: 2c7b 302c 302c 317d 7d2c 206d 6174 7269 ,{0,0,1}}, matri │ │ │ │ -00043270: 7820 7b7b 302c 302c 317d 2c7b 302c 312c x {{0,0,1},{0,1, │ │ │ │ -00043280: 307d 2c7b 312c 302c 307d 7d2c 7c0a 7c20 0},{1,0,0}},|.| │ │ │ │ -00043290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000432a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00043260: 3220 3a20 4c20 3d20 7b6d 6174 7269 7820 2 : L = {matrix │ │ │ │ +00043270: 7b7b 302c 312c 307d 2c7b 312c 302c 307d {{0,1,0},{1,0,0} │ │ │ │ +00043280: 2c7b 302c 302c 317d 7d2c 206d 6174 7269 ,{0,0,1}}, matri │ │ │ │ +00043290: 7820 7b7b 302c 302c 317d 2c7b 302c 312c x {{0,0,1},{0,1, │ │ │ │ +000432a0: 307d 2c7b 312c 302c 307d 7d2c 7c0a 7c20 0},{1,0,0}},|.| │ │ │ │ 000432b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000432c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000432d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000432e0: 3220 3d20 7b7c 2030 2031 2030 207c 2c20 2 = {| 0 1 0 |, │ │ │ │ -000432f0: 7c20 3020 3020 3120 7c2c 207c 2031 2030 | 0 0 1 |, | 1 0 │ │ │ │ -00043300: 2030 207c 7d20 2020 2020 2020 2020 2020 0 |} │ │ │ │ -00043310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043320: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00043330: 2020 2020 207c 2031 2030 2030 207c 2020 | 1 0 0 | │ │ │ │ -00043340: 7c20 3020 3120 3020 7c20 207c 2030 2030 | 0 1 0 | | 0 0 │ │ │ │ -00043350: 2031 207c 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ -00043360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043370: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00043380: 2020 2020 207c 2030 2030 2031 207c 2020 | 0 0 1 | │ │ │ │ -00043390: 7c20 3120 3020 3020 7c20 207c 2030 2031 | 1 0 0 | | 0 1 │ │ │ │ -000433a0: 2030 207c 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ -000433b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000433c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000432d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000432e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000432f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00043300: 3220 3d20 7b7c 2030 2031 2030 207c 2c20 2 = {| 0 1 0 |, │ │ │ │ +00043310: 7c20 3020 3020 3120 7c2c 207c 2031 2030 | 0 0 1 |, | 1 0 │ │ │ │ +00043320: 2030 207c 7d20 2020 2020 2020 2020 2020 0 |} │ │ │ │ +00043330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043340: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00043350: 2020 2020 207c 2031 2030 2030 207c 2020 | 1 0 0 | │ │ │ │ +00043360: 7c20 3020 3120 3020 7c20 207c 2030 2030 | 0 1 0 | | 0 0 │ │ │ │ +00043370: 2031 207c 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ +00043380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043390: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000433a0: 2020 2020 207c 2030 2030 2031 207c 2020 | 0 0 1 | │ │ │ │ +000433b0: 7c20 3120 3020 3020 7c20 207c 2030 2031 | 1 0 0 | | 0 1 │ │ │ │ +000433c0: 2030 207c 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ 000433d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000433e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000433e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000433f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043410: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00043420: 3220 3a20 4c69 7374 2020 2020 2020 2020 2 : List │ │ │ │ -00043430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043430: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00043440: 3220 3a20 4c69 7374 2020 2020 2020 2020 2 : List │ │ │ │ 00043450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043460: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -00043470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043480: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00043490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000434a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000434b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6d ------------|.|m │ │ │ │ -000434c0: 6174 7269 7820 7b7b 312c 302c 307d 2c7b atrix {{1,0,0},{ │ │ │ │ -000434d0: 302c 302c 317d 2c7b 302c 312c 307d 7d20 0,0,1},{0,1,0}} │ │ │ │ -000434e0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ -000434f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043500: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00043510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000434b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000434c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000434d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6d ------------|.|m │ │ │ │ +000434e0: 6174 7269 7820 7b7b 312c 302c 307d 2c7b atrix {{1,0,0},{ │ │ │ │ +000434f0: 302c 302c 317d 2c7b 302c 312c 307d 7d20 0,0,1},{0,1,0}} │ │ │ │ +00043500: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00043510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043520: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00043530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00043540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00043560: 3320 3a20 4720 3d20 6669 6e69 7465 4163 3 : G = finiteAc │ │ │ │ -00043570: 7469 6f6e 284c 2c20 5229 2020 2020 2020 tion(L, R) │ │ │ │ -00043580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000435a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00043550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00043580: 3320 3a20 4720 3d20 6669 6e69 7465 4163 3 : G = finiteAc │ │ │ │ +00043590: 7469 6f6e 284c 2c20 5229 2020 2020 2020 tion(L, R) │ │ │ │ +000435a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000435b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000435c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000435c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000435d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000435e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000435f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00043600: 3320 3d20 5220 3c2d 207b 7c20 3020 3120 3 = R <- {| 0 1 │ │ │ │ -00043610: 3020 7c2c 207c 2030 2030 2031 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